text stringlengths 1 2.12k | source dict |
|---|---|
As with an average of averages, here we are reconstructing the individual numbers (10 and 400 out of 50 and 500), then finding the actual percentages of the whole. That’s what a weighted average is.
I closed by referring to some of the answers we saw last week.
## Average price per square foot
For a specific real-life example, consider this 2003 question:
Average of Ratios vs. Ratio of Averages
I write and maintain software for real estate agents, and we include a calculation called, "Average dollars per square foot." We currently calculate this as the ratio of the average price divided by the average square footage of all the homes in the list. It seems to me that it should be calculated as the average of the price-per-square-foot ratio of each house. Can you think of any reason why the ratio of the averages would be more useful than the average of the ratios? Is there a technical name for this ratio of averages?
I answered again, pointing out as before that different calculations can both be meaningful, though they mean different things:
Hi, Laure.
Interesting question!
What you are currently calculating actually does make sense; you are just averaging over all the square feet of houses, rather than over all houses, and that may be just the right thing to do -- or it might not. Here's what I mean:
Suppose that N houses are sold; the sum of all their prices is P, and the sum of all their areas is A. (That is, if the individual prices are P1, P2, ..., Pn, and the individual areas are A1, A2, ..., An, then P is the sum of P1 through Pn, and A is the sum of A1 through An.)
Then the average price of a house is P/N, and the average area of a house is A/N; and you are calculating
P/N
--- = P/A
A/N
as the average price per square foot. And that is exactly what it is: the total price of all those square feet, divided by the number of square feet. | {
"domain": "themathdoctors.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.985496423290417,
"lm_q1q2_score": 0.8508693504563144,
"lm_q2_score": 0.863391617003942,
"openwebmath_perplexity": 590.2567744497204,
"openwebmath_score": 0.746770441532135,
"tags": null,
"url": "https://www.themathdoctors.org/weighted-averages-averaging-averages-or-rates/"
} |
In terms of the individual numbers, this is $$\frac{P_1+P_2+\dots+P_n}{A_1+A_2+\dots+A_N}$$ Imagine laying out tiles representing each square foot of each house, and dollar bills representing the price of each house. This average spreads all the money equally over all the tiles, to find an overall price per square root. That sounds perfectly reasonable.
What you envision is
P1/A1 + P2/A2 + ... + Pn/An
---------------------------
N
which would average the price per square foot of all the houses. This puts the focus on the individual houses, rather than the individual square feet. How would this be different?
Again, this is $$\frac{\frac{P_1}{A_1}+\frac{P_2}{A_2}+\dots+\frac{P_N}{A_N}}{N}$$ Imagine laying out tiles representing each square foot of each house, keeping each house separate, and dollar bills representing the price of each house, spread equally among the tiles of that house. Now we take one tile and its cost from each house, and average those, spreading the money equally among the selected tiles. What have we found this time?
The way to see the difference is to take an extreme example. (Don’t check whether the sizes or prices are realistic; they aren’t meant to be!)
Well, let's take a simple case with N = 2. Suppose we have a big, well-built house of 10,000 square feet, and that it costs $2,000,000 ($200 per square foot), and a little house of 1,000 square feet that costs $20,000 ($20 per square foot). Then the total cost of the houses P is \$2,020,000, and the total area A is 11,000 square feet. The average price per square foot is
P/A = 2,020,000/11,000 = 183.6
(closer to the more expensive price) while the average of the two price-per-square-foot numbers is
average(Pn/An) = (200 + 20)/2 = 110 | {
"domain": "themathdoctors.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.985496423290417,
"lm_q1q2_score": 0.8508693504563144,
"lm_q2_score": 0.863391617003942,
"openwebmath_perplexity": 590.2567744497204,
"openwebmath_score": 0.746770441532135,
"tags": null,
"url": "https://www.themathdoctors.org/weighted-averages-averaging-averages-or-rates/"
} |
average(Pn/An) = (200 + 20)/2 = 110
(which is considerably lower). What pulled the first number up is the fact that the bigger house had the bigger price per square foot; since we counted each square foot equally, the numerous high-cost ones won. The second calculation treats all 10,000 of the expensive square feet equally with the mere 1,000 square feet of the little house, so the little house pulled the average down.
Which do you think is the better calculation?
Both numbers are meaningful. The first fully deserves the name you are giving it (though there is definitely some ambiguity in the English!); but the second may better reflect what the average homeowner (as opposed to the "average square foot of floor space") can expect. Call it, perhaps, the "cost per square foot of the average house", where the number you are currently calculating is the "average cost of a square foot" or "cost of an average square foot".
Perhaps the builder would consider the first calculation more appropriate, since they put more effort into the bigger house, and every square foot they built matters to them; but the homeowner might care only about his own house — which neither calculation really reflects!
So, again, both numbers can reasonably be called "average cost per square foot"; which is more useful to you depends on how you want to use it. Do you want a number that is pulled up by big fancy houses, or one that shows what the average house is worth? Or would separate numbers for different categories of houses make more sense? Perhaps you can gather data that shows how costs per square foot are distributed, and how each average reflects that.
Any kind of average is an attempt to reduce a lot of data to a single number, and will never give a full picture of a diverse population. Other numbers, or a graph of the distribution, are often more useful.
Laure chose to leave the calculation as it was.
## Average speed | {
"domain": "themathdoctors.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.985496423290417,
"lm_q1q2_score": 0.8508693504563144,
"lm_q2_score": 0.863391617003942,
"openwebmath_perplexity": 590.2567744497204,
"openwebmath_score": 0.746770441532135,
"tags": null,
"url": "https://www.themathdoctors.org/weighted-averages-averaging-averages-or-rates/"
} |
Laure chose to leave the calculation as it was.
## Average speed
Let’s look at one more 2003 question, which is about a ratio in disguise:
Weighted Average of the Velocities
A truck on a straight road starts from rest and accelerates at 2.0 m/s2 until it reaches a speed of 20 m/s. Then the truck travels for 20 s at constant speed until the brakes are applied, stopping the truck in a uniform manner in an additional 5.0 s.
(a) How long is the truck in motion?
(b) What is the average velocity of the truck for the motion described?
Hi, Garin.
The trick here is to get the average velocity for each of the three phases of the trip, then do a weighted average of the three velocities.
Average velocity for phase 2 is easy, 20 m/s. During phases 1 and 3, the velocity changes linearly. That means that we can just average the starting and ending velocities to get the velocity for the entire phase.
This sort of average becomes most meaningful when you apply calculus, but for uniform acceleration, it requires only algebra. Doctor Edwin chose to leave that part for Garin to handle (or to ask for more help, which he never did), and focused just on the idea of the average. It turns out that the acceleration phase takes 10 seconds, at an average speed of 10 m/s; the cruise phase takes 20 seconds at 20 m/s; and the deceleration phase takes 5 seconds at an average speed of 10 m/s.
Once you've got those, you'll do a weighted average with respect to the time spent in each phase:
(v_1 * t_1) + (v_2 * t_2) + (v_3 * t_3)
v_avg = ---------------------------------------
t_total
The calculation looks like this: $$\frac{v_1\cdot t_1 + v_2\cdot t_2 + v_3\cdot t_3}{t_1+t_2+t_3} = \frac{10\cdot 10 + 20\cdot 20 + 10\cdot 5}{10+20+5} = \frac{100 + 400 + 50}{10+20+5} = \frac{550}{35} = 15.7\text{ m/s}$$ | {
"domain": "themathdoctors.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.985496423290417,
"lm_q1q2_score": 0.8508693504563144,
"lm_q2_score": 0.863391617003942,
"openwebmath_perplexity": 590.2567744497204,
"openwebmath_score": 0.746770441532135,
"tags": null,
"url": "https://www.themathdoctors.org/weighted-averages-averaging-averages-or-rates/"
} |
Just for fun, I'll point out that the same set of equations has another interpretation that works just as well. Average velocity is just distance over time, right? So if we figure out how far the truck went in each phase, add them all up, and divide by the total time, you'll also get the same answer:
d_1 + d_2 + d3
v_avg = ---------------------------------------
t_total
But the middle terms in those two equations are the same:
d_2 = v_2 * t_2
and so are the first terms:
a(t_1)^2
d_1 = --------
2
v
a = -
t
v_1_end * (t_1)^2
d_1 = -----------------
2(t_1)
v_1_end * t_1
d_1 = -------------
2
But since the starting velocity is zero, we can add it in wherever we want:
(v_1_start + v_1_end)
d_1 = ---------------------- * t_1
2
which means that our first terms are the same as well, and so are our third terms.
Anyway, there you have it. Total distance traveled divided by time, or a weighted average of velocities with respect to time, it works out to the same thing.
In other words, each “average velocity · time” is a distance, so our average velocity calculation was really just total distance over total time. We’ve seen previously that when distances are the same, we can use a harmonic mean to find the average speed; this confirms that when times are known but different, we can use a weighted arithmetic mean, using times as weights.
This site uses Akismet to reduce spam. Learn how your comment data is processed. | {
"domain": "themathdoctors.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.985496423290417,
"lm_q1q2_score": 0.8508693504563144,
"lm_q2_score": 0.863391617003942,
"openwebmath_perplexity": 590.2567744497204,
"openwebmath_score": 0.746770441532135,
"tags": null,
"url": "https://www.themathdoctors.org/weighted-averages-averaging-averages-or-rates/"
} |
# Function with arbitrary small period
Is there a function f: $$\mathbb{R} \to \mathbb{R}$$ with arbitrary small period different from $$f(x) = k$$? ($$\forall \epsilon >0 \exists a < \epsilon$$ such that f(x) has a periodicity $$a$$) I think the function is the Dirichlet function but I don't know how to prove it properly.
• F(x)=1 for numbers with a halting decimal expansion, for example $3.452$ and not $1/3$ Oct 4, 2018 at 12:00
• @Empy2 Nice: your example also works for domain $\mathbb{Q}$. Oct 4, 2018 at 12:38
• I would avoid saying “such that the period of $f(x)$ is $a$” here. The term period would usually denote the smallest $a>0$ that fulfills $f(x+a) = f(x)$. Here you should rather just say “such that $f$ has a periodicity $a$”, or “such that $f$ is $a$-periodic”, which does assert $f(x+a) = f(x)$ but does not make any statement as to whether there exists also $b \in ]0,a[$ with $f(x+b)=f(x)$. Oct 4, 2018 at 13:55
• Of course you want $0<a<\epsilon$
– MPW
Oct 4, 2018 at 18:07
• You can construct a lot of such functions. The recipe: 1) Take any sequence $a_k$ which has infinite number of values in any neighbourhood of zero; 2) Make a set consisting of all finite sums of $a_k$ with arbitrary integer coefficients; 3) The characteristic function of this set will satisfy your requirements for $f$. If $a_k=1/k$ then you're getting rationals. If $a_k=10^{-k}$ then you're getting Empy2's set. If $a_k=k^{-\pi}$ then you're getting new set which is hard to imagine :-) If $a_k=\sin{k}$ then you're getting new interesting set. Oct 5, 2018 at 7:40
You're right. The characteristic function of the rationals is periodic of period $$1/n$$ for all $$n \in \mathbb N$$ because $$x$$ is rational iff $$x+1/n$$ is rational.
You're correct.
Let $$\epsilon$$ be arbitrarily small. You need to prove that there exists some $$0 such that $$D(x)=D(x+p)$$ for all $$x\in \mathbb{R}$$. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9854964220125032,
"lm_q1q2_score": 0.8508693372286261,
"lm_q2_score": 0.8633916047011594,
"openwebmath_perplexity": 121.84249319753401,
"openwebmath_score": 0.8838260173797607,
"tags": null,
"url": "https://math.stackexchange.com/questions/2942016/function-with-arbitrary-small-period/2942235"
} |
We know that $$\epsilon$$ is some positive real number, so there exists some $$p\in \mathbb{Q}$$ such that $$0. Let's look at an arbitrary $$x\in \mathbb{R}$$ and see if our property is satisfied or not:
If $$x$$ is rational, then $$D(x)=1$$. Since the sum of two rationals is rational, then $$D(x+p)=1$$ too.
If $$x$$ is irrational, then $$D(x)=0$$. Since the sum of a rational and an irrational is irrational, then $$D(x+p)=0$$ too.
So if we choose our period to be $$p$$, our property is satisfied.
The main question has been answered in other answers well enough, but I would like to address a few natural follow-up questions. What about continuous functions $$f$$ with this property?
It turns out that in this case there are no nontrivial solutions - every such function is constant. Here's a topological proof: Let $$K=\{x\in\Bbb R\mid \forall y, f(y)=f(x+y)\}$$ be the set of periods of $$f$$. If $$f$$ is continuous, then this is an intersection of the sets $$\{x\in\Bbb R\mid f(y)=f(x+y)\}$$, which is closed (it is the preimage of $$\{0\}$$ under the function $$g(x)=f(y)-f(x+y)$$), so $$K$$ itself is closed. $$K$$ is also dense in $$\Bbb R$$, because it is an additive group with arbitrarily small elements, so $$K=\Bbb R$$ and hence $$f(x)=f(y)$$ for all $$x,y\in \Bbb R$$.
If we consider discontinuous functions again, then we know $$K$$ is a dense additive subgroup of $$\Bbb R$$. Does every dense additive subgroup generate such a function? Yes, we can just take the characteristic function of $$K$$. For a fixed $$K$$, the space of such functions is just all functions $$\Bbb R/K\to \Bbb R$$. This is another way to get at the constancy result, since as a topological group, $$\Bbb R/K$$ has the indiscrete topology, because any open set will cover $$\Bbb R$$ if copied around with translations by $$K$$. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9854964220125032,
"lm_q1q2_score": 0.8508693372286261,
"lm_q2_score": 0.8633916047011594,
"openwebmath_perplexity": 121.84249319753401,
"openwebmath_score": 0.8838260173797607,
"tags": null,
"url": "https://math.stackexchange.com/questions/2942016/function-with-arbitrary-small-period/2942235"
} |
Of course $$\Bbb R/K$$ can be uncountable, for example if $$K=\Bbb Q$$ or any other countable subgroup. Can it be countable or finite? It can be countable assuming some choice, as observed in TomGrubb's answer. If we consider a Hamel basis $$B$$ of $$\Bbb R$$ over $$\Bbb Q$$, then the set of all real numbers with zero first projection is a subgroup $$K$$ of $$\Bbb R$$ for which $$\Bbb R/K\simeq \Bbb Q$$.
But it can't be finite (unless it is trivial). In other words, there is no coherent way to talk about real numbers being partitioned into the "even" and "odd" ones. If $$\Bbb R/K$$ has $$n>1$$ elements, then that means that every number which is a multiple of $$n$$ is in $$K$$; but every real number is a multiple of $$n$$, to wit, $$x=n(x/n)$$.
Another way to look at it is to think about the set of periods, i.e.
$$P = \{ p \in \mathbb{R} | f(x + p) = f(x) \text{ for all } x \in \mathbb{R} \}$$
Zero is clearly a member of this set no matter what f is. P is closed under addition and negation. So clearly P is a group over addition. So, if you want to find a function that has arbitrarily small periods, you want to find a subgroup of $$\mathbb{R}$$ that has arbitrarily small values. $$\mathbb{Q}$$ is the obvious choice, so the characteristic function for $$\mathbb{Q}$$ works, as stated in another answer.
Here's a different function with the same property (which relies on a fair bit of choice). Choose a Hamel basis for $$\mathbb{R}$$ over $$\mathbb{Q}$$ and pick a basis vector $$v$$. Let $$f$$ be the function which projects onto the $$v$$ coordinate. Then for any other basis vector $$u$$ and any integer $$n$$, $$f(x+u/n)=f(x).$$ More can be found on these functions in the article "Discontinuous additive functions" by Bernardi.
• Hope you don't mind my edit, since it's not that much choice needed for what you want. =) Oct 4, 2018 at 17:10 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9854964220125032,
"lm_q1q2_score": 0.8508693372286261,
"lm_q2_score": 0.8633916047011594,
"openwebmath_perplexity": 121.84249319753401,
"openwebmath_score": 0.8838260173797607,
"tags": null,
"url": "https://math.stackexchange.com/questions/2942016/function-with-arbitrary-small-period/2942235"
} |
# Expected number of groups of 3 consecutive wins in 200 rounds
If I know the probability of winning an individual round (while playing, say, a slot machine), call it $p_0$, what is the expected number of groups of 3 consecutive wins in 200 rounds? Or, to make it more general, what is the expected number of $k$-consecutive-win groups in $n$ rounds?
The groups cannot overlap, so 4 consecutive wins do not count as 2 different groups of 3 wins, while 6 consecutive wins do count as 2 different groups.
I was thinking of looking for the total number of possible $k$-group configurations in $n$ rounds, but I'm not too sure.
Explaining how you got this result would be most welcome, though the final formula would do as well. Thank you very much.
My solution will not only answer your question exactly (within linear algebra roundoff error) for the general case, but actually gives you the entire probability distribution of number of successful (i.e., completed) streaks.
This can readily be solved by using a discrete time discrete state (time-homogeneous) Markov Chain, in an approach similar in spirit to, but with larger state space, than the methods I used in https://math.stackexchange.com/questions/383704/probability-of-streaks/1739987#1739987 . After you read my answer in that thread plus this thread, you should be a wizard in using Markov Chains to solve all manner of streaks problems.
I will let
p = probability of winning per round
k = number of wins for successful streak
n = number of rounds
Then g is determined to be the upper bound on largest number of streaks, calculated as
g = floor(n/k) | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9854964228644457,
"lm_q1q2_score": 0.8508693362321365,
"lm_q2_score": 0.863391602943619,
"openwebmath_perplexity": 644.4102924561752,
"openwebmath_score": 0.7503558993339539,
"tags": null,
"url": "https://stats.stackexchange.com/questions/207460/expected-number-of-groups-of-3-consecutive-wins-in-200-rounds"
} |
g = floor(n/k)
Define the states as bivariate pairs(i,j), in which i denotes the current number of successful (i.e., completed) streaks and j is the length of the current streak. i will range from 0 to g, and j will range from 0 to k-1, except for when i = g, in which case j only goes from 0 to 0, and we make that state, (g,0), an absorbing state because we can't get any additional successful streaks. Order the states with j increasing fastest, then i increasing. I.e., with k = 3, the states would be (0,0),(0,1),(0,2),(1,0),(1,1),(1,2), etc.
Once the Markov Chain one step transition matrix, M, has been populated (see below), we compute the n step Markov Chain transition matrix, Mn as $M^n$. Given that we start in state (0,0) before the first of n rounds, the first row of Mn contains the probabilities of being in the various states after n rounds. For each possible value of number of successful streaks i, the sum of the probabilities over j for all states (i,j) provides the probability of exactly i successful streaks having occurred. It is then trivial to compute the expected number of successful streaks.
The one step transition matrix is populated as follows (note that w.p. is short for "with probability"):
For each i from 0 to g-1 (I'll show here for the case k = 3)
State (i,0) transitions to (i,0) w.p. 1-p and to (i,1) w.p. p
State (i,1) transitions to (i,0) w.p. 1-p and to (i,2) w.p. p
State (i,2) transitions to (i,0) w.p. 1-p and to (i+1,0) w.p. p
The absorbing state (g,0) transitions w.p. 1 to (g,0)
The population of the one step Markov Chain for a general value of k (and n) is shown in my MATLAB code below. Everything on a line after % is a comment. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9854964228644457,
"lm_q1q2_score": 0.8508693362321365,
"lm_q2_score": 0.863391602943619,
"openwebmath_perplexity": 644.4102924561752,
"openwebmath_score": 0.7503558993339539,
"tags": null,
"url": "https://stats.stackexchange.com/questions/207460/expected-number-of-groups-of-3-consecutive-wins-in-200-rounds"
} |
k = 3; n = 200; p = .6; % set k, n, and p to particular values
g = floor(n/k);
B = [(1-p)*ones(k,1),p*eye(k)]; % recurring block in transition matrix M.
% B is a k by (k+1) matrix, consisting of a column vector of (1-p) 's,
% right-horizontally concatenated with p times the k by k identity matrix
% Start building up one step transition matrix, M
M = zeros(g*k+1); % (g*k+1) by (g*k+1) matrix of zeros
for i=0:g-1, M(k*i+1:k*(i+1),k*i+1:k*(i+1)+1) = B; end;
M(g*k+1,g*k+1) = 1; % Construction of M is now complete
Mn = M^n; % n step transition matrix
% Calculate array of probabilities of number of successful streaks
% and place in prob_array
prob_array = zeros(g+1,1);
for i=0:g-1, prob_array(i+1) = sum(Mn(1,k*i+1:k*i+k)); end;
prob_array(g+1) = Mn(1,g*k+1); % prob_array is now complete
expected_number_streaks = (0:g)*prob_array
Here are example results for n = 200 rounds, and streaks of length k = 3, with bonus results for streaks of length k = 5.
Here is an example one step transition matrix, for p = 0.6, n = 11, k = 3, which results in g = 3.
• Great. Yes, this agrees with observations... By the by, do you recommend any books to learn (relatively) advanced probability? I hadn't met the use of Markov Chains like this before. Apr 18 '16 at 7:02
• @Kristian D'Amato , many so-called advanced probability books deal with rigorous theory - it sounds like that may not be what you want. It sounds like a book on stochastic processes might be good for you, or perhaps applied probability with significant stochastic processes content, but I don't have a specific book of such type to recommend. Apr 18 '16 at 17:12 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9854964228644457,
"lm_q1q2_score": 0.8508693362321365,
"lm_q2_score": 0.863391602943619,
"openwebmath_perplexity": 644.4102924561752,
"openwebmath_score": 0.7503558993339539,
"tags": null,
"url": "https://stats.stackexchange.com/questions/207460/expected-number-of-groups-of-3-consecutive-wins-in-200-rounds"
} |
Since you're talking about a slot-machine, I assume that the probabilities are independent. The probability of $k$ consecutive wins is $p_0^k$. That probability is the expected portion of the groups with size $k$ that contain all wins. The number of groups with size $k$ (that does not overlap) is $200/k$, and we can call the number of all rounds $n$ (already alluded to in your question). So, the expected number of groups with size $k$ that contains consecutive wins would be
$p_0^k \cdot (n/k)$
$p_0^3 \cdot (200/3)$
Edit:
Apparently this problem is deceptively hard, like really hard. Just answering what the probability is that one group of length $k$ will appear is hard. The probability $S(n,k)$ of a group of consecutive wins of length $k$ in a chain of $n$ events, with the probability of a win $p$ and probability of a fail $1-p=q$ is
$S(n,k) = p^k \sum_{i=0}^{\infty} {n - (i+1)k \choose i}(-qp^k)^i - \sum_{i=1}^{\infty} {n - ik \choose i}(-qp^k)^i$
Here, the groups can even overlap, which is not what you asked for. I hope this can point you in the right direction.
(I apologize for the wrong answer earlier, I hope I did not lead you astray too much) | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9854964228644457,
"lm_q1q2_score": 0.8508693362321365,
"lm_q2_score": 0.863391602943619,
"openwebmath_perplexity": 644.4102924561752,
"openwebmath_score": 0.7503558993339539,
"tags": null,
"url": "https://stats.stackexchange.com/questions/207460/expected-number-of-groups-of-3-consecutive-wins-in-200-rounds"
} |
(I apologize for the wrong answer earlier, I hope I did not lead you astray too much)
• Are you sure this is right? Observed information is not matching with this prediction (observations are about 50% higher) Apr 15 '16 at 10:47
• Are you sure that the groups may not overlap? Because, if they are allowed to overlap I can understand why the estimate is missing by 50% and in that case you should use the formula n + 1 - k to estimate the number of possible blocks. Apr 15 '16 at 11:41
• Yep. Keep in mind that if they are allowed to overlap the chance of getting a second win-group overlapping with the first is not independent of the first (since you only need one more round to have another win-group)... Anyway they don't overlap. Apr 15 '16 at 12:35
• I'm doubting myself, but I don't see how this could be wrong. p^k is the expected proportion of groups with all wins, n/k is the maximum amount of consecutive, non overlapping groups that can fit within the chain. Are you running simulations and could you supply some code in that case? Apr 15 '16 at 13:40
• I will when I'm able to later on. There are other k-chains in the entire sequence, since a group can start at position 2, for instance, covering 2-4, etc... Apr 15 '16 at 13:51 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9854964228644457,
"lm_q1q2_score": 0.8508693362321365,
"lm_q2_score": 0.863391602943619,
"openwebmath_perplexity": 644.4102924561752,
"openwebmath_score": 0.7503558993339539,
"tags": null,
"url": "https://stats.stackexchange.com/questions/207460/expected-number-of-groups-of-3-consecutive-wins-in-200-rounds"
} |
# Taylor Theorem inequality
Prove that for all $f\in C^2([0,1])$ with $f(0)=f(1)=0$ and $|f''(x)| \le 1$ $$|f(x)| \le \frac{1}{2}x(1-x)$$ $\forall x \in [0,1]$.
-
If your function is only continuous, how do you know for sure that it has a second derivative? – Mercy Mar 30 '14 at 1:01
Mistake in the post, sorry! Just corrected… – Brontolo Mar 30 '14 at 1:05
We have, $\displaystyle (1-x)\int_0^x tf''(t)\,dt -x \int_x^1 (t-1)f''(t)\,dt = -f(x)$, for $x \in [0,1]$ (just apply integration by parts on LHS to ease the simplification).
Since, $t\ge 0$ for $t\in [0,1]$ and $(t-1) \le 0$ for $t \in [0,1]$, taking modulus on both sides,
$|f(x)| = \displaystyle \left|(1-x)\int_0^x tf''(t)\,dt -x \int_x^1 (t-1)f''(t)\,dt\right|$
$\le \displaystyle \left|(1-x)\int_0^x tf''(t)\,dt\right| + \left|-x \int_x^1 (t-1)f''(t)\,dt\right|$
$\le \displaystyle \sup\limits_{t\in[0,1]}|f''(t)|. \bigg( (1-x)\int_0^x t\,dt -x \int_x^1 (t-1)\,dt \bigg)$
$= \dfrac{x(1-x)}{2}.\sup\limits_{t\in[0,1]}|f''(t)| \le \dfrac{x(1-x)}{2}$.
Aliter: Define $g(t)=f(t)-\dfrac{t(t-1)}{x(x-1)}f(x)$, on $[0,1]$.
Then, $g(0)=g(x)=g(1)=0$.
Applying Rolle's Theorem twice on $(0,1)$, $\exists \alpha \in (0,1)$ such that $g''(\alpha)=0$.
That is $g''(\alpha) = f''(\alpha) - \dfrac{2}{x(x-1)}f(x)=0$
or, $|f(x)|=\dfrac{x(1-x)}{2}|f''(\alpha)| \le \dfrac{x(1-x)}{2}$. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9854964194566753,
"lm_q1q2_score": 0.8508693350219458,
"lm_q2_score": 0.8633916047011595,
"openwebmath_perplexity": 643.1677334210763,
"openwebmath_score": 0.944036602973938,
"tags": null,
"url": "http://math.stackexchange.com/questions/731928/taylor-theorem-inequality"
} |
or, $|f(x)|=\dfrac{x(1-x)}{2}|f''(\alpha)| \le \dfrac{x(1-x)}{2}$.
-
How do you pull $\sup|f''(t)|$ out of the integrals while maintaining the subtraction between the integrals? Suppose $f''(t)=1$ on $[0,x]$ and $f''(t)=-1$ on $[x,1]$ (or a continuous function very close to that). – robjohn Mar 30 '14 at 2:00
I think if you add the absolute values of the integrals you get what you want. That is, $$\frac12x^2(1-x)+\frac12x(1-x)^2=\frac12x(1-x)$$ – robjohn Mar 30 '14 at 2:03
@robjohn fixed typo and added a line .. thanks for pointing it out :) – r9m Mar 30 '14 at 2:08
Nice way the second one! I thought that it would be good to do with Taylor Theorem (therefore the title) but I found no way. – Brontolo Mar 30 '14 at 10:17
For some $\xi_k\in[0,1]$, $$f(1)=f(0)+f'(0)+\frac12f''(\xi_1) \quad\text{and}\quad f(0)=f(1)-f'(1)+\frac12f''(\xi_2)$$ implies $$|f'(0)|\le\frac12 \quad\text{and}\quad |f'(1)|\le\frac12$$ Let $g(x)=f(x)-\frac12x(1-x)$ and $h(x)=f(x)+\frac12x(1-x)$.
$g(0)=0$, $g'(0)\le0$, and $g''(x)\le0$; therefore, $g(x)\le0$.
$h(0)=0$, $h'(0)\ge0$, and $h''(x)\ge0$; therefore, $h(x)\ge0$.
Thus, $$\overbrace{-\frac12x(1-x)\le}^{h(x)\ge0}f(x)\overbrace{\le\frac12x(1-x)}^{g(x)\le0}$$
-
Can you explain why $g(0)=0, g′(0)≤0$, and $g″(x)≤0$; therefore, $g(x)≤0$? – Brontolo Mar 30 '14 at 10:09
@TheMaker94: Mean Value Theorem on $[0,1]$. Since $g'(0)\le0$ and $g''(x)\le0$, we have $g'(x)\le0$. Since $g(0)=0$ and $g'(x)\le0$, $g(x)\le0$. – robjohn Mar 30 '14 at 12:14
Perfect, clear thanks! – Brontolo Mar 30 '14 at 12:28
We want to prove that for each $x$:$$\exists c\ \ f(x) =\frac 12 f''(c) x(1-x)$$
We want to find such a $c$ via the Rolle theorem (or via the mean value theorem, but we can always go back to the Rolle version).
We can already apply the Rolle theorem, which gives an annulation for $f'$.
Let us modify $f$ to go to $0$ once more: assuming $0<x<1$,
$$g(u) := f(u) + A_xu(1-u)\\ g(x) = 0\Leftarrow A_x = - \frac{f(x)}{x(1-x)}$$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9854964194566753,
"lm_q1q2_score": 0.8508693350219458,
"lm_q2_score": 0.8633916047011595,
"openwebmath_perplexity": 643.1677334210763,
"openwebmath_score": 0.944036602973938,
"tags": null,
"url": "http://math.stackexchange.com/questions/731928/taylor-theorem-inequality"
} |
$$g(u) := f(u) + A_xu(1-u)\\ g(x) = 0\Leftarrow A_x = - \frac{f(x)}{x(1-x)}$$
Now $g(1)=g(x) = g(0)$ hence, applying several times the Rolle theorem:
$$\exists c \ \ 0=g''(c) = f''(c) - \frac{f(x)}{x(1-x)}(-2)\\ f(x) = -\frac 12 x(1-x)f''(c)$$
NB: the error in the sign does not change the final inequality.
-
clever idea (+1) – robjohn Mar 30 '14 at 8:09 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9854964194566753,
"lm_q1q2_score": 0.8508693350219458,
"lm_q2_score": 0.8633916047011595,
"openwebmath_perplexity": 643.1677334210763,
"openwebmath_score": 0.944036602973938,
"tags": null,
"url": "http://math.stackexchange.com/questions/731928/taylor-theorem-inequality"
} |
# Finding Area using Integrals
As part of my Calculus 2 class, I have created a practice exam to help my students study. In doing so, I felt that providing the questions and the solutions in a blog series would be helpful not just to my students, but also to anyone that may be taking, or teaching, calculus 2. I hope you find the following solutions helpful in your studies.
You can also view me presenting the solution on YouTube at
## Find the area bounded by the function $$f(x)=8x$$, $$g(x)=\frac{8}{x}$$, and $$h(x)=x^{2}$$.
Drawing a sketch of these graphs, we get the following picture.
In order to find the area, we cut this into smaller cross sections, then add up the areas of the cross sections. Here we will cut out rectangles as shown below.
Note that the height of the rectangles has a different function value in the first portion than it does in the second portion, so we will split this into two parts. In the first section, we get that
\begin{align*} A_{cs}&=h*w \\ &=(y_{b}-y_{s})\Delta x, \end{align*} where $$y_{b}$$ is the bigger $$y$$ value and $$y_{s}$$ is the smaller $$y$$ value. In this case, we note from the graph that $$y_{b}=8x$$ and $$y_{s}=x^{2}$$. Therefore,
\begin{align*} A_{cs}=(8x-x^{2})\Delta x. \end{align*}
In order to find the total area over this interval, we then find that
\begin{align*} A_{T1}=\int_{a}^{b}(8x-x^{2})dx. \end{align*} Where is $$a$$ is the smallest $$x$$ value and $$b$$ is the largest $$x$$ value where these cross sections work.
We find $$a$$ as the intersection of $$8x$$ and $$x^{2}$$. That is, we solve
\begin{align*} 8x&=x^{2} \\ 0&=x^{2}-8x \\ 0&=x(x-8) \\ x&=0,8. \end{align*} From the picture, we note that the point we are interested for this example is actually $$a=0$$.
We then find $$b$$ as the intersection of $$8x$$ and $$\frac{8}{x}$$, so we get
\begin{align*}
8x&=\frac{8}{x} \\
8x^{2}&=8 \\
x^{2}&=1 \\
x&=\pm 1.
\end{align*}
Again, by looking at the picture, we note that the correct choice is $$b=1$$. | {
"domain": "dralb.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9919380077567411,
"lm_q1q2_score": 0.8508527815745291,
"lm_q2_score": 0.8577681013541613,
"openwebmath_perplexity": 504.4114249531262,
"openwebmath_score": 0.9998196959495544,
"tags": null,
"url": "https://dralb.com/2019/02/02/finding-area-using-integrals/"
} |
Therefore,
\begin{align*} A_{T1}&=\int_{0}^{1}(8x-x^{2})dx \\ &=4x^{2}-\frac{x^{3}}{3}|_{0}^{1} \\ &=4-\frac{1}{3}-0=\frac{11}{3}. \end{align*}
If we follow the same process for the second area, we note that we will have $$y_{b}=\frac{8}{x}$$ and $$y_{s}=x^{2}$$. Therefore, we will get that
\begin{align*} A_{cs}&=h*w \\ &=(y_{b}-y_{s})\Delta x \\ &=(\frac{8}{x}-x^{2})\Delta x. \end{align*}
Therefore, the total area of the second section will be
\begin{align*} A_{T2}=\int_{b}^{c}(\frac{8}{x}-x^{2})dx. \end{align*}
We have already found that $$b=1$$, so now we need to find $$c$$. Here, this is the intersection of $$\frac{8}{x}$$ and $$x^{2}$$. Therefore, we get that
\begin{align*} \frac{8}{x}&=x^{2} \\ 8&=x^{3} \\ x&=2. \end{align*} Hence, $$c=2$$.
We then get that
\begin{align*} A_{T2}&=\int_{1}^{2}(\frac{8}{x}-x^{2})dx \\ &=8\ln|x|-\frac{x^{3}}{3}|_{1}^{2} \\ &=8\ln(2)-\frac{8}{3}-(8\ln(1)-\frac{1}{3}) \\ &=8\ln(2)-\frac{7}{3}. \end{align*}
Now that we have found both areas, the total area of the region will be given as the sum of the two areas. Therefore, the total area is
\begin{align*} A_{T}&=A_{T1}+A_{T2} \\ &=\frac{11}{3}+8\ln(2)-\frac{7}{3} \\ &=\frac{4}{3}+8\ln(2). \end{align*}
## Other Resources
I have it linked above, but you can find the practice exam here. You can find the other solutions to the practice exam in the other posts available here.
If you found the post, or the YouTube video helpful, please like the post or the video. Also remember to follow the blog and subscribe to the YouTube channel. By doing so you will not only be able to find future content, you will also make it easier for other students to find the content when they are studying for Calculus. Thank you.
## 1 thought on “Finding Area using Integrals”
This site uses Akismet to reduce spam. Learn how your comment data is processed. | {
"domain": "dralb.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9919380077567411,
"lm_q1q2_score": 0.8508527815745291,
"lm_q2_score": 0.8577681013541613,
"openwebmath_perplexity": 504.4114249531262,
"openwebmath_score": 0.9998196959495544,
"tags": null,
"url": "https://dralb.com/2019/02/02/finding-area-using-integrals/"
} |
# Kronecker Product (G Dataflow)
Calculates the Kronecker product of two input matrices.
## matrix A
The first input matrix.
This input accepts a 2D array of double-precision, floating point numbers or 2D array of complex double-precision, floating point numbers.
Default: Empty array
## matrix B
The second input matrix.
This input accepts a 2D array of double-precision, floating point numbers or 2D array of complex double-precision, floating point numbers.
Default: Empty array
## error in
Error conditions that occur before this node runs. The node responds to this input according to standard error behavior.
Default: No error
## kronecker product
Matrix containing the Kronecker product of the first and second input matrices.
The number of rows in kronecker product is the product of the number of rows in the first and second input matrices. The number of columns in kronecker product is the product of the number of columns in the first and second input matrices.
## error out
Error information. The node produces this output according to standard error behavior.
## Algorithm for Calculating the Kronecker Product
If A is an n-by-m matrix and B is a k-by-l matrix, the Kronecker product of A and B, C = AB, results in a matrix C with dimensions nk-by-ml. This node calculates the Kronecker product using the following equation.
$C={\left[\begin{array}{cccc}{a}_{11}B& {a}_{12}B& \dots & {a}_{1m}B\\ {a}_{21}B& {a}_{22}B& \dots & {a}_{2m}B\\ ⋮& ⋮& \ddots & ⋮\\ {a}_{n1}B& {a}_{n2}B& \dots & {a}_{nm}B\end{array}\right]}_{nk×ml}$
For example, if
$\begin{array}{cc}A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]& B=\left[\begin{array}{cc}5& 6\\ 7& 8\end{array}\right]\end{array}$
then | {
"domain": "ni.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9875683465856102,
"lm_q1q2_score": 0.8508278685190617,
"lm_q2_score": 0.8615382129861583,
"openwebmath_perplexity": 1527.5243900412706,
"openwebmath_score": 0.346889466047287,
"tags": null,
"url": "http://www.ni.com/documentation/en/labview-comms/2.0/node-ref/kronecker-product/"
} |
then
$\begin{array}{ccc}{a}_{11}B=\left[\begin{array}{cc}5& 6\\ 7& 8\end{array}\right]& {a}_{12}B=\left[\begin{array}{cc}10& 12\\ 14& 16\end{array}\right]& C=\left[\begin{array}{cc}{a}_{11}B& {a}_{12}B\\ {a}_{21}B& {a}_{22}B\end{array}\right]=\left[\begin{array}{cccc}5& 6& 10& 12\\ 7& 8& 14& 16\\ 15& 18& 20& 24\\ 21& 24& 28& 32\end{array}\right]\end{array}$
Where This Node Can Run:
Desktop OS: Windows
FPGA: Not supported | {
"domain": "ni.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9875683465856102,
"lm_q1q2_score": 0.8508278685190617,
"lm_q2_score": 0.8615382129861583,
"openwebmath_perplexity": 1527.5243900412706,
"openwebmath_score": 0.346889466047287,
"tags": null,
"url": "http://www.ni.com/documentation/en/labview-comms/2.0/node-ref/kronecker-product/"
} |
moment of inertia of thin disc about a parallel axis, distance away So the moment of inertia for all such thin discs becomes when The cone is lying on its side with the vertex at the origin so which gives. Physics 111 Lecture 21 (Walker: 10. The spheres have negligible size, and the rod has negligible mass. ( 2 pt ) The distance @ that would be used in the parallel axis theorem to find the mass moment of inertia about the origin, O, of the 0. Question: Show That The Moment Of Inertia Of A Uniform Solid Sphere Rotating About A Diameter Is 2/5 M R^2. We will be using mainly a cylindrical ring and a sphere in our experaments. (a) Show that the moment of inertia about a diameter of a uniform spherical shell of inner radius Rio outer radius R and density p is 1 = p(πm/15)(R5/2 – R5/1'). What is the angular momentum of the sphere?. (1) Moment of inertia of a Solid Sphere : (a) About an axis passing through its diameter : Consider a solid sphere of mass M and radius R. (ii) about a tangent: A tangent drawn to the sphere at any point, will obviously be parallel to one of its diameters and the distance between the axes is equal to R, the radius of the sphere. perpendicular to xy-plane passing through a point on the x-axis at a distance x. 1562 kg m 2 and torque applied is 0. 8 Solid sphere rotating about the central axis. (This also assumes we are rotating the bodies around the same axis. (b) Given the moment of inertia of a disc of mass M and radius R about any of its diameters to 1 be 1/4 MR 2, find the moment of inertia about an axis normal to the disc passing through a point on its edge. calculate moment of inertia of any object, rotating about an arbitrary axis. ) of a sphere about its diameter = 2MR 2 /5 According to the theorem of parallel axes, the moment of inertia of a body about any axis is equal to the sum of the moment of inertia of the body about a parallel axis passing through its centre of mass and the product of its mass and the square of the distance. For | {
"domain": "venetoacquaeterme.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9875683465856102,
"lm_q1q2_score": 0.850827865008135,
"lm_q2_score": 0.8615382094310357,
"openwebmath_perplexity": 240.6116665009223,
"openwebmath_score": 0.5893383622169495,
"tags": null,
"url": "http://vkwf.venetoacquaeterme.it/show-that-the-moment-of-inertia-of-a-uniform-solid-sphere-rotating-about-a-diameter-is.html"
} |
passing through its centre of mass and the product of its mass and the square of the distance. For simple solid objects, one can calculate the moment of inertia from the mass, size, and shape. Torque The turning effect of a force with respect to some axis, is called moment of force or torque due to the force. (b) Given the moment of inertia of a disc of mass M and radius R about any of its diameters to 1 be 1/4 MR 2, find the moment of inertia about an axis normal to the disc passing through a point on its edge. Since moment of inertia is a scalar quantity, a compound object made up of several objects joined together has a moment of inertia which is the A rotating door is made from four rectangular glass panes, as shown in the drawing. The moment of inertia is a value that describes the distribution. ( 1 pt each ) Object A is a solid sphere. 0 kg and radius 0. By parallel axes theorem; This is an expression for M. Sphere #1 will arrive first B. The power transmitted by the shaft is. (This also assumes we are rotating the bodies around the same axis. 4 Radius of Gyration 3. A machine part has the shape of a solid uniform sphere of mass 225 g and diameter 3. • Rotating objects tend to keep rotating, while non-rotating objects tend to remain non-rotating. Moment of inertia of solid sphere when it is rotating about its diameter can be determined using integration process and about different axes can be. ( 1 pt ) A disc in the x-y plane is rotating about an axis perpendicular to the x-y plane. In mathematical notation, the moment of inertia is often symbolized by I, and the radius is symbolized by r. For the sake of one more bit of integration practice, we shall now use the same argument to show that the moment of inertia of a uniform circular disc about a. The moment of inertia for some common shapes are given below. where I is the moment of inertia. mass m and radius a, about a diameter of its plane face. We would expect the moment of inertia to be smaller about an | {
"domain": "venetoacquaeterme.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9875683465856102,
"lm_q1q2_score": 0.850827865008135,
"lm_q2_score": 0.8615382094310357,
"openwebmath_perplexity": 240.6116665009223,
"openwebmath_score": 0.5893383622169495,
"tags": null,
"url": "http://vkwf.venetoacquaeterme.it/show-that-the-moment-of-inertia-of-a-uniform-solid-sphere-rotating-about-a-diameter-is.html"
} |
a, about a diameter of its plane face. We would expect the moment of inertia to be smaller about an axis through the center of mass than the endpoint axis, just as it was for the barbell example at the start of this section. For example, if two disks have the same mass but one has all the mass around the rim and the other is solid, then the disks would have different moments of inertia. , an axle through the center and perpendicular to the disk, the moment of inertia is calculated by carrying out the. A uniform solid sphere with mass M and radius R has a moment of inertia I = 2/5MR{eq}^2 {/eq} about an axis through its center. (a) Find the angular momentum of the skater. The moment of inertia may be defined as, I = sum m_ir_i^2 and if the system is continuous, then I = int r^2dm If rho is the mass density then, dm = rhodV where dV is an elementary volume. Thin walled cylinder rotating about the central axis. Experiments show that, if we compare bodies of similar shape and size but having different masses, the moment of inertia, I is directly proportional to the mass. Moment of Inertia (Rotational Inertia) I:- Moment of Inertiaof a body, about a given axis, is Moment of inertia of a solid disc:- (a) About an axis passing through its center and perpendicular Motion of a point mass attached to a string would over a cylinder capable of rotating about its axis of symmetry. Point P is midway between the center and the rim of the disk, and point Q is on the rim. Learn how to calculate moment of inertia of different shapes or objects using the several formulas. The moment of inertia of a solid sphere of radius of 0. Labels: Circular Disc and solid sphere, Mass Moment of Inertia, Mass Moment of I think This blog is very interesting, I hope you feel same like me about my link @ moment of inertia calculator online It's a good post about moment of inertia. Use Newton's second law to obtain two equations in a and T that we can solve simultaneously. A solid sphere with a mass of | {
"domain": "venetoacquaeterme.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9875683465856102,
"lm_q1q2_score": 0.850827865008135,
"lm_q2_score": 0.8615382094310357,
"openwebmath_perplexity": 240.6116665009223,
"openwebmath_score": 0.5893383622169495,
"tags": null,
"url": "http://vkwf.venetoacquaeterme.it/show-that-the-moment-of-inertia-of-a-uniform-solid-sphere-rotating-about-a-diameter-is.html"
} |
to obtain two equations in a and T that we can solve simultaneously. A solid sphere with a mass of 8. So when the masses are placed at r= 0, I= I0. Show that the moments of inertia of a uniform rod f mass M and length 2a about an axis through its centre perp. The translational velocity is slow enough to make easy accurate measurements. (****) Find the moment of inertia of a uniform, thin-walled sphere of radius R and mass M. A solid sphere, disc and solid cylinder all of same mass and made up of same material are allowed to roll down (from rest) on an inclined plane, then (a) solid sphere reaches the bottom late (b) solid sphere reaches the bottom first (c) disc will reach the bottom first (d) all will reach the bottom at the same time. 16 in order to remain upright under the in uence of gravity. Does it have a larger moment of inertia for an axis through the thicker end of the rod and perpendicular to the length of the rod, or for an axis through the thinner end of the rod. 00-m-diameter wagon. 115m and a mass of 12. A uniform solid S is generated by fully rotating R in the x axis. A particle of mass M is attached to one end of the stick. Moment of inertia states that:The product mass and the square of perpendicular distance from the axis of rotation is known as moment of inertia. not the hypotenuse. A constant tension of 23. Two solid uniform spheres roll down a ramp without slipping or sliding. 2 1 3 Where M is the mass and R is the radius of the sphere. 37 x 106 meters. 00 m long and has mass 4. 1 Rotational Inertia. Moment of Inertia: Sphere. This is determined by summing the moments of inertia of the thin discs that form the sphere. A hollow cylinder and a solid cylinder have the same diameter. The translational velocity is slow enough to make easy accurate measurements. (6) About what axis will a uniform, balsa-wood sphere have R the same moment of inertia as does a thin-walled, hollow, lead Rsphere of the same mass and radius, with the axis along a diameter? | {
"domain": "venetoacquaeterme.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9875683465856102,
"lm_q1q2_score": 0.850827865008135,
"lm_q2_score": 0.8615382094310357,
"openwebmath_perplexity": 240.6116665009223,
"openwebmath_score": 0.5893383622169495,
"tags": null,
"url": "http://vkwf.venetoacquaeterme.it/show-that-the-moment-of-inertia-of-a-uniform-solid-sphere-rotating-about-a-diameter-is.html"
} |
a thin-walled, hollow, lead Rsphere of the same mass and radius, with the axis along a diameter? Use th e Parallel Axis Theorem. The density is then (1) and the moment of inertia tensor is (2) (3) (4). The moment of inertia of the sphere is I = 2 5 MR2. 84 m diameter solid sphere can be rotated about an axis through its center by a torque of 10. Sphere 2 has twice the radius of sphere 1. Thin walled cylinder rotating about the central axis. Circular Disk Rotating About Its Diameter The moment of inertia for the same circular disk rotating about an axis in the plane of the disk, passing through its center, is given by Thus, the uniform disk's moment of inertia in its own plane is twice that about its diameter. The moment of inertia is a physical quantity which describes how easily a body can be rotated about a given axis. Moment of inertia of a sphere can be explained in two parts (1) Solid Sphere (2)Hollow Sphere. (a) Show that the moment of inertia of a uniform hollow cylinder of inner radius R1, outer radius R2, and mass M, is I ½ M(R12 R22), if the rotation axis is through the center along the axis of symmetry. ( 2 pt ) The distance @ that would be used in the parallel axis theorem to find the mass moment of inertia about the origin, O, of the 0. show more decimal digits. 7 cm in diameter. Labels: Circular Disc and solid sphere, Mass Moment of Inertia, Mass Moment of I think This blog is very interesting, I hope you feel same like me about my link @ moment of inertia calculator online It's a good post about moment of inertia. The axis of rotation in the question is a tangent to the ring. 01 18-Jun-2003 1. A solid disk with a mass of 0. The moment of inertia of an object changes if the axis of rotation is changed. calculate its moment of inertia about any axis through its centre. The following links are to calculators which will calculate the Section Area Moment of Inertia Properties of common shapes. In this section we show how the idea of integration as the | {
"domain": "venetoacquaeterme.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9875683465856102,
"lm_q1q2_score": 0.850827865008135,
"lm_q2_score": 0.8615382094310357,
"openwebmath_perplexity": 240.6116665009223,
"openwebmath_score": 0.5893383622169495,
"tags": null,
"url": "http://vkwf.venetoacquaeterme.it/show-that-the-moment-of-inertia-of-a-uniform-solid-sphere-rotating-about-a-diameter-is.html"
} |
of Inertia Properties of common shapes. In this section we show how the idea of integration as the limit of a sum can be used to find the moment of inertia of a lamina. The ratio of the larger Sphere moment of inertia to that of the smaller sphere is 4 Consider two uniform solid spheres were one has twice the mass and what is the diameter of the other. A solid disk will have a different moment than a washer, and there are formulas derived for calculating the moments of many common shapes. 6 mm when it bears a load. 6 F z z P. 132 kg m 2 about an axis which is found to be. 8 of Newman 1977 gives the added inertia for coefficient for spheroids of varying aspect ratio, referred to the moment of inertia of the displaced mass. Show that the moments of inertia of a uniform rod f mass M and length 2a about an axis through its centre perp. As a consequence, the flow path in a rotating chute deviates considerably from that in a non-rotating chute. 3 Physical Significance of Moment of Inertia 3. Harm to minors, violence or threats, harassment or privacy invasion, impersonation or misrepresentation, fraud or phishing, show. Two uniform solid spheres have the same mass, but one has twice the radius of the other. Not the earth going around the sun, but the earth rotating on its axis, then you'd have to say that the moment of inertia for that amount of rotation is 2/5 mr squared, because it's a sphere rotating through an axis that goes through its center. Find the moment of inertia of the rod and solid sphere combination about the two axes as shown below. Sphere 2 has three times the radius of sphere 1. (b) Find the final rotation rate of the skater. Moments of inertia of rigid bodies∗ - Similar to Moments of inertia of rigid bodies∗ May 23, 2011 Moment of inertia of rigid body depends on the distribution of mass dimensional bodies like cylinder and sphere. [ In this question, you may assume standard results for the moment of inertia of uniform circular discs. Since moment of | {
"domain": "venetoacquaeterme.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9875683465856102,
"lm_q1q2_score": 0.850827865008135,
"lm_q2_score": 0.8615382094310357,
"openwebmath_perplexity": 240.6116665009223,
"openwebmath_score": 0.5893383622169495,
"tags": null,
"url": "http://vkwf.venetoacquaeterme.it/show-that-the-moment-of-inertia-of-a-uniform-solid-sphere-rotating-about-a-diameter-is.html"
} |
you may assume standard results for the moment of inertia of uniform circular discs. Since moment of inertia is a scalar quantity, a compound object made up of several objects joined together has a moment of inertia which is the A rotating door is made from four rectangular glass panes, as shown in the drawing. A hoop a solid sphere a flat disk a hollow sphere Each of the objects has mass M and radius R. The moment of inertia of a uniform rod about an axis through its center is. A homogeneous solid cylinder of mass m, length L, and radius R rotates about an axis through point P, which is parallel to the cylinder axis. 84 m diameter solid sphere can be rotated about an axis through its center by a torque of 10. EXAMPLE: MOMENT OF INERTIA / EARTH EXAMPLE: 6The Earth has mass and radius 5. 16 The variation of angular position θ, of a point on a rotating rigid body, with time t is shown in Fig. Next, we calculate the moment of inertia for the same uniform thin rod but with a different axis choice so we can compare the results. The moment of inertia of a circular ring about a diameter is ½ mr2, with usual notations. 115m and a mass of 12. For a different rotation point of an object—say a rod rotating around one end, like a turnstile, instead of around its center—we use the parallel axis theorem to find the object's moment of inertia. In this case, the moment of inertia of the mass in this system is a scalar known as the polar moment of inertia. • Subdivide body into small volume elements • Add the moment of inertia contributed by all these amounts of massAdd the moment of inertia contributed by all these amounts of mass • I = M ⋅(average value of R2) 2. In this section, we show how to calculate the moment of inertia for several standard types of objects, as well as how to use known moments of inertia to find the moment of inertia for a shifted axis or for a compound object. For a thin uniform homogenous rectangular plate, the mass moment of inertia about the rectangular | {
"domain": "venetoacquaeterme.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9875683465856102,
"lm_q1q2_score": 0.850827865008135,
"lm_q2_score": 0.8615382094310357,
"openwebmath_perplexity": 240.6116665009223,
"openwebmath_score": 0.5893383622169495,
"tags": null,
"url": "http://vkwf.venetoacquaeterme.it/show-that-the-moment-of-inertia-of-a-uniform-solid-sphere-rotating-about-a-diameter-is.html"
} |
For a thin uniform homogenous rectangular plate, the mass moment of inertia about the rectangular coordinate axes, a and b, passing through the centre of gravity of the circular plate can be obtained from the area moment of inertia. Moment of inertia of solid disk Period of small oscillation of sphere rolling in cylinder. • Subdivide body into small volume elements • Add the moment of inertia contributed by all these amounts of massAdd the moment of inertia contributed by all these amounts of mass • I = M ⋅(average value of R2) 2. (iii) Moment of inertia of a body should always be referred to as about a given axis, since it depends upon distribution of mass about that axis. Each sphere is a distance R+L/2 from the axis of rotation, so we must use the parallel axis theorem. Physics 100A Homework 10 – Chapter 10 (part 2) 10. 36 Determining Moments of Inertia. The moment of inertia of a uniform rod about an axis through its center is. The Brick Solid block adds to the attached frame a solid element with geometry, inertia, and color. Furthermore, because of the symmetry of the sphere, each principal moment is the same, so the moment of inertia of the sphere taken about any diameter is. • Parallel axis theorem for products of inertia: Product of inertia is useful in calculating MI @ inclined axes. The rotational kinetic energy is the kinetic energy of rotation of a rotating rigid body or system of particles, and is given by $K=\frac{1}{2}I{\omega }^{2}$, where I is the moment of inertia, or “rotational mass” of the rigid body or system of particles. 00 m long and has mass 4. Solid sphere, radius r, about diameter. Answer is in kg⋅m2/s 2. The moment of inertia of a thin spherical shell of mean radius 0. Moment of inertia of solid shere about any diameter is (2/5)MR^2 , where M is mass and R is radius of sphere. The sphere is rotated about a diameter with an angular speed ω. 8(a) in side view. The radius of the sphere is 20. Show that the magnetic moment 'u' and the | {
"domain": "venetoacquaeterme.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9875683465856102,
"lm_q1q2_score": 0.850827865008135,
"lm_q2_score": 0.8615382094310357,
"openwebmath_perplexity": 240.6116665009223,
"openwebmath_score": 0.5893383622169495,
"tags": null,
"url": "http://vkwf.venetoacquaeterme.it/show-that-the-moment-of-inertia-of-a-uniform-solid-sphere-rotating-about-a-diameter-is.html"
} |
ω. 8(a) in side view. The radius of the sphere is 20. Show that the magnetic moment 'u' and the angular momentum 'L' of the sphere are related as: u=Lq/2m Pls help with this question. A man stands on a rotating platform that has an angular speed of 6. I have derived moment of inertia of solid sphere along diameter but my textbook says that moment of inertia is "Homework-like questions should ask about a specific physics concept and show some effort to work through the problem. Moments of Inertia and Angular Momentum. So it applies no torque, Since this is the only force acting on the system, the net torque is zero. It should not be confused with the second moment of area. The moment of inertia of a body rotating around an arbitrary axis is equal to the moment of inertia of a body rotating around a parallel axis through the center of mass plus the mass times the perpendicular distance between the axes h squared. 2: Angular momentum of a sphere Question: A uniform sphere of mass and radius spins about an axis passing through its centre with period. then the vertical gravitation would be x. 2 m is set into rotation about an axis passing through its centre and perpendicular to its plane by applying torque 10 Nm. We were discussing "Method to determine the area moment of inertia for a hollow rectangular section", "The theorem of parallel axis Let us consider one hollow circular section, where we can see that D is the diameter of main section and d is the diameter of cut-out section as displayed in following figure. Moment of Inertia of a solid body • Mass continuously distributed throughout its volume. 5 kg of a cold metal at a temperature of 258 K is immersed in 2. Find the moment of inertia of this combination about each of the following axes: (a) an axis pelvendicular to the bar through. Rotational Motion. M -mass, R -Radius. Vocabulary Angular Momentum: The measure of. If the surface of the ball is defined by the equation: 1301 + + =,. – What is the rotational | {
"domain": "venetoacquaeterme.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9875683465856102,
"lm_q1q2_score": 0.850827865008135,
"lm_q2_score": 0.8615382094310357,
"openwebmath_perplexity": 240.6116665009223,
"openwebmath_score": 0.5893383622169495,
"tags": null,
"url": "http://vkwf.venetoacquaeterme.it/show-that-the-moment-of-inertia-of-a-uniform-solid-sphere-rotating-about-a-diameter-is.html"
} |
of. If the surface of the ball is defined by the equation: 1301 + + =,. – What is the rotational kinetic energy of a 450-g solid sphere with a diameter of 23 cm rotating at rate of 17 rpm? – What is the total rotational kinetic energy of a 18-kg child riding on the edge of a merry-go-round of mass 160 kg and r = 2. A solid uniform L-shaped plate has mass $m$ and dimensions as shown. You have two steel spheres. [You may assume, without proof, that the moment of inertia of a uniform circular disc, of mass m and radius r, about a diameter is 1 4 mr2. diameter of the solid cylinder is large b. 6 F z z P. And needs to solve in both spherical & cylindrical coordinate system. The moment of inertia of a circular ring about a diameter is ½ mr2, with usual notations. MOTION OF SYSTEM OF PARTICLES AND RIGID BODY CONCEPTS. Moment of inertia - Parallel-Axis Theorem Pendulum Moment of Inertia of a Rotating Body Collision Moment of inertia tensor Moment of Inertia of a thin uniform rod by integration A 15--diameter CD has a mass of 24. The moment of inertia of a uniform rod about an axis through its center is. 0cm spins about the axle through its center. In the first part of our lab a rotating solid cylindrical drum with a hollow body drum given a rotational velocity from a falling mass. 0 cm and has mass 1. Sponsored Links. mass m and radius a, about a diameter of its plane face. You have two steel spheres. ( 2 pt ) The distance @ that would be used in the parallel axis theorem to find the mass moment of inertia about the origin, O, of the 0. mass of the solid cylinder is large d. We will compare our results for a uniform, solid disk and a uniform ring with those derived from theory. Using the definition of moment of inertia, I = r 2 dm, one can show that theory predicts I disk = ½ MR2 (4) I ring = ½ M(R IN 2 + R OUT 2) (5). If they all are released from rest. Hollow sphere of radius r and mass m Similar to the solid sphere, only this time considering a stack of infinitesimal | {
"domain": "venetoacquaeterme.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9875683465856102,
"lm_q1q2_score": 0.850827865008135,
"lm_q2_score": 0.8615382094310357,
"openwebmath_perplexity": 240.6116665009223,
"openwebmath_score": 0.5893383622169495,
"tags": null,
"url": "http://vkwf.venetoacquaeterme.it/show-that-the-moment-of-inertia-of-a-uniform-solid-sphere-rotating-about-a-diameter-is.html"
} |
radius r and mass m Similar to the solid sphere, only this time considering a stack of infinitesimal thin, circular hoops. KE = (I * w^2)/2. The mass of the hub can be ignored. (b) Obtain the moment of inertia for a solid cylinder. Solid Versus Hollow Edit A hoop (hollow object) has a greater moment of inertia than a greater moment of inertia than a solid disk of the same mass because all of the mass of the hoop is at a large radius. Although mass is defined in terms of inertia, it is conventionally interpreted as. Its moment of inertia about an axis tangent to it and perpendicular to its plane is ?. A solid sphere rolls (without slipping) down a plane inclined at 30˚ to the horizontal. Now if the two masses are each. 10 (a) Find the moment of inertia of a sphere about a tangent to the sphere, given the moment of inertia of the sphere about any of its diameters to be 2MR2/5, where M is the mass of the sphere and R is the radius of the sphere. The moments of inertia of common shapes (such as a uniform rod, a uniform or a hollow cylinder, a uniform or a hollow sphere) are well known and readily accessible in any mechanics textbook. A solid uniform sphere of radius R and mass M has a rotational inertia about a diameter that is given by (2 = 5) MR 2. Recall that the moment of inertia of a rod about its centre is and that the moment of inertia of a solid sphere about its centre is. disk can be considered as a uniform solid disk of radius 25 cm and mass of 1. (2R/ O15) from the center of the sphere) (7) A frictionless pulley has the shape of a uniform solid di sk of mass 2. Moment of inertia, denoted by I, measures the extent to which an object resists rotational acceleration about a particular axis, and is the rotational analogue to mass. Learn how to calculate moment of inertia of different shapes or objects using the several formulas. ] (10) (b) Hence find the moment of inertia of a uniform solid sphere, of mass M and radius a, about a diameter. The moment of inertia | {
"domain": "venetoacquaeterme.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9875683465856102,
"lm_q1q2_score": 0.850827865008135,
"lm_q2_score": 0.8615382094310357,
"openwebmath_perplexity": 240.6116665009223,
"openwebmath_score": 0.5893383622169495,
"tags": null,
"url": "http://vkwf.venetoacquaeterme.it/show-that-the-moment-of-inertia-of-a-uniform-solid-sphere-rotating-about-a-diameter-is.html"
} |
inertia of a uniform solid sphere, of mass M and radius a, about a diameter. The moment of inertia of a uniform rod about an axis through its center is. Derive the expression for moment of inertia of a uniform ring about an axis passing through the center and perpendicular to the plane. Calculate the moment of inertia of a thin plate in the shape of a right triangle, about an axis that passes through one end of the hypotenuse and is parallel to the opposite leg of the triangle, as in Figure. Assume the ball is a uniform, solid sphere. The moment of inertia of a uniform object depends not only on the size and shape of that object but on the location of the axis about which the object is rotating. Annapolis MD. The spheres have negligible size, and the rod has negligible mass. Determine the moment of inertia for a solid cylinder with mass m and radius R with a non-uniform mass density given by p= ar^2. m what is her final moment of inertia'? How does she physi- cally accomplish this change? (Il) A potter's wheel is rotating around a vertical axis through its center at a frequency of 1-5 rev/s The wheel can be considered a uniform disk of mass 5. 94 m/s2 down the ramp (b) 3. For axis A, the rod is rotating about its centre of mass. Not the earth going around the sun, but the earth rotating on its axis, then you'd have to say that the moment of inertia for that amount of rotation is 2/5 mr squared, because it's a sphere rotating through an axis that goes through its center. A thin uniform bar has two small balls glued to its ends. a) Calculate the torque applied to the disk by the rope. 60-cm-diameter sprocket? Three objects of uniform density—a solid sphere, a solid cylinder, and a hollow cylinder—are placed at the top of an incline (Fig. 1 kg m2 as the skater draws his arms and legs inward toward the axis of rotation. Find the moment of inertia about a diameter?. 85kg and diameter 45. 16 The variation of angular position θ, of a point on a rotating rigid body, with | {
"domain": "venetoacquaeterme.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9875683465856102,
"lm_q1q2_score": 0.850827865008135,
"lm_q2_score": 0.8615382094310357,
"openwebmath_perplexity": 240.6116665009223,
"openwebmath_score": 0.5893383622169495,
"tags": null,
"url": "http://vkwf.venetoacquaeterme.it/show-that-the-moment-of-inertia-of-a-uniform-solid-sphere-rotating-about-a-diameter-is.html"
} |
and diameter 45. 16 The variation of angular position θ, of a point on a rotating rigid body, with time t is shown in Fig. The moment of inertia of the sphere is I = 2 5 MR2. For a ring let’s assume an element of mass dm on the ring. Answer: (a) Moment of inertia of sphere about any diameter = 2/5 MR 2. The density is then (1) and the moment of inertia tensor is (2) (3) (4). moment of inertia is the same about all of them. none of the above are necessary A common thread spool rests on a flat table. For a solid uniform sphere and a thin hoop, each of mass Mand radius R and rotating about their respective centers of mass, the moment of inertia of the hoop is larger than that of the sphere. I have been asked what the rotational inertia difference is between a hollow Aluminium driveshaft of diameter 44. To look for the angle of inclination which is b we do the following steps: 90 - b = a. (Hint: Form the shell by superposition of a sphere of density p and a smaller sphere of density -p. A small solid marble of mass m and radius r rolls without slipping along a loop-the-loop track shown in Figure 12. Step 2: The moment of inertia of the balsa wood sphere (solid) about the diameter is, 2 2 I2= M5 We have to choose an axis through which these two moment of inertias would be same. 500 kg and can be treated as point masses. 00-cm-diameter, 330 sphere is released from rest at the top of a 2. (1) Moment of inertia of a Solid Sphere : (a) About an axis passing through its diameter : Consider a solid sphere of mass M and radius R. Cotufa is doing homework on "moment of inertia" of uniform solid sphere and a uniform solid cylinder. 00-cm-diameter sprocket if the wheel is to attain an acceleration of 4. 0 kg and R — -MR2. A uniform solid S is generated by fully rotating R in the x axis. The moment of inertia about an axis at one end is. The inertia tensor of the disk alone about its center of mass is: MR2 4 1 0 0 0 1 0 0 0 2 (A) Find the inertia tensor of the disk alone about the | {
"domain": "venetoacquaeterme.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9875683465856102,
"lm_q1q2_score": 0.850827865008135,
"lm_q2_score": 0.8615382094310357,
"openwebmath_perplexity": 240.6116665009223,
"openwebmath_score": 0.5893383622169495,
"tags": null,
"url": "http://vkwf.venetoacquaeterme.it/show-that-the-moment-of-inertia-of-a-uniform-solid-sphere-rotating-about-a-diameter-is.html"
} |
center of mass is: MR2 4 1 0 0 0 1 0 0 0 2 (A) Find the inertia tensor of the disk alone about the point A. 150 m has a moment of inertia for rotation through its central axis. Find the value of the spin in revolutions per second for a= 10cm and b= 1cm. Derive the expression for moment of inertia of a uniform ring about an axis passing through the center and perpendicular to the plane. moment of inertia of thin disc about a parallel axis, distance away So the moment of inertia for all such thin discs becomes when The cone is lying on its side with the vertex at the origin so which gives. If we had a sphere, a solid sphere, then So here you have a solid sphere, and I rotate it about an axis through its center. If her initial moment of inertia was 4. It is a convention to write to indicate the moment of inertia with respect to an axis passing though the center of mass of the rotating rigid body. Obtain the moment of inertia of a hollow solid sphere of inner and outer radii r1 and r2 Products. The expression for the moment of inertia of a sphere can be developed by summing the moments of infintesmally thin disks about the z axis. Mass Moment of Inertia Angular Momentum Rotational Kinetic Energy rad rad N-m kg-m rad sec sec2 o kg-m2 kg-m2 sec Hz sec sec2 sec 14. 18) After fixing a flat tire on a bicycle you give the wheel a spin. The links will open a new browser window. A bug of mass m lands at the center of the disc and then walks outward. Can someone please show me show more The rotational inertia of a solid uniform sphere about a diameter is (2/5)MR2, where M is its mass and R is its radius. Problem- 2 Find the moment of inertia of a uniform solid sphere of mass M and radius R about a diameter. Let us consider a sphere of radius R and mass M. It is a convention to write to indicate the moment of inertia with respect to an axis passing though the center of mass of the rotating rigid body. Although mass is defined in terms of inertia, it is conventionally interpreted | {
"domain": "venetoacquaeterme.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9875683465856102,
"lm_q1q2_score": 0.850827865008135,
"lm_q2_score": 0.8615382094310357,
"openwebmath_perplexity": 240.6116665009223,
"openwebmath_score": 0.5893383622169495,
"tags": null,
"url": "http://vkwf.venetoacquaeterme.it/show-that-the-moment-of-inertia-of-a-uniform-solid-sphere-rotating-about-a-diameter-is.html"
} |
rotating rigid body. Although mass is defined in terms of inertia, it is conventionally interpreted as. About what axis will a uniform, balsa-wood sphere have the same moment of inertia as does a thin-walled, hollow, lead sphere of the same mass and radius, with the axis along a diameter Students also viewed these Mechanics questions. The results for centroid, moment of inertia, statical moment section modulus and torsion constant will display on your right. The bar is 2. The moment of inertia is the torque required to start an angle of acceleration along a rotating axis. 97 x 1024 kg and 6. Recall that the moment of inertia of a rod about its centre is and that the moment of inertia of a solid sphere about its centre is. Now for a sphere with uniform density rotating around its axis I is, I = (2MR^2)/5. If they are both released from the same height and at the same time, which one will arrive at the bottom of the ramp first? A. A solid sphere, disc and solid cylinder all of same mass and made up of same material are allowed to roll down (from rest) on an inclined plane, then (a) solid sphere reaches the bottom late (b) solid sphere reaches the bottom first (c) disc will reach the bottom first (d) all will reach the bottom at the same time. The work-energy theorem for a rigid body rotating around a fixed axis is where —10) and the rotational work done by a net force rotating a body from point A to point B is (10. The force F = 15 N is applied to the rope for a duration of 3 seconds. Physics 100A Homework 10 – Chapter 10 (part 2) 10. 6 kg object is found to have a moment of inertia of. 1) Find its angular acceleration. 0 is applied to the rope and the sphere starts to roll without slipping on the show more A uniform 8. 00 kg m2, but this is reduced to 2. For a thin uniform homogenous rectangular plate, the mass moment of inertia about the rectangular coordinate axes, a and b, passing through the centre of gravity of the circular plate can be obtained from the area | {
"domain": "venetoacquaeterme.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9875683465856102,
"lm_q1q2_score": 0.850827865008135,
"lm_q2_score": 0.8615382094310357,
"openwebmath_perplexity": 240.6116665009223,
"openwebmath_score": 0.5893383622169495,
"tags": null,
"url": "http://vkwf.venetoacquaeterme.it/show-that-the-moment-of-inertia-of-a-uniform-solid-sphere-rotating-about-a-diameter-is.html"
} |
a and b, passing through the centre of gravity of the circular plate can be obtained from the area moment of inertia. Moment of inertia of a Uniform Hollow Cylinder -. moment of inertia about its center of mass 𝐼𝐼. Here are some of the most common moments of inertia: Solid cylinder or disk of radius r rotating about its axis of symmetry. (b) Obtain the moment of inertia for a solid cylinder. A uniform disk of mass m is not as hard to set into rotational motion as a \dumbbell" with the same mass and radius. 15 • As shown in the figure, solid sphere rolls on a horizontal surface at 20 m/s and then rolls up the incline. If you are allowed to use the fact that the rotational inertia of a spherical shell is (2/3)mr2 [and I can derive that if necessary], all you have to do is take your solid. I recommend not to post that. Furthermore, because of the symmetry of the sphere, each principal moment is the same, so the moment of inertia of the sphere taken about any diameter is. Problem- 2 Find the moment of inertia of a uniform solid sphere of mass M and radius R about a diameter. m what is her final moment of inertia'? How does she physi- cally accomplish this change? (Il) A potter's wheel is rotating around a vertical axis through its center at a frequency of 1-5 rev/s The wheel can be considered a uniform disk of mass 5. 85kg and diameter 45. Explain why the moment of inertia is larger about the end than about the center. Moment of inertia of this disc about the diameter of the rod is, Moment of inertia of the disc about axis is given by parallel axes theorem is, Hence, the moment of inertia of the cylinder is given as, Solid Sphere a) About its diameter Let us consider a solid sphere of radius and mass. Recall that the moment of inertia of a rod about its centre is and that the moment of inertia of a solid sphere about its centre is. not the hypotenuse. 571 radians) between the strips, the angular velocity ω is computed. angular acceleration? An object remains in a state | {
"domain": "venetoacquaeterme.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9875683465856102,
"lm_q1q2_score": 0.850827865008135,
"lm_q2_score": 0.8615382094310357,
"openwebmath_perplexity": 240.6116665009223,
"openwebmath_score": 0.5893383622169495,
"tags": null,
"url": "http://vkwf.venetoacquaeterme.it/show-that-the-moment-of-inertia-of-a-uniform-solid-sphere-rotating-about-a-diameter-is.html"
} |
the strips, the angular velocity ω is computed. angular acceleration? An object remains in a state of uniform rotational motion unless acted on. As the solid sphere's non uniformity is not mentioned,it should be considered to be uniform and hence spherically symmetric body. ( 1 pt ) A disc in the x-y plane is rotating about an axis perpendicular to the x-y plane. In the first part of our lab a rotating solid cylindrical drum with a hollow body drum given a rotational velocity from a falling mass. Please explain. The moment of inertia of a solid cylinder of radius r is given by: J = mr2 2 By comparison, the moment of inertia of a hollow cylinder, of inner and outer radii respectively, is as follows: J = m(r o 2 - r i 2)2 It can be seen that, for a given outer radius, the moment of inertia of a hollow cylinder is greater than that of a solid cylinder of. I deal with stars, and stars have rotational kinetic energy. Icm = moment of inertia for rotation around an axis through the center of mass () M = total mass of the object (kg) d = distance between the two rotation axes (m) Parallel Axis Theorem Formula Questions: 1) A solid sphere with mass 60. The net torque acting on this sphere as it is slowing down is closest to: A. Using Moment of Inertia The moment of inertia of an object rotating around a fixed object is useful in calculating two key quantities in rotational motion:. Moment of inertia shows the tendency of an object to stay in its state of rotatory motion. If her initial moment of inertia was 4. 3 Physical Significance of Moment of Inertia 3. Moment of inertia of solid sphere about its diameter by. Moment of Inertia--Cylinder : Consider a uniform solid cylinder of mass M, radius R, height h. Let M and R be mass and radius of the hollow cylinder and the solid sphere, then, Moment of inertia of the hollow cylinder about its axis of symmetry, l 1, = MR 2 Moment of inertia of the solid sphere about from ω ω 0 + at, we find that for given ω 0 and t, ω 2 > ω 1, | {
"domain": "venetoacquaeterme.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9875683465856102,
"lm_q1q2_score": 0.850827865008135,
"lm_q2_score": 0.8615382094310357,
"openwebmath_perplexity": 240.6116665009223,
"openwebmath_score": 0.5893383622169495,
"tags": null,
"url": "http://vkwf.venetoacquaeterme.it/show-that-the-moment-of-inertia-of-a-uniform-solid-sphere-rotating-about-a-diameter-is.html"
} |
of inertia of the solid sphere about from ω ω 0 + at, we find that for given ω 0 and t, ω 2 > ω 1, angular speed of solid sphere will be greater than the angular speed of. Related: Beam Deflection Stress Equation Calculators. We are allowed to use the standard result that the moment of inertia about the axis running down its centre is 1/2 m r^2. 15m from is center of mass. then prove that-omega =9/14 omega not. In a rotating body Torque is equal to the moment of Inertia multiplied by angular acceleration. 46 car on an incline A car on an incline is timed from release until the end of a measured distance. We want our questions to be useful to the broader community, and. We would expect the moment of inertia to be smaller about an axis through the center of mass than the endpoint axis, just as it was for the barbell example at the start of this section. Its rotational inertia about the point of attachment at the ceiling is: A) (2/5)MR2 B) 4MR2 C) (7/5)MR2 D) (22/5)MR2 E) (47. Its moment of inertia about an axis of rotation passing through its diameter is I = MR 2. Apply the parallel axis theorem From this result, we can conclude that it is twice as hard to rotate the barbell about the end than Next, we calculate the moment of inertia for the same uniform thin rod but with a different axis. L as Y O C) 2017 Akaa Daniel Ayangeakaa. R about any of its diameters to be MR2/4, find its moment of inertia about an axis normal to the disc and passing through a point on its edge. Sphere #2 will arrive first C. Some of the moments of inertia are given in the table below: slender rod: axis through center axis through end rectangular plane: axis through center axis along edge sphere thin-walled hollow solid cylinder hollow solid walled thin-hollow. What is the moment of inertia of the system of. More of the sphere's mass is far away from the center of rotation, so the hollow one has a big moment of inertia. Obtain the moment of inertia of a hollow solid sphere of inner and outer | {
"domain": "venetoacquaeterme.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9875683465856102,
"lm_q1q2_score": 0.850827865008135,
"lm_q2_score": 0.8615382094310357,
"openwebmath_perplexity": 240.6116665009223,
"openwebmath_score": 0.5893383622169495,
"tags": null,
"url": "http://vkwf.venetoacquaeterme.it/show-that-the-moment-of-inertia-of-a-uniform-solid-sphere-rotating-about-a-diameter-is.html"
} |
a big moment of inertia. Obtain the moment of inertia of a hollow solid sphere of inner and outer radii r1 and r2 Products. For the sake of one more bit of integration practice, we shall now use the same argument to show that the moment of inertia of a uniform circular disc about a. 98 x 10 24 kg) (6. A light string of length 3 R is 45 ± to the tangent ans: D Section: 10{8; Di±culty: E 174 Chapter 10: ROTATION 59. In systems that are both rotating and translating, conservation of mechanical energy can be used if there are no nonconservative forces at work. Get the linear acceleration of center of sphere denoting it as y. Show that the moments of inertia of a uniform rod f mass M and length 2a about an axis through its centre perp. Answer: E 14) A uniform solid sphere has a moment of inertia I about an axis tangent to its surface. What is the moment of inertia of a uniform circular disc and circular ring of radius R and mass M. Since moment of inertia is a scalar quantity, a compound object made up of several objects joined together has a moment of inertia which is the A rotating door is made from four rectangular glass panes, as shown in the drawing. A solid sphere with a mass of 8. Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. Problem- 2 Find the moment of inertia of a uniform solid sphere of mass M and radius R about a diameter. The volume of such a layer is. (ii) about a tangent: A tangent drawn to the sphere at any point, will obviously be parallel to one of its diameters and the distance between the axes is equal to R, the radius of the sphere. I = ∑mr2 Rotational Kinetic Energy:- K r. What is the direction of its angular momentum vector? 15. 18) After fixing a flat tire on a bicycle you give the wheel a spin. The moment of inertia of this element is: dr. 16 The variation of angular position θ, of a point on a rotating rigid body, with time t is shown in Fig. The only data for 3D solids we are aware | {
"domain": "venetoacquaeterme.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9875683465856102,
"lm_q1q2_score": 0.850827865008135,
"lm_q2_score": 0.8615382094310357,
"openwebmath_perplexity": 240.6116665009223,
"openwebmath_score": 0.5893383622169495,
"tags": null,
"url": "http://vkwf.venetoacquaeterme.it/show-that-the-moment-of-inertia-of-a-uniform-solid-sphere-rotating-about-a-diameter-is.html"
} |
on a rotating rigid body, with time t is shown in Fig. The only data for 3D solids we are aware of are for spheroids: figure 4. As a leading global manufacturer of crushing, grinding and mining equipments, we offer advanced, reasonable solutions for any size-reduction requirements including, Obtain the moment of inertia of a hollow solid sphere of inner and outer radii r1 and r2, quarry, aggregate, and different kinds of. How is the moment of inertia related to. 98 x 1024 kg and an average radius of 6. | {
"domain": "venetoacquaeterme.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9875683465856102,
"lm_q1q2_score": 0.850827865008135,
"lm_q2_score": 0.8615382094310357,
"openwebmath_perplexity": 240.6116665009223,
"openwebmath_score": 0.5893383622169495,
"tags": null,
"url": "http://vkwf.venetoacquaeterme.it/show-that-the-moment-of-inertia-of-a-uniform-solid-sphere-rotating-about-a-diameter-is.html"
} |
# How many six digit numbers start with the same two digits and end with the same three digits?
Say that there is a 6 digit number the first digit is not allowed to be 0 or 1
so
How many number combinations start with the same two digits and end with the same three digits
ie.119333, 448222, 889888 etc..
My thoughts are 8_ 1_ 10_ 10_ 1_ 1_ = 8*10*10=800?
and how would I do it if instead of the and it said Or?
first of all, I'm I on the correct path here? any other examples similar would help.
• Your title should be specific to the problem at hand so that someone who has a similar question can find it if she or he searches the site. – N. F. Taussig Oct 14 '15 at 9:54
Now let us find the number of possibilities if our number starts with the same two digits OR ends with the same three digits. Again we assume that $0$ and $1$ are forbidden as first digit. But they may occur as second digit, for example in $506888$.
There are $(8)(10^4)$ numbers that begin with two equal digits, the first (and therefore second) digit being neither $0$ or $1$.
There are $(8)(10^3)$ numbers that start with an allowed digit and whose last three digits are the same.
If we add the two numbers above, we will have double-counted the numbers in which the first two digits are the same, and the last three are the same.
It follows that the required number is $(8)(10^4)+(8)(10^3)-(8)(10^2)$.
Remark: This is a relatively simple instance of a technique called Inclusion/Exclusion. There are more elaborate versions. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9875683517080176,
"lm_q1q2_score": 0.8508278641548946,
"lm_q2_score": 0.8615382040983515,
"openwebmath_perplexity": 223.72988956123527,
"openwebmath_score": 0.7084540724754333,
"tags": null,
"url": "https://math.stackexchange.com/questions/1479267/how-many-six-digit-numbers-start-with-the-same-two-digits-and-end-with-the-same"
} |
• so Just minus the AND part? – learnmore Oct 14 '15 at 5:48
• Yes. A general formula for two sets is $|A\cup B|=|A|+|B|-|A\cap B|$. Here $|X|$ means the number of elements in the set $X$. – André Nicolas Oct 14 '15 at 5:53
• how would one go about in determine the count if at least one digit appears more then once in the combination.. i.e 252169 – learnmore Oct 14 '15 at 6:01
• This is a different problem, since we no longer have two identical digits at the beginning, or three at the end. I will assume that $0$ is forbidden at the beginning. Then if we had no further restrictions, the answer would be $(9)(10^5)$. Now we subtract the bad numbers, in which all digits are distinct. Let us count the bad numbers. I will stop here and continue in the next comment. – André Nicolas Oct 14 '15 at 6:11
• (Continued) For counting the bad numbers, there are $9$ choices for the first digit. For each of these there are $9$ choices for the second digit, since $0$ is now allowed. For each choice of first two digits, there are $8$ choices for third digit, and so on, for a total of $(9)(9)(8)(7)(6)(5)$. – André Nicolas Oct 14 '15 at 6:15 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9875683517080176,
"lm_q1q2_score": 0.8508278641548946,
"lm_q2_score": 0.8615382040983515,
"openwebmath_perplexity": 223.72988956123527,
"openwebmath_score": 0.7084540724754333,
"tags": null,
"url": "https://math.stackexchange.com/questions/1479267/how-many-six-digit-numbers-start-with-the-same-two-digits-and-end-with-the-same"
} |
A triangular pyramid is constructed using spherical balls so that each ball rests on exactly three balls of the next lower level.
Then, we calculate the number of paths leading from the apex to each position: A path starts at the apex and progresses downwards to any of the three spheres directly below the current position. Consequently, the number of paths to reach a certain position is the sum of the numbers immediately above it (depending on the position, there are up to three numbers above it).
The result is Pascal’s pyramid and the numbers at each level n are the coefficients of the trinomial expansion $(x + y + z)^n$. How many coefficients in the expansion of $(x + y + z)^{200000}$ are multiples of $10^{12}$?
## Solution Using the Multinomial Theorem
The generalization of the binomial theorem is the multinomial theorem. It says that multinomials raised to exponents can be expanded using the formula
$(x_1+x_2+\cdots+x_m)^n=\sum_{{k_1+k_2+\cdots+k_m=n}\atop{0\le k_i\le n}}\left({n}\atop{k_1,k_2,\ldots,k_m}\right)\prod_{1\le t\le m}x_t^{k_t}$
where
$\left({n}\atop{k_1,k_2,\ldots,k_m}\right)=\frac{n!}{k_1!k_2!\cdots k_m!}.$
Of course, when m=2 this gives the binomial theorem. The sum is taken over all partitions $k_1+k_2+\cdots+k_m=n$ for integers $k_i$. If n=200000 abd m=3, then the terms in the expansion are given by
$\left({200000}\atop{k_1,k_2,k_3}\right)x_1^{k_1}x_2^{k_2}x_3^{k_3}=\frac{200000!}{k_1!k_2!k_3!}x_1^{k_1}x_2^{k_2}x_3^{k_3}$
where $k_1+k_2+k_3=200000$. It’s worth pointing out that each of the coefficients is an integer, and thus has a unique factorization into products of prime integers. Of course, there’s no way that we’re going to calculate these coefficients. We only need to know when they’re divisible by $10^{12}$. Thus, we only need to consider how many factors of 2 and 5 are involved. | {
"domain": "jasonbhill.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9875683469514965,
"lm_q1q2_score": 0.8508278618124334,
"lm_q2_score": 0.8615382058759129,
"openwebmath_perplexity": 1737.7390952399246,
"openwebmath_score": 0.4279261529445648,
"tags": null,
"url": "http://code.jasonbhill.com/c/project-euler-154/"
} |
First, we’ll create a function $p(n,d)$ that outputs how many factors of $d$ are included in $n!$. We have that
$p(n,d)=\left\lfloor\frac{n}{d}\right\rfloor+\left\lfloor\frac{n}{d^2}\right\rfloor+\left\lfloor\frac{n}{d^3}\right\rfloor+ \cdots+\left\lfloor\frac{n}{d^r}\right\rfloor,$
where $d^r$ is the highest power of $d$ dividing $n$. For instance, there are 199994 factors of 2 in 200000!. Since we’re wondering when our coefficients are divisible by $10^{12}=2^{12}5^{12}$, we’ll be using the values provided by $p(n,d)$ quite a bit for $d=2$ and $d=5$. We’ll store two lists:
$p2=[p(i,2)\text{ for }1\le i\le 200000]\quad\text{and}\quad p5=[p(i,5)\text{ for }1\le i\le 200000].$
For a given $k_1,k_2,k_3$, the corresponding coefficient is divisible by $10^{12}$ precisely when
$p2[k_1]+p2[k_2]+p2[k_3]<199983\ \text{and}\ p5[k_1]+p5[k_2]+p5[k_3]<49987.$
That is, this condition ensures that there are at least 12 more factors of 2 and 5 in the numerator of the fraction defining the coefficients.
Now, we know that $k_1+k_2+k_3=200000$, and we can exploit symmetry and avoid redundant computations if we assume $k_1\le k_2\le k_3$. Under this assumption, we always have
$k_1\le\left\lfloor\frac{200000}{3}\right\rfloor=66666.$
We know that $k_1+k_2+k_3=200000$ is impossible since 200000 isn't divisible by 3. It follows that we can only have (case 1) $k_1=k_2 < k_3$, or (case 2) $k_1 < k_2=k_3$, or (case 3) $k_1 < k_2 < k_3$.
In case 1, we iterate $0\le k_1\le 66666$, setting $k_2=k_1$ and $k_3=200000-k_1-k_2$. We check the condition, and when it is satisfied we record 3 new instances of coefficients (since we may permute the $k_i$ in 3 ways).
In case 2, we iterate $0\le k_1\le 66666$, and when $k_1$ is divisible by 2 we set $k_2=k_3=\frac{200000-k_1}{2}$. When the condition holds, we again record 3 new instance. | {
"domain": "jasonbhill.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9875683469514965,
"lm_q1q2_score": 0.8508278618124334,
"lm_q2_score": 0.8615382058759129,
"openwebmath_perplexity": 1737.7390952399246,
"openwebmath_score": 0.4279261529445648,
"tags": null,
"url": "http://code.jasonbhill.com/c/project-euler-154/"
} |
In case 3, we iterate $0\le k_1\le 66666$, and we iterate over $k_2=k_1+a$ where $1\le a < \left\lfloor\frac{200000-3k_1}{2}\right\rfloor$. Then $k_3=200000-k_1-k_2$. When the condition holds, we record 6 instances (since there are 6 permutations of 3 objects).
## Cython Solution
I’ll provide two implementations, the first written in Cython inside Sage. Then, I’ll write a parallel solution in C.
%cython import time from libc.stdlib cimport malloc, free head_time = time.time() cdef unsigned long p(unsigned long k, unsigned long d): cdef unsigned long power = d cdef unsigned long exp = 0 while power <= k: exp += k / power power *= d return exp cdef unsigned long * p_list(unsigned long n, unsigned long d): cdef unsigned long i = 0 cdef unsigned long * powers = <unsigned long *>malloc((n+1)*sizeof(unsigned long)) while i <= n: powers[i] = p(i,d) i += 1 return powers run_time = time.time() # form a list of number of times each n! is divisible by 2. cdef unsigned long * p2 = p_list(200000,2) # form a list of number of times each n! is divisible by 5. cdef unsigned long * p5 = p_list(200000,5) cdef unsigned long k1, k2, k3, a cdef unsigned long long result = 0 k1 = 0 while k1 <= 66666: # case 1: k1 = k2 < k3 k2 = k1 k3 = 200000 - k1 - k2 if 199982 >= (p2[k1]+p2[k2]+p2[k3]) and 49986 >= (p5[k1]+p5[k2]+p5[k3]): result += 3 # case 2: k1 < k2 = k3 if k1 % 2 == 0: k2 = (200000 - k1)/2 k3 = k2 if 199982 >= (p2[k1]+p2[k2]+p2[k3]) and 49986 >= (p5[k1]+p5[k2]+p5[k3]): result += 3 # case 3: k1 < k2 < k3 a = 1 while 2*a < (200000 - 3*k1): k2 = k1 + a k3 = 200000 - k1 - k2 if 199982 >= (p2[k1]+p2[k2]+p2[k3]) and 49986 >= (p5[k1]+p5[k2]+p5[k3]): result += 6 a += 1 k1 += 1 free(p2) free(p5) elapsed_run = round(time.time() - run_time, 5) elapsed_head = round(time.time() - head_time, 5) print "Result: %s" % result print "Runtime: %s seconds (total time: %s seconds)" % (elapsed_run, elapsed_head) | {
"domain": "jasonbhill.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9875683469514965,
"lm_q1q2_score": 0.8508278618124334,
"lm_q2_score": 0.8615382058759129,
"openwebmath_perplexity": 1737.7390952399246,
"openwebmath_score": 0.4279261529445648,
"tags": null,
"url": "http://code.jasonbhill.com/c/project-euler-154/"
} |
When executed, we find the correct result relatively quickly.
Result: 479742450 Runtime: 14.62538 seconds (total time: 14.62543 seconds)
## C with OpenMP Solution | {
"domain": "jasonbhill.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9875683469514965,
"lm_q1q2_score": 0.8508278618124334,
"lm_q2_score": 0.8615382058759129,
"openwebmath_perplexity": 1737.7390952399246,
"openwebmath_score": 0.4279261529445648,
"tags": null,
"url": "http://code.jasonbhill.com/c/project-euler-154/"
} |
## C with OpenMP Solution
#include <stdio.h> #include <stdlib.h> #include <malloc.h> #include <omp.h> /*****************************************************************************/ /* function to determine how many factors of 'd' are in 'k!' */ /*****************************************************************************/ unsigned long p(unsigned long k, unsigned long d) { unsigned long power = d; unsigned long exp = 0; while (power <= k) { exp += k/power; power *= d; } return exp; } /*****************************************************************************/ /* create a list [p(0,d),p(1,d),p(2,d), ... ,p(n,d)] and return pointer */ /*****************************************************************************/ unsigned long * p_list(unsigned long n, unsigned long d) { unsigned long i; unsigned long * powers = malloc((n+1)*sizeof(unsigned long)); for (i=0;i<=n;i++) powers[i] = p(i,d); return powers; } /*****************************************************************************/ /* main */ /*****************************************************************************/ int main(int argc, char **argv) { unsigned long k1, k2, k3, a; unsigned long long result = 0; unsigned long * p2 = p_list(200000, 2); unsigned long * p5 = p_list(200000, 5); #pragma omp parallel for private(k1,k2,k3,a) reduction(+ : result) for (k1=0;k1<66667;k1++) { // case 1: k1 = k2 < k3 k2 = k1; k3 = 200000 - k1 - k2; if (p2[k1]+p2[k2]+p2[k3]<199983 && p5[k1]+p5[k2]+p5[k3]<49987) { result += 3; } // case 2: k1 < k2 = k3 if (k1 % 2 == 0) { k2 = (200000 - k1)/2; k3 = k2; if (p2[k1]+p2[k2]+p2[k3]<199983 && p5[k1]+p5[k2]+p5[k3]<49987) { result += 3; } } // case 3: k1 < k2 < k3 for (a=1;2*a<(200000-3*k1);a++) { k2 = k1 + a; k3 = 200000 - k1 - k2; if (p2[k1]+p2[k2]+p2[k3]<199983 && p5[k1]+p5[k2]+p5[k3]<49987) { result += 6; } } } free(p2); free(p5); printf("result: %lld\n", result); return 0; }
This can be compiled and optimized using GCC as follows. | {
"domain": "jasonbhill.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9875683469514965,
"lm_q1q2_score": 0.8508278618124334,
"lm_q2_score": 0.8615382058759129,
"openwebmath_perplexity": 1737.7390952399246,
"openwebmath_score": 0.4279261529445648,
"tags": null,
"url": "http://code.jasonbhill.com/c/project-euler-154/"
} |
This can be compiled and optimized using GCC as follows.
$gcc -O3 -fopenmp -o problem-154-omp problem-154-omp.c When executed on a 16-core machine, we get the following result. $ time ./problem-154-omp result: 479742450 real 0m1.487s
This appears to be the fastest solution currently known, according to the forum of solutions on Project Euler. The CPUs on the 16-core machine are pretty weak compared to modern standards. When running on a single core on a new Intel Core i7, the result is returned in about 4.7 seconds. | {
"domain": "jasonbhill.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9875683469514965,
"lm_q1q2_score": 0.8508278618124334,
"lm_q2_score": 0.8615382058759129,
"openwebmath_perplexity": 1737.7390952399246,
"openwebmath_score": 0.4279261529445648,
"tags": null,
"url": "http://code.jasonbhill.com/c/project-euler-154/"
} |
# Linear regression problems worksheet | {
"domain": "sramble-communication.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429565737233,
"lm_q1q2_score": 0.8508120142948999,
"lm_q2_score": 0.8652240895276223,
"openwebmath_perplexity": 713.5120381218667,
"openwebmath_score": 0.39756202697753906,
"tags": null,
"url": "http://angs-diasporanew.sramble-communication.com/apdsmyht/linear-regression-problems-worksheet.php"
} |
4. Quadratic regression produces a more accurate quadratic model than the procedure in Example 3 because it uses all the data points. 10) 5. A, B, C, D, E, F. Activity—Twizzlers Linear Regression Pre-Activity & Worksheet Answer Key 2 6. Like half the models in statistics, standard linear regression relies on an assumption of normality. Biology: Five additional weeks of sunshine the sugar concentration in vine grapes will rise by X %. 2. Let the model be Y = 0 + 1X 1 + 2X 2 + ", where E("jX 1;X 2) = 0, and assume that we have a sample AP Stats: Section 3. d. Linear regression estimates the regression coefficients β 0 and β 1 in the equation Y j =β 0 +β 1 X j +ε j where X is the independent variable, Y is the dependent 4. 3 Inferences on the Slope Rarameter ββββ1111 NIPRL 1 12. That is, the equation of the best linear t. Linear reg. 9. A regression with two or more predictor variables is called a multiple regression. 02 (Simple Linear Regression) is based on the identical data set to the paired t -test example above. 1. regression. 32 0. There are two common ways to deal with nonlinear relationships: 1. Multiple Linear Regression Model We consider the problem of regression when the study variable depends on more than one explanatory or independent variables, called a multiple linear regression model. Part 1. 15. 6. CHAPTER 5. The general model can be estimated by grid search or by non-linear maximization of the likelihood and a maximum likelihood estimate for a obtained. Quiz. . Use the regression model to predict the credit card volume in 2003 and in 2010. ac. Going back to our original data, we can try to fit a line through the points that we have; this is called a “trend line”, “linear regression” or “line of best fit” (as we said earlier, the line that’s the “closest fit” to the points – the best trend line). 8 – Trigonometry & Regression Linear Correlations 1. The estimated Under Output Options, choose "New Worksheet Ply," then click OK. (2006) - Chpt 6 zQuinn & | {
"domain": "sramble-communication.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429565737233,
"lm_q1q2_score": 0.8508120142948999,
"lm_q2_score": 0.8652240895276223,
"openwebmath_perplexity": 713.5120381218667,
"openwebmath_score": 0.39756202697753906,
"tags": null,
"url": "http://angs-diasporanew.sramble-communication.com/apdsmyht/linear-regression-problems-worksheet.php"
} |
estimated Under Output Options, choose "New Worksheet Ply," then click OK. (2006) - Chpt 6 zQuinn & Keough (2002) - Chpt 5 zRowntree (1981) - Chpts 12 Question 1 - Simple linear regression Here is an example from Fowler, Cohen and Parvis (1998). To learn more about Nonlinear Regression with data linearization, see the Nonlinear Regression worksheet. In this section we will first discuss correlation analysis, which is used to quantify the association between two continuous variables (e. For the reason that we should supply everything you need in a true along with dependable origin, most people provide valuable details on many themes in addition to topics. Then we perform linear regression on the data. 2 HSS-ID. 355 Problems Predictor Coef SE Coef T P Constant 44. For each of the following tables, treat the left-hand column as the independent variable (input) and the right-hand column as the dependent variable (output), and answer each of the following questions, along with any additional questions related to the actual problem. ’ Here are some of the common Linear Regression Interview Questions that pop up in interviews all over the world. Linear and Quadratic . Regression analysis is the art and science of fitting straight lines to patterns of data. Make a table that shows data from If both the regression coefficients are negative, r would be negative and if both are positive, r would assume a positive value. So, in Excel, you do linear regression using the least squares method and seek coefficients a and b such that: y = bx + a Student Worksheets Created by Matthew M. 4 and 8 2) The difference of two numbers is 3. Linear regression simply refers to creating a best fit for a linear relationship between two variables from observed data. Last ride. 1751(𝑌𝑌𝑌𝑌𝑌𝑌/𝐴𝐴) 𝑊𝑊𝑊𝑊 𝑦𝑦 = −70. Word problems on ages. The equation to represent this data is . In the data, x is the number of seconds after the missile is launched and y is the number of feet above water for the missile. | {
"domain": "sramble-communication.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429565737233,
"lm_q1q2_score": 0.8508120142948999,
"lm_q2_score": 0.8652240895276223,
"openwebmath_perplexity": 713.5120381218667,
"openwebmath_score": 0.39756202697753906,
"tags": null,
"url": "http://angs-diasporanew.sramble-communication.com/apdsmyht/linear-regression-problems-worksheet.php"
} |
of seconds after the missile is launched and y is the number of feet above water for the missile. 5 Prediction Intervals for Future Response Values 12. Property 4 : The two lines of regression coincide i. The least squares method is generally used with a linear regression, but Regression is the process by which the relationship between two variables is determined. The problem of determining the best values of a and b involves the principle of 2 is the sum of squares due to the linear regression SSR, with mean square. The least squares method is generally used with a linear regression, but Aug 01, 2018 · The linear regression equation always has an error term because, in real life, predictors are never perfectly precise. image0. Regression is used to assess the contribution of one or more “explanatory” variables (called independent variables) to one “response” (or dependent ) variable. The data in the table below show different X (depth in feet) 50 Y (maximum dive time) 80 depths with the maximum dive times in minutes. 4¯6, s2 Example of a regression equation Y = $0 +$ 1 (Age - 40) + $2 Gender + , Salary = 50 + 1 (Age - 40) - 3 Gender + , Salary in$1,000s, Age in years and Gender = 0 if male and 1 if female What is the average salary for 50 year old males? Ave(Y) = 50 + 1 (50-40) - 3(0) = $60K following form: y=alpha+beta*x+epsilon (we hypothesize a linear relationship) • The regression analysis „estimates“ the parameters alpha and beta by using the given observations for x and y. 13 Feb 2014 Linear Regression and Correlation - Example. In your story, interpret the slope of the line, the y-intercept, and the x-intercept. In the final chapter, I'll show you how to make the most of your results by changing parameters by hand, performing sensitivity analysis and creating scenarios. Word problems on sets and venn diagrams. By comparing the values of, determine the function that best fits the data. Simple Linear Regression and Correlation 12. • The simplest form of | {
"domain": "sramble-communication.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429565737233,
"lm_q1q2_score": 0.8508120142948999,
"lm_q2_score": 0.8652240895276223,
"openwebmath_perplexity": 713.5120381218667,
"openwebmath_score": 0.39756202697753906,
"tags": null,
"url": "http://angs-diasporanew.sramble-communication.com/apdsmyht/linear-regression-problems-worksheet.php"
} |
that best fits the data. Simple Linear Regression and Correlation 12. • The simplest form of estimating alpha and beta is called ordinary least squares (OLS) regression Problem 3: Let X and Y be two variables in a study. Find the residual amount for a person who is 42. 902. LAB ACTIVITIES FOR SIMPLE LINEAR REGRESSION: TWO VARIABLES 1. This one-page worksheet contains seven problems. A student who waits on tables at a restaurant recorded the cost of meals and the tip b. In case you need guidance on exponents or even radical expressions, Algebra1help. Worksheet 13. dta. Outcomes Students will use a graphing calculator to find a quadratic curve of best fit. Some of the worksheets for this concept are Linear equations work, Solving linear equations, Linear regression work 1, Depends y dependent variable x independent variable y m x, Writing linear equationslinear regression, Real world applications of linear equations, Slope intercept form word problems, Y mx b A statistics Worksheet: The student will calculate and construct the line of best fit between two variables. Next, students will use their calculator to fit a simple linear regression equation with poverty predicting obesity. Y. In this tutorial, […] In Chapter 2 you used a graphing calculator to perform linear regression on a data set in order to find a linear model for the data. The data is shown below. 7 p1 The regression equation is Sales = 116 - 97. 26721 × (8) = 2. Apr 16, 2018 · Note The Regression tool alerts you to this problem and does not continue. Regression interpretation 3. This part of the program will fit a linear function of the form: Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with step-by-step explanations, just like a math tutor. This lesson focuses on two variables which have a linear association. 7. Round to two Statistical Analysis 6: Simple Linear Regression. The population (in thousands) for Alpha City, t years after | {
"domain": "sramble-communication.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429565737233,
"lm_q1q2_score": 0.8508120142948999,
"lm_q2_score": 0.8652240895276223,
"openwebmath_perplexity": 713.5120381218667,
"openwebmath_score": 0.39756202697753906,
"tags": null,
"url": "http://angs-diasporanew.sramble-communication.com/apdsmyht/linear-regression-problems-worksheet.php"
} |
Analysis 6: Simple Linear Regression. The population (in thousands) for Alpha City, t years after January 1, 2004 is modeled by the quadratic function P (t) 0. Old Faithful. To better understand the future strategies, you can visually represent the predicted values in a line chart. Detailed instructions on how to use the calculator are provided. The distribution for each important later. Under some conditions for the observed data, this problem can be solved numerically. variable is (ya - 1)/a, so that with a = 1, the regression is linear, with a = 0, it is logarithmic, these cases being only two possibilities out of an infinite range as a varies. Use the Input Y Range text box to identify the worksheet range holding your Write the linear regression equation for these data where miles driven is the independent Explain what the y-intercept means in the context of the problem. (1998) -Chpts 14 & 15 zHolmes et al. Transform the data so that there is a linear relationship between the transformed variables. Please be sure to check your answers as part of your homework assignment and be prepared with questions for next class. Worksheets are Work 3, Chapter 9 correlation and regression solutions, Writing linear equationslinear regression, Work 1, Linear regression work 1, Linear reg correlation coeff work, Work on correlation and regression, Algebra ii exponential regression work value 10. Make a scatter plot and determine the value of r. not significantly different from zero we conclude that: a) X is a good predictor of Y b) there is no linear relationship between X and Y. 01165 + 0. Mintzberg’s classic book The Nature of Managerial Work (1973) identified the roles found in all managerial jobs. 1 Linear Predictors Before computers became fast, linear regression was almost the only way of at-tacking certain prediction problems. 1751(6. 827 4. 3. Loading. . e. 2. c) BIOSTATS 540 - Fall 2018 Simple Linear Regression and Correlation Page 1 of 54 Nature Population/ Sample | {
"domain": "sramble-communication.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429565737233,
"lm_q1q2_score": 0.8508120142948999,
"lm_q2_score": 0.8652240895276223,
"openwebmath_perplexity": 713.5120381218667,
"openwebmath_score": 0.39756202697753906,
"tags": null,
"url": "http://angs-diasporanew.sramble-communication.com/apdsmyht/linear-regression-problems-worksheet.php"
} |
540 - Fall 2018 Simple Linear Regression and Correlation Page 1 of 54 Nature Population/ Sample Observation/ Data Relationships/ Modeling Analysis/ Synthesis Unit 12 Simple Linear Regression and Correlation “ Assume that a statistical model such as a linear model is a good first start only” Lab!10:!Exploring!Linear!Regression! Objective:!In! this! lab,! you! will examine! relationships! between! two! quantitative! variables! using! a! graphical!tool May 31, 2016 · Regression analysis makes use of mathematical models to describe relationships. The regression coefficient estimated with a linear regression equation y = a + b*x can then tell the researchers b the life expectancy (y) is when smoking x cigarettes a day. Worksheet 3 Universidad Carlos III de Madrid Worksheet 3 The Multiple Regression Model Note: In those problems that include estimations and have a reference to a data set the students should check the outputs obtained with Gretl. That is why it is also termed "Ordinary Least Squares" regression. Linear inequalities word problems. That is, there is lack of fit in the simple linear regression model. However, some programs, including Excel, do the error term calculation behind the scenes. They believe that the number of books that will ultimately be sold for any particular course is related to the number of students registered for the course when the books are ordered. a. Feb 29, 2016 - Explore theboss1000's board "LINEAR REGRESSION", followed by 431 people on Pinterest. In linear regression, you are looking for a hyperplane "near" most of the points; with SVMs, you will be looking for a thick hyperplane, as thin as possible, that contains all the observations. In linear equation in two variables distance problems you have to use two variables and you can solve using any method such as substitution or elimination. Lab Activity: Linear Regression and Correlation In this lab activity, you will collect sample data of two variables, determine if a linear | {
"domain": "sramble-communication.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429565737233,
"lm_q1q2_score": 0.8508120142948999,
"lm_q2_score": 0.8652240895276223,
"openwebmath_perplexity": 713.5120381218667,
"openwebmath_score": 0.39756202697753906,
"tags": null,
"url": "http://angs-diasporanew.sramble-communication.com/apdsmyht/linear-regression-problems-worksheet.php"
} |
In this lab activity, you will collect sample data of two variables, determine if a linear correlation exists between the two variables, and perform linear regression. correlation coefficients, and obtain linear regression equations. For this linear regression worksheet, students solve linear regression problems using the TI-86 calculator. Last year, Walmart conducted a study as to the amount of waiting in time in checkout lanes its customers had to wait. Scaffolded questions that start relatively easy and end with some real challenges. 391 + 17. regression project worksheet: Did the student show the material learned in this course can be useful in a topic that is relevant to the individual students? (explain in one sentence how your data is either interesting to you personally OR related to you major of study) The following are tables of data to be used for linear regression exercises. Find the linear and quadratic regression equations and correlation coefficients. This data set has n=31 observations of boiling points (Y=boiling) and temperature (X=temp). Linear Regression Displaying all worksheets related to - Linear Regression. 1. There are 2 types of factors in regression analysis: Dependent variable (y) : It’s also called the ‘criterion variable’ , ‘response’ , or ‘outcome’ and is the factor being solved. g. A correlation analysis provides information on the strength and direction of the linear relationship between two variables, while a simple linear regression analysis estimates parameters in a linear equation that can be used to predict values of one variable based on Quadratic Models and Quadratic Regression Worksheet 1. P. Consider the following scatter plots: (a) Write the new regression model. Writing Linear Equations/Linear Regression Write the slope-intercept form of the equation of each line given the slope and y-intercept. Graph the linear equation by 2. Regression problems are supervised learning problems in which the response is continuous. A | {
"domain": "sramble-communication.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429565737233,
"lm_q1q2_score": 0.8508120142948999,
"lm_q2_score": 0.8652240895276223,
"openwebmath_perplexity": 713.5120381218667,
"openwebmath_score": 0.39756202697753906,
"tags": null,
"url": "http://angs-diasporanew.sramble-communication.com/apdsmyht/linear-regression-problems-worksheet.php"
} |
by 2. Regression problems are supervised learning problems in which the response is continuous. A researcher has collected data on the price of gasoline from 1990 to 2010 and has found that the price in dollars after t years can be predicted using the equation: y xx− += +0. Problem 2. The user can STAT 2215 Worksheet 5 – Chapter 7: Simple Linear Regression Problem 23, page 198. If there is not a linear relationship between x and y, then $$\mu_{i} ≠ \beta_{0} + \beta_{1}X_{i}$$. ! 2 4 6 8 0 50 100 150 200 250 Calories vs Alcohol Content Alcohol Content (%) Worksheets that accompany this lesson can be located under related documents, worksheets, Data Analysis #1-#8. Figure #10. You may notice that Linear regression where the sum of vertical distances d1 + d2 + d3 + d4 between observed and predicted (line and its equation) values is minimized. introduce problems that are relevant to the fitting of nonlinear regression func- Title: Linear Regression Grade: 8th Lesson Summary: This short lesson plan is to describe students learn how to find the best line to fit the data of two variables they collect and be able to predicate the data by using the regression equation. Simple linear regression is a statistical method that allows us to summarize and study relationships between two continuous (quantitative) variables. In these worksheets, problems are presented as word problems. Simple Linear Regression To describe the linear association between quantitative variables, a statistical procedure called regression often is used to construct a model. The Simple Linear Regression Model is summarized by the equation $y=\beta _1x+\beta _0+\varepsilon$ Identify the deterministic part and the random part. mtpfrom the CD-ROM. SCUBA divers have maximum dive times they cannot exceed when going to different depths. Find r2. Linear regression is a type of machine learning algorithm that is used to model the relation between scalar dependent and one or more independent variables. To use | {
"domain": "sramble-communication.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429565737233,
"lm_q1q2_score": 0.8508120142948999,
"lm_q2_score": 0.8652240895276223,
"openwebmath_perplexity": 713.5120381218667,
"openwebmath_score": 0.39756202697753906,
"tags": null,
"url": "http://angs-diasporanew.sramble-communication.com/apdsmyht/linear-regression-problems-worksheet.php"
} |
is used to model the relation between scalar dependent and one or more independent variables. To use K-nearest neighbors regression, or KNN regression for short, we must start with a data set. Motivation and Objective: We’ve spent a lot of time discussing simple linear regression, but simple linear regression is, well, “simple” in the sense that there is usually more than one variable that helps “explain” the variation in the response variable. write a regression equation and interpret the meaning of the slope and y-intercepts in the context of the problem; make predictions based on the correct mathematical models; and; solve linear equations. Even those who problem. However, we only calculate a regression line if one of the vari-ables helps to explain or predict the other variable. Superimpose the regression curve on the scatter plot. see and learn about curve fitting for multiple linear regression using method of least errors is as small as possible. At home: Read Chapter 5 and work the problems at the end of each short section as you go through them. Estimate the Blood pressure for a person who is 50. Chapter 12. Aug 07, 2014 · Search this site. 7 p1 + 109 p2 Remember: -97. If Y denotes the Excel Linear Regression. (10 marks) C). In linear regression analysis, the dependent variable is thought to be related to the independent variable or variables in a linear way. We would expect the ratio MSLF/MSPE to be close to 1. Time and work word problems. Then, we graph the linear regression equation with the scatterplot data. They compute the percent of the variability. For our problem, we would need only one dummy variable (since K = 2), the dichotomous variable coding level of idealism. Go to Overview Index. In simple linear regression, when β is . Do the Linear Regression ws. a and b are the constants of the regression model. It is a staple of statistics and is often considered a good introductory machine learning method. The elements in X are non-stochastic, meaning | {
"domain": "sramble-communication.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429565737233,
"lm_q1q2_score": 0.8508120142948999,
"lm_q2_score": 0.8652240895276223,
"openwebmath_perplexity": 713.5120381218667,
"openwebmath_score": 0.39756202697753906,
"tags": null,
"url": "http://angs-diasporanew.sramble-communication.com/apdsmyht/linear-regression-problems-worksheet.php"
} |
a good introductory machine learning method. The elements in X are non-stochastic, meaning that the Linear Regression and NORMAL Curve Advanced Placement AAP Review will be held in room 315 and 312 on Tuesdays and Thursdays. Feb 26, 2018 · Linear regression is used for finding linear relationship between target and one or more predictors. Rewrite the equation into finding the x- and y-intercepts. Multiple Linear Regression. If you're seeing this message, it means we're having trouble loading external resources on our website. Plot four points so that the regression line is horizontal. Customize the worksheets to include one-step, two-step, or multi-step equations, variable on both sides, parenthesis, and more. Students will interpret the r-value of the data and write a summary of its meaning. Big Ideas: Bivariate quantitative variables can be represented by a table, graph, and a prediction equation, and estimates can be made from each. GAISE Components One step equation word problems. ) Another very serious problem is the lack of any provision for forecasting from additional values of the independent variables . Throughout the module, you will find many real-world appli cations of these two important topics: least-squares regression line and the correlation coefficient. Assumptions in the Linear Regression Model 2. Some of the worksheets for this concept are Linear regression work 1, Writing linear equationslinear regression, Chapter 9 correlation and regression solutions, Work 1, Work 3, , Quadratic regression, Kuta software. Such an equation can be used for prediction: given a new x-value, this equation can predict the y-value that is consistent with the information known about the data. The sample must be representative of the population 2. Jun 01, 2020 · Statistics Q&A Library WORKSHEET 24 Linear Regression and Correlation Name: 15. 8. This lesson builds on students work in the 8th grade. problem to be solved is reduced to a quadratic programming problem in | {
"domain": "sramble-communication.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429565737233,
"lm_q1q2_score": 0.8508120142948999,
"lm_q2_score": 0.8652240895276223,
"openwebmath_perplexity": 713.5120381218667,
"openwebmath_score": 0.39756202697753906,
"tags": null,
"url": "http://angs-diasporanew.sramble-communication.com/apdsmyht/linear-regression-problems-worksheet.php"
} |
work in the 8th grade. problem to be solved is reduced to a quadratic programming problem in which the objective function is the residual sum of the squares in regression, and the constraints are linear ones imlx~ed on the regression coefficients. 3 times as important as Unconventional. b) Use the calculator and the equation of the linear regression line to complete the following table: Year 1960 1968 1988 1999 2005 2008 Time (s) c) Use the values in the table to draw the linear regression function on the scatter plot. To get a better feel for the regression line, try the following tasks. In Class: Practice Linear Regression HW: Obesity Problem (Hints included in key) r_9. EXTRAS. 45. Displaying all worksheets related to - Regression Analysis. Math 137 Quadratic Regression Classwork2 . Explain worksheet - regression inference 1. It allows the mean function E()y to depend on more than one explanatory variables Like correlation coefficients, linear regression analyzes the relationship between two variables, x and y. 1) Slope = −1, y-intercept = 0 y = −x 2) Slope = 1 4, y-intercept = 1 y = 1 4 x + 1 Write the slope-intercept form of the equation of the line through the given point with the given slope. Start by creating a visual model of the data. Bivariate Data - Correlation - Linear Regression - Correlation Coefficient r Jun 2, 2020 - Are you looking for Algebra 2 worksheets WITH answers? Making keys is super time consuming, so peruse this board for resources that have the hard part done for you!. 5 and 8 This is because the correlation value for the cubic regression is about 0. The multiple linear regression result implies that Reliable is around 1. The table below lists the total estimated numbers of United States AIDS cases, by year of diagnosis. 3582 1. The first is done using the Tools menu, and results in a tabular output that contains Shodor > Interactivate > Lessons > Linear Regression and Correlation omit the scatter plot worksheet; As a class, before | {
"domain": "sramble-communication.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429565737233,
"lm_q1q2_score": 0.8508120142948999,
"lm_q2_score": 0.8652240895276223,
"openwebmath_perplexity": 713.5120381218667,
"openwebmath_score": 0.39756202697753906,
"tags": null,
"url": "http://angs-diasporanew.sramble-communication.com/apdsmyht/linear-regression-problems-worksheet.php"
} |
> Lessons > Linear Regression and Correlation omit the scatter plot worksheet; As a class, before splitting them into groups, have the If you have a continuous dependent variable, linear regression is probably the To address these problems, statisticians have developed several advanced variants: ordered data (assuming you record them in your worksheet in time order). A graphing calculator can also be used to perform quadratic regression. In this correlation and linear regression worksheet, students examine data to determine the statistic mean. 2 Fitting the Regression Line 12. Brandon Foltz 296,888 views Linear regression simply refers to creating a best fit for a linear relationship between two variables from observed data. Ratio and proportion word problems. Although a linear regression can be quite helpful in understanding data, it can sometimes be misleading, as Anscombe's Quartet shows . H. If the degree of correlation between variables is high enough, it can cause problems when you fit the model and interpret the results. (When we need to note the difference, a regression on a single predic-tor is called a simple regression. simple linear regression A college bookstore must order books two months before each semester starts. widely used; runs fast; easy to use (not a lot of tuning The linear regression model that I’ve been discussing relies on several assumptions. A. In other words, the SS is built up as each variable is added, in the order they are given in the command. 12A2 - HW Calendar Linear Regression Problems Q. LINEAR REGRESSION 6 1. The course website page REGRESSION AND CORRELATION has some examples of code to produce regression analyses in STATA. use functions fitted to data to solve problems in the context of the data. Model Summaries Worksheet Every workbook produced by RegressIt contains not only the data analysis and regression worksheets but also a model summaries worksheet that keeps an audit trail of all regression models fitted so far and | {
"domain": "sramble-communication.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429565737233,
"lm_q1q2_score": 0.8508120142948999,
"lm_q2_score": 0.8652240895276223,
"openwebmath_perplexity": 713.5120381218667,
"openwebmath_score": 0.39756202697753906,
"tags": null,
"url": "http://angs-diasporanew.sramble-communication.com/apdsmyht/linear-regression-problems-worksheet.php"
} |
a model summaries worksheet that keeps an audit trail of all regression models fitted so far and allows side-by-side comparison of models fitted to the same dependent variable, suitable for framing. the meaning of slope in a linear regression equation the relationship of the slope to analyzed data whether or not the y-intercept of the linear regression equation is a relevant value EU 2 Econometrics. Enter the data to answer this question. com. To see why, consider a model such as this Y = β 0 +β 1eβ 2X + , (4. Curve Fitting Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to constraints. 48. It also plots the experimental points and the equation y = a x + b where a and b are given by the formulas above. 9) 𝑦𝑦 = 48. where a and b are given by. Round r to the nearest 3 decimal places. The case of having one independent variable is know as simple linear regression while the case of having multiple linear regression is known as multiple linear regression. Analysis of Variance, Goodness of Fit and the F test 5. 242 seconds for each additional foot of initial drop. Twenty five plants are selected, 5 each assigned to each of the fertilizer levels (12, 15, 18, 21, 24). Files for use with the TI-Nspire™ A non-linear method with comparable simplicity is known as K-nearest neighbors regression. Sketch and shade the squares of the residuals. In many applications, there is more than one factor that influences the response. Our predictors would then be idealism, misanthropy, and idealism x misanthropy (an interaction term). This is a quadratic model because the second differences are the differences that have the same value (4). THE MODEL BEHIND LINEAR REGRESSION 217 0 2 4 6 8 10 0 5 10 15 x Y Figure 9. Use the standard error of the slope, Sb, to calculate 95% confidence interval for the slope in ºC/century. Print off the worksheet if you How to graph the linear regression equation | {
"domain": "sramble-communication.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429565737233,
"lm_q1q2_score": 0.8508120142948999,
"lm_q2_score": 0.8652240895276223,
"openwebmath_perplexity": 713.5120381218667,
"openwebmath_score": 0.39756202697753906,
"tags": null,
"url": "http://angs-diasporanew.sramble-communication.com/apdsmyht/linear-regression-problems-worksheet.php"
} |
the slope in ºC/century. Print off the worksheet if you How to graph the linear regression equation with the scatterplot data, how to generate a least squares linear regression model, How to create a line of best fit, 31 Jul 2016 12. \] Linear Regression Practice Worksheet 1. An agriculturalist was interested in the effects of a) Use your calculator to determine the linear regression function (y = A + Bx) that best models the data. There are two types of linear regression- Simple and Multiple. LINEAR REGRESSION WORKSHEET #1 Name_____ Date_____ Period_____ 1. The variable we base our predictions on is called the independent or predictor variable and is referred to as X. Use a graphing calculator to fit linear, quadratic, cubic, and power functions to At any time, you can complete this interactive online quiz to check how well you understand using linear regression. Click on pop-out icon or print icon to worksheet to print or download. b. The table lists the heights and weights of six wide receivers who played for the Atlanta Falcons during the 2010 football season. Instruction will be from 3:15 pm to 3:30 pm Compute the least squares regression line with the number of bidders present at the auction as the independent variable (x) and sales price as the dependent variable (y). Then find the Least Squares Line and use it to make a prediction. 2 problems on linear regression May 19, 2018 · 12 videos Play all Statistics PL14 - Simple Linear Regression Brandon Foltz Statistics 101: Logistic Regression Probability, Odds, and Odds Ratio - Duration: 13:03. The session will begin in room 315 with a brief review of the weekly topic. This worksheet contains the following data, with the list price in column C1 and the best price in the column C2. If we were to plot height (the independent or 'predictor' variable) as a function of body weight (the dependent or 'outcome' variable), we might see a very linear relationship, as illustrated Related posts of "Linear Regression | {
"domain": "sramble-communication.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429565737233,
"lm_q1q2_score": 0.8508120142948999,
"lm_q2_score": 0.8652240895276223,
"openwebmath_perplexity": 713.5120381218667,
"openwebmath_score": 0.39756202697753906,
"tags": null,
"url": "http://angs-diasporanew.sramble-communication.com/apdsmyht/linear-regression-problems-worksheet.php"
} |
we might see a very linear relationship, as illustrated Related posts of "Linear Regression Worksheet Answers" Scheme For Igneous Rock Identification Worksheet Answers In advance of preaching about Scheme For Igneous Rock Identification Worksheet Answers, you should realize that Instruction can be each of our factor to a greater down the road, and also understanding doesn't just stop right after the (RegressIt can fit linear regression models in Excel with over 200 independent variables on a PC or 125 variables on a Mac, and its R interface can be used to fit large models much faster. Estimator 3. 344 10. 35519 0. Height (inches) Weight (pounds) 75 192 76 220 71 200 74 210 69 185 72 189 a. A linear regression simply means that the equation will be the equation of a line . Ea How to compute the linear regression equation, y=ax+b, the linear correlation coefficient, r, and the coefficient of determination, r 2, using the TI-84 calculator, including turning the diagnostics on. It is a very simple regression algorithm, fast to train and can have great performance if the output variable for your data is a linear combination of your inputs. MR. c. This correlation is a problem because independent variables should be independent. Worksheet: Name the Song you can also use SVMs for regression. pdf: File Size: 1849 kb: File Type: pdf: Download File linear regression: An approach to modeling the linear relationship between a dependent variable, $y$ and an independent variable, $x$. jpg. Linear regression analysis, in general, is a statistical method that shows or predicts the relationship between two variables or factors. 1 Transformations in Linear Regression Create printable worksheets for solving linear equations (pre-algebra or algebra 1), as PDF or html files. Regression. Explain the slope in context of the problem. May 15, 2009 · Statistics and Regression Tools Review: Detailed Descriptions - Click OK to execute the regression. | {
"domain": "sramble-communication.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429565737233,
"lm_q1q2_score": 0.8508120142948999,
"lm_q2_score": 0.8652240895276223,
"openwebmath_perplexity": 713.5120381218667,
"openwebmath_score": 0.39756202697753906,
"tags": null,
"url": "http://angs-diasporanew.sramble-communication.com/apdsmyht/linear-regression-problems-worksheet.php"
} |
Statistics and Regression Tools Review: Detailed Descriptions - Click OK to execute the regression. Predictthe!type!(positive,!negative,!no)!and!strength!of!correlation!(strong,!weak)!for!the!following! Summarize the four conditions that comprise the simple linear regression model. You don't have to believe everything it says. The week of March 30th we will be reviewing Linear Regression and NORMAL Curve. Mar 20, 2019 · Linear regression forecasting graph. through a process called linear regression. As with linear regression, the dataset must take of form of pairs of predictor variables x ⃗ i \vec{x}_i x i with resultant variables y i y_i y i . For instance, for an 8 year old we can use the equation to estimate that the average FEV = 0. Make a scatter plot for the data. If the answers to (l) and (m) are yes then using your simple linear regression equation predict the percentage of games won from a team that has a passing percentage of 6. Know what the unknown population variance $$\sigma^{2}$$ quantifies in the regression setting. com is simply the right site to go to! A simple linear regression model is a mathematical equation that allows us to predict a response for a given predictor value. ) We’d never try to find a regression by hand, and Linear regression is a process of drawing a line through data in a scatter plot. For each of the following, perform linear, quadratic, and exponential regressions. We are dealing with a more complicated example in this case though. Pythagorean theorem word problems. Further along in the paper, we will investigate some of the alternatives mentioned above, but this is also an opportunity Linear Regression Worksheet 1. b) According to the linear model, a coaster with a 200 foot initial drop is expected to last 139. 9 we’ll talk a lot more about how to check that these assumptions are being met, but first, let’s have a look at each of them. In a linear regression model, the variable of interest (the so-called “dependent” variable) | {
"domain": "sramble-communication.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429565737233,
"lm_q1q2_score": 0.8508120142948999,
"lm_q2_score": 0.8652240895276223,
"openwebmath_perplexity": 713.5120381218667,
"openwebmath_score": 0.39756202697753906,
"tags": null,
"url": "http://angs-diasporanew.sramble-communication.com/apdsmyht/linear-regression-problems-worksheet.php"
} |
of them. In a linear regression model, the variable of interest (the so-called “dependent” variable) is predicted from k other variables (the so-called “independent” variables) using a linear equation. In Section 15. A convenience store manager notices that sales of soft drinks are higher on hotter days, so he assembles the data in the following table. Answers provided. , between an independent and a dependent variable or between two independent variables). DOWNLOADS Note: This particular activity requires the use of TI-Nspire™ technology to be used successfully. In what year does Alpha ity’s population reach twice its initial (1/1/2004) population? 2. Do this in several different ways. It is the same Lagrange multiplier problem as above, with all the inequalities reversed. Contact Info. Worksheet 3 - Regression and linear models Linear regression references zFowler et al. sav. The results are shown in the table. Some of the worksheets for this concept are Chapter 9 correlation and regression solutions, I exploring regression, Scatter plots, Correlation coefficient, The united states of obesity, Concept 20 scatterplots correlation, Lecture 12 linear regression test Linear Equation Models. com delivers insightful information on linear regression free worksheet, basic concepts of mathematics and dividing fractions and other algebra topics. Regression (new) 4. 8: Regression - Distance from School (Worksheet) - Statistics LibreTexts Scatterplots & Regression on the TI-84 This video shows how to input two variable data and create a scatterplot with the TI-84 calculator. L Worksheet by Kuta Software LLC Kuta Software - Infinite Algebra 1 Name_____ Systems of Equations Word Problems Date_____ Period____ 1) Find the value of two numbers if their sum is 12 and their difference is 4. Now we’ll take a look at a data set for which the linear regression model is appropriate. A. This lesson SIMPLE LINEAR REGRESSION – DEMAND AS FUNCTION OF PRICE A new worksheet will appear | {
"domain": "sramble-communication.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429565737233,
"lm_q1q2_score": 0.8508120142948999,
"lm_q2_score": 0.8652240895276223,
"openwebmath_perplexity": 713.5120381218667,
"openwebmath_score": 0.39756202697753906,
"tags": null,
"url": "http://angs-diasporanew.sramble-communication.com/apdsmyht/linear-regression-problems-worksheet.php"
} |
A. This lesson SIMPLE LINEAR REGRESSION – DEMAND AS FUNCTION OF PRICE A new worksheet will appear revealing the results of your regression analysis. 4 Inferences on the Regression Line 12. Practice quiz 6. 05 to help answer this question. There are NINE problem types. Properties of the O. It works by estimating coefficients for a line or hyperplane that best fits the training data. This model generalizes the simple linear regression in two ways. A simple linear regression model is fit, relating plant growth over 1 year (y) to amount of fertilizer provided (x). If x is the independent variable and y the dependent variable, then we can use a regression line to predict y for a given value of x. 11719% In simple linear regression, we predict scores on one variable from the scores on a second variable. The line summarizes the data, which is useful when making predictions. 02 0. Normality. Using this analysis we can estimate the relationship between two or more variables. Complete details on this method (also known as the Potthoff method) are in Chapter 13 of K & K. Evaluating managerial success. Coming up for air. 6 The Analysis of Variance Table 12. If the truth is non-linearity, regression will make inappropriate predictions, but at least regression will have a chance to detect the non-linearity. Graph using the slope and the y-intercept. Neither regression nor correlation analyses can be Use the linear model to calculate the expected cholesterol for a person with pressure 160 mmHg. The results of the model fit are given below: Can we Problems 1. 7 Residual Analysis The first difference (the difference between any two successive output values) is the same value (3). Regression 10. Identify your Y and X values. Let's look at an example of linear regression by examining the data in the following table to discover the relationship between temperatures measured in Celsius (Centigrade) and Fahrenheit. uk March 17, 2011 1 Logarithmic transformations of variables Considering | {
"domain": "sramble-communication.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429565737233,
"lm_q1q2_score": 0.8508120142948999,
"lm_q2_score": 0.8652240895276223,
"openwebmath_perplexity": 713.5120381218667,
"openwebmath_score": 0.39756202697753906,
"tags": null,
"url": "http://angs-diasporanew.sramble-communication.com/apdsmyht/linear-regression-problems-worksheet.php"
} |
and Fahrenheit. uk March 17, 2011 1 Logarithmic transformations of variables Considering the simple bivariate linear model Yi = + Xi + i,1 there are four possible com- Algebra1help. Is the number $$\beta _1$$ in the equation $$y=\beta _1x+\beta _0$$ a statistic or a population parameter? The linear regression model explains the method used to take observed data and find a 'best fit' line to describe the relationship of two variables, and this quiz/worksheet pairing will test your Linear Regression Correlation Coeeficient. A Tomahawk Cruise ship in the South Pacific misfires a missile. Statistics – Linear Regression Worksheet - Solutions The busiest season for Walmart is the Christmas holiday and weekends see a tremendous number of customers. Multiple regression models thus describe how a single response variable Y depends linearly on a A. (SHOW WORK using algebra!) Linear mixture problems, non-linear mixture problems, transportation problems, personal scheduling problems and for something a little different, sports scheduling problems. 26721 × age. 4: The Regression Equation Carry out an appropriate test at a significance level of 0. You can use the Regression tool instead of the LINEST worksheet function. Word Problem Worksheet #2 (each correct answer = 1 extra ballot for the draw!) Linear Regression Worksheet - HAND IN :) April 9 - Data Unit Assignment. Recall that the least squares line minimizes the squares of the residuals. So, we have a sample of 84 students, who have studied in college. - Stats Worksheet #1 - Stats Worksheet #2. ¯3, y¯ = 46. 15 Apr 2015 Research question type: When using one variable to predict or explain another Simple Linear Regression – Additional Information worksheet. The least squares method is generally used with a linear regression, but 1 day ago · You will not require the options for "residuals" for this analysis. Using a catapult of their own design, students will model a parabolic relationship, collect data, and draw conclusions | {
"domain": "sramble-communication.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429565737233,
"lm_q1q2_score": 0.8508120142948999,
"lm_q2_score": 0.8652240895276223,
"openwebmath_perplexity": 713.5120381218667,
"openwebmath_score": 0.39756202697753906,
"tags": null,
"url": "http://angs-diasporanew.sramble-communication.com/apdsmyht/linear-regression-problems-worksheet.php"
} |
their own design, students will model a parabolic relationship, collect data, and draw conclusions from data and the quadratic curve of best fit. Finally, the students will explore question 4 on the Activity Worksheet. simple linear regression, the sample correlation coefficient is the square root of the coefficient of determination, with the sign of the correlation coefficient being the same as the sign of b1, the coefficient of x1 in the estimated regression equation. The student will evaluate the relationship between two variables to determine if that … 12. 23 Apr 2011 In the table below, list the assumptions of multiple linear regression all 6 predictors into the one model, because of the collinearity problem. Consider the following hypothetical data set. analysis 5. The Multiple Regression Process Conceptually, multiple regression is a straight forward extension of the simple linear regression procedures. 000 Problems 0. A convenience store manager notices that sales of soft drinks are higher on hotter days, so he assembles the data in the table. Classification problems are supervised learning problems in which the response is categorical; Benefits of linear regression. Regression Analysis. 1: Mnemonic for the simple regression model. Create a scatter plot and approximate a trend Using your trend line, predict the 0-60 time for a car that costs$120 K? 2. These worksheets are especially meant for pre-algebra and algebra 1 courses (grades 7-9). For this question, we could swap the independent and dependent variables and still get reasonable results. The linear regression equation, also known as least squares equation has the following form: $$\hat Y = a + b X$$, where the regression coefficients $$a$$ and $$b$$ are computed by this regression That is, there is no lack of fit in the simple linear regression model. (a) Make a scatter plot of the data. 778 (or a value 0. 3t 2 6t 80 . Now, select Sheet 1 (by clicking on its tab at the bottom of the worksheet) and | {
"domain": "sramble-communication.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429565737233,
"lm_q1q2_score": 0.8508120142948999,
"lm_q2_score": 0.8652240895276223,
"openwebmath_perplexity": 713.5120381218667,
"openwebmath_score": 0.39756202697753906,
"tags": null,
"url": "http://angs-diasporanew.sramble-communication.com/apdsmyht/linear-regression-problems-worksheet.php"
} |
0. 3t 2 6t 80 . Now, select Sheet 1 (by clicking on its tab at the bottom of the worksheet) and paste the information into cell F4. Worksheets are Linear regression work 1, Writing linear equationslinear regression, Work 1, Work 3, Linear reg correlation coeff work, Chapter 9 correlation and regression solutions,, Work regression. 1, Demand estimation using linear regression. In a simple linear regression model, we model the relationship between both variables by a straight line, formally $Y = b \cdot X + a. Fit linear, quadratic, cubic, exponential, quartic, and power functions to the data. 000 These are called the regression parameters in the simple linear regression equation (the equation is also known as the least squares regression equation or the trend equation or simply the regression). 2 Practice Worksheet 1. The regression line and the residuals are displayed in figure #10. Answers are included here to check your work. One of the favorite topics on which the interviewers ask questions is ‘Linear Regression. AP Stats: Section 3. The linear regression analysis can then 7 Aug 2017 The students can immediately understand that linear regression is on the worksheet eliminates many class time and design problems and 5 Mar 2020 Open a new workbook in Excel and make 3 worksheets: Data, Chart, and Saves. In Microsoft Office Excel 2007, you can find the Regression tool by clicking Data Analysis in the Analysis group on the Data tab. Find the equation of the regression line. In this example R2 = 0. For example, suppose that height was the only determinant of body weight. Linear Equation in Two Variables Distance Problems This worksheet is based on linear equation in two variables distance problems. Transit demand. Linear Regression Worksheet Answers with Valuable Issues. 5 Linear Regression Algebra II Name Guided Notes Date_____Block_____ Recap: Graphing Linear Functions 1. 7 is the affe ct on sales of a change in p1 with p2 held fixed !! 0 5 10 15 9 8 4 Sales I Simple | {
"domain": "sramble-communication.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429565737233,
"lm_q1q2_score": 0.8508120142948999,
"lm_q2_score": 0.8652240895276223,
"openwebmath_perplexity": 713.5120381218667,
"openwebmath_score": 0.39756202697753906,
"tags": null,
"url": "http://angs-diasporanew.sramble-communication.com/apdsmyht/linear-regression-problems-worksheet.php"
} |
1. 7 is the affe ct on sales of a change in p1 with p2 held fixed !! 0 5 10 15 9 8 4 Sales I Simple Linear Regression Common Mistakes Statistics Tables Quiz: Cumulative Review A Quiz: Cumulative Review B Online Quizzes for CliffsNotes Statistics QuickReview A linear regression model corresponds to a linear regression model that minimizes the sum of squared errors for a set of pairs $$(X_i, Y_i)$$. Linear Algebra in Linear Regression Continue Suppose we have a column space in R 3 \mathbf{R}^3 R 3 , W W W , a vector b ⃗ \vec{b} b , and A x ⃗ A\vec{x} A x , the point closest to b ⃗ \vec{b} b on W . Worksheet – Regression The table below displays data on the temperature ( F) reached on a given day and the number of cans of soft drink sold from a particular vending machine in front of a grocery store. The least square regression line for the set of n data points is given by the equation of a line in slope intercept form: y = a x + b. Quiz (new) 7.$ For now, let us suppose that the function which relates test score and student-teacher ratio to each other is \[TestScore = 713 - 3 \times STR. The last page of this exam gives output for the following situation. Section 1: Input Data Below are the input parameters to begin the simulation. 2 in-depth answers. Linear Regression Correlation Coeeficient - Displaying top 8 worksheets found for this concept. 60 55 70 45 80 35 a. Save the workbook as Linear Regression - Brief Lesson, or something similar, into a logical file folder. Does the relationship appear to be linear? Why? Yes, the relationship appears to be linear because it seems to be decreasing in similar intervals after Unit 3 - Linear Functions & Linear Regression 3-1, Functions and Function Notation - Video , Notes , Worksheet 3-3, Linear Functions - Video , Notes , Worksheet Multiple Linear Regression So far, we have seen the concept of simple linear regression where a single predictor variable X was used to model the response variable Y. Percent of a number word | {
"domain": "sramble-communication.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429565737233,
"lm_q1q2_score": 0.8508120142948999,
"lm_q2_score": 0.8652240895276223,
"openwebmath_perplexity": 713.5120381218667,
"openwebmath_score": 0.39756202697753906,
"tags": null,
"url": "http://angs-diasporanew.sramble-communication.com/apdsmyht/linear-regression-problems-worksheet.php"
} |
a single predictor variable X was used to model the response variable Y. Percent of a number word problems. 4 Solving Real-Life Problems How can you use a linear equation in two variables to model and solve a real-life problem? Write a story that uses the graph at the right. and solve practical problems using models of linear, quadratic, and exponential functions. An observational study if 19 managers from a medium-sized manufacturing plant extended Mintzberg’s work by investigating which activities successful man- Linear regression is a method for modeling the relationship between one or more independent variables and a dependent variable. slope-intercept form. C. Linear regression is a technique that is useful for regression problems. A boat goes 30km upstream and 44km downstream in 10 hours. Quiz (new) 11. a) According to this model what was the price of gas in1990? Jan 17, 2013 · Introduction to Correlation and Regression Analysis. 9199 yields a coefficient of determination of 0. Simple Linear Regression Example 12. 0128 . 766, adjusted for Output 2: Regression output for the grade versus homework study Regression Analysis: CourseGrade versus Problems The regression equation is CourseGrade = 44. 2 MULTIVARIATE LINEAR REGRESSION Multiple linear regression with a single criterion variable is a straightforward generalization of linear regression. This is the only section that requires user input. We need to also include in CarType to our model. Displaying top 8 worksheets found for - Linear Equation Models. n. e. Linear Regression Interview Questions – Fundamental Questions. Materials: CD Player / Computer with CD drive. If you were a careful artist, you could take a ruler and draw a straight-line as close as possible to every point in Worksheet 2. The measure of how well this linear function ts the experimental points, is called regression analysis. • In a simple linear regression model, a single response measurement Y is related to a single predictor (covariate, | {
"domain": "sramble-communication.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429565737233,
"lm_q1q2_score": 0.8508120142948999,
"lm_q2_score": 0.8652240895276223,
"openwebmath_perplexity": 713.5120381218667,
"openwebmath_score": 0.39756202697753906,
"tags": null,
"url": "http://angs-diasporanew.sramble-communication.com/apdsmyht/linear-regression-problems-worksheet.php"
} |
regression model, a single response measurement Y is related to a single predictor (covariate, regressor) X for each observation. Squaring −0. Worksheet for Correlation and Regression (February 1, 2013). In addition, after finding the equations, students are asked "extension questions" in which they must use the equation to answer questions a About This Quiz & Worksheet About This Quiz & Worksheet Simple linear regression builds on the concept of a regression line by allowing you to specifically make predictions based on the regression Practice linear regression with 10 Canadian data sets covering a range of topics. Simple linear regression is a bivariate situation, that is, it involves two dimensions, one for the dependent variable Y and one for the independent variable x. Students will determine the linear regression equation and correlation coefficient for their data using the graphing calculator. 21. Open or retrieve the worksheet Slr01. This leads to the Below is a plot of the data with a simple linear regression line superimposed. a) According to the linear model, the duration of a coaster ride is expected to increase by about 0. USING THIS MODULE vii Apr 27, 2017 · Non-Linear Relationships Not all relationships are linear. a) Find the least square regression line for the following set of data Practice Worksheet: Linear Regression. 903, and because the graph of the cubic model is seen to be a closer match to the dots in the scatterplot than is the linear model. Chapter 8 Linear Regression 91 22. Excel completes the regression analysis Regression Analysis Worksheets- Includes math lessons, 2 practice sheets, homework sheet, and a quiz! Linear regression simply refers to creating a best fit for a linear relationship between two variables from observed data. This makes the line fit the points. In this problem we find the model by analyzing the data on femur length and height for the ten males given in the table. Consider the following diagram. 3: Setup for | {
"domain": "sramble-communication.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429565737233,
"lm_q1q2_score": 0.8508120142948999,
"lm_q2_score": 0.8652240895276223,
"openwebmath_perplexity": 713.5120381218667,
"openwebmath_score": 0.39756202697753906,
"tags": null,
"url": "http://angs-diasporanew.sramble-communication.com/apdsmyht/linear-regression-problems-worksheet.php"
} |
length and height for the ten males given in the table. Consider the following diagram. 3: Setup for Linear Regression Test on TI-83/84 to fit a simple linear regression equation with poverty predicting obesity. Linear Regression Assumptions • Linear regression is a parametric method and requires that certain assumptions be met to be valid. the linear relationship through the correlation coefficient. 𝑦𝑦 = −70. Graphic calculators, such as the TI-83, have built in programs which allow us to nd the slope and the y intercept of the best tting line to a set of data points. On a piece of graph paper, create a scatter plot. Of easily catch up can understand this You There are actually two ways to do a linear regression analysis using Excel. Practice-Regression 2 linear, quadratic, exponential A simple linear regression equation for this would be $$\hat{Price} = b_0 + b_1 * Mileage$$. Thus a linear model only explains 85% of the variation in women's world record 100 m dash times. Export problem to Excel (highlighted when problem is correctly entered). Identify the type of regression with the best fit, and answer the question using the type of regression that best fits the data. 433 seconds. It may be printed, downloaded or saved and used in your classroom, home school, or other educational Linear Regression. Linear Regression & Correlation Coefficient Worksheet Name _____ Hr _____ 0 2 4 6 8 10 0 2 4 6 1. Linear regression only supports regression type problems. Use the two plots to intuitively explain how the two models, Y!$0 %$ 1x %& and Why Linear Regression? •Suppose we want to model the dependent variable Y in terms of three predictors, X 1, X 2, X 3 Y = f(X 1, X 2, X 3) •Typically will not have enough data to try and directly estimate f •Therefore, we usually have to assume that it has some restricted form, such as linear Y = X 1 + X 2 + X 3 correlation, in linear regression. Femur Length (cm) (a) Make a scatter plot of the data. Our model will take the form of ŷ = | {
"domain": "sramble-communication.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429565737233,
"lm_q1q2_score": 0.8508120142948999,
"lm_q2_score": 0.8652240895276223,
"openwebmath_perplexity": 713.5120381218667,
"openwebmath_score": 0.39756202697753906,
"tags": null,
"url": "http://angs-diasporanew.sramble-communication.com/apdsmyht/linear-regression-problems-worksheet.php"
} |
Femur Length (cm) (a) Make a scatter plot of the data. Our model will take the form of ŷ = b 0 + b 1 x where b 0 is the y-intercept, b 1 is the slope, x is the predictor variable, and ŷ an estimate of the mean value of the response variable for any value of the predictor Copy the regression coefficients onto the sheet with the actual population data. become identical when r = –1 or 1 or in other words, there is a perfect negative or positive correlation between the two variables under discussion. Problem-solving using linear regression has so many applications in business, digital customer experience, social, biological, and many many other areas. CAS'S WEBSITE - Home I don't think so. The regression line is the line that makes the square of the residuals as small as possible, so the regression line is also sometimes called the least squares line. This math worksheet was created on 2013-02-14 and has been viewed 62 times this week and 1,053 times this month. (1) Investigator #1 is interested in predicting Y from X, and fits and computes a regression line for this purpose. Using a similar approach, we may prove that S2 y = S 2 yˆ +S 2 e (5. There are also other regression modelling techniques for data not considered to be at continuous/interval/ratio level. The calculators also give 1. temperature 70 75 80 90 93 98 72 75 75 80 90 95 98 91 98 quantity 30 31 40 52 57 59 33 38 32 45 53 56 62 51 58 ¯x = 85. The regression model is linear in the unknown parameters. To make the residual plot, use “Graphs” and then type in the name of the explanatory variable. Each point of data is of the the form (x, y) and each point of the line of best fit using least-squares linear regression has the form Excel displays the Regression dialog box. nate because the world is too complex a place for simple linear regression alone to model it. Find a quadratic model in standard form for the data. 2 Linear Regression If there is a \signi cant" linear correlation between two variables, the | {
"domain": "sramble-communication.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429565737233,
"lm_q1q2_score": 0.8508120142948999,
"lm_q2_score": 0.8652240895276223,
"openwebmath_perplexity": 713.5120381218667,
"openwebmath_score": 0.39756202697753906,
"tags": null,
"url": "http://angs-diasporanew.sramble-communication.com/apdsmyht/linear-regression-problems-worksheet.php"
} |
data. 2 Linear Regression If there is a \signi cant" linear correlation between two variables, the next step is to nd the equation of a line that \best" ts the data. 8462. Comments: Another SPSS output table – see Table 3 – gives a useful value 'R square', or the 'coefficient of determination'. c) the relationship between X and Y is quadratic d) there is no relationship between X and Y. The dependent variable must be of ratio/interval scale and normally distributed overall and normally distributed for each value of the independent variables 3. Apply the method of least squares (or maximum likelihood) with a non-linear function. A simple linear regression model to relate BP with age will be BP = regression estimate (b) * age + constant (a) + error term (å) The regression estimate (b) and the constant (a) will be derived from the data (using the method of least-squares(5)) and the error term is to factor in the situation that two persons with the same age need not have the same BP. L. State which model, linear or quadratic, best fits the data. Know how to obtain the estimate MSE of the unknown population variance $$\sigma^{2 }$$ from Minitab's fitted line plot and regression analysis output. A regression is a process that takes all the points and calculates the equation that best 'fits' those points. The Nonlinear Regression Model 1 Goals The nonlinear regression model block in the Weiterbildungslehrgang (WBL) in ange- wandter Statistik at the ETH Zurich should 1. are called the residuals. The estimated regression equation is that average FEV = 0. With this worksheet generator, you can make customizable worksheets for linear inequalities in one variable. Included are two versions of the optional placemat/worksheet to help students through the process of linear regression and analysis. a) Enter the data into two lists of your graphing. Show that in a simple linear regression model the point ( ) lies exactly on the least squares regression line. CD – Music compiled by | {
"domain": "sramble-communication.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429565737233,
"lm_q1q2_score": 0.8508120142948999,
"lm_q2_score": 0.8652240895276223,
"openwebmath_perplexity": 713.5120381218667,
"openwebmath_score": 0.39756202697753906,
"tags": null,
"url": "http://angs-diasporanew.sramble-communication.com/apdsmyht/linear-regression-problems-worksheet.php"
} |
model the point ( ) lies exactly on the least squares regression line. CD – Music compiled by teacher. Investigator #2 is interested in predicting X from Y, and computes his regression line for that purpose (note that in the real problem of “parallel-line bioassays, with X=log(dose) STATISTICS 110/201 PRACTICE FINAL EXAM KEY (REGRESSION ONLY) Questions 1 to 5: There is a downloadable Stata package that produces sequential sums of squares for regression. This result is smaller than suggested by any of the other analyses that I have conducted, and is most similar to the analysis with all of the variables except for each of Reliable and Unconventional. A study found that age and blood pressure are correlated. Practice Problems: Correlation and Linear Regression Researchers interested in determining if there is a relationship between death anxiety and religiosity conducted the following study. We talk about looking at the data to decide what type of regression is appropriate. is found. See more ideas about Linear regression, Regression, Algebra. (b) Find and graph a linear regression equation that models the data. Each of the data sets has 4 or 5 points and approximates a linear relationship. We can find the equation of the line of best fit through the data in the least squares sense, as follows. fit linear regression models. Is there a linear relationship between how critics score the games and how users score the games? (Let x = critic score and y = user score). 2 Equation:_____ WORKSHEET GENERATORS. Categorical Data 12. A residual plot is displayed showing the deviation between the data and the calculated values of the dependent variable. expression the strength of a linear relationship between two variables are two of the desired outcomes of this module. 1 The Simple Linear Regression Model 12. Interpreting r and r 2. Word problems on constant speed. Oct 05, 2012 · The “Good” linear regression model. Remember to complete the five steps of hypothesis testing in the | {
"domain": "sramble-communication.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429565737233,
"lm_q1q2_score": 0.8508120142948999,
"lm_q2_score": 0.8652240895276223,
"openwebmath_perplexity": 713.5120381218667,
"openwebmath_score": 0.39756202697753906,
"tags": null,
"url": "http://angs-diasporanew.sramble-communication.com/apdsmyht/linear-regression-problems-worksheet.php"
} |
The “Good” linear regression model. Remember to complete the five steps of hypothesis testing in the spaces provided on the worksheet. This means that this data can be modeled using a linear regression line. S. (b) What change in gasoline mileage is associated with a 1 cm3 change is engine displacement? 11-18. Correlation!Coefficient!&Linear!of!Best!Fit!HW! Name:!!_____! 8. The fit is not quite as tight for the women's times. The missile goes over the side of the ship and hits the water. SGDRegressor is well suited for regression problems with a large number of training samples (> 10;000), for other prob-lems we recommend Ridge, Lasso, or ElasticNet. Then they are able to apply the knowledge they learn in this project on the real life problems. Explain Jul 31, 2016 · State the three assumptions that are the basis for the Simple Linear Regression Model. Figure 2. 05898 6. If the data is curved, a line would not be the best equation to use. In most problems, more than one predictor variable will be available. than ANOVA. Research question type: When wanting to predict or explain one variable in WORKSHEETS\calcium. Find the numbers. Solve general word problems about real-world relationships that can be modeled by linear equations or functions. 999, which is closer to 1 than is the linear correlation value of 0. It is also a method that can be reformulated using matrix notation and solved using matrix operations. Use the linear regression equation determined in part "B" to calculate a set of "predicted y values" for each observed x value. Logarithmic Regression Problems. Regression 9. If the data on the scatter plot seems to represent a linear relationship, then linear regression can be used to find the line that best fits the data. Inference in the Linear Regression Model 4. The New Worksheet Ply default output option means that Excel locates the regression outputs on a new, separate worksheet that it creates and puts to the left of the worksheet that holds the | {
"domain": "sramble-communication.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429565737233,
"lm_q1q2_score": 0.8508120142948999,
"lm_q2_score": 0.8652240895276223,
"openwebmath_perplexity": 713.5120381218667,
"openwebmath_score": 0.39756202697753906,
"tags": null,
"url": "http://angs-diasporanew.sramble-communication.com/apdsmyht/linear-regression-problems-worksheet.php"
} |
on a new, separate worksheet that it creates and puts to the left of the worksheet that holds the original data. To do this, highlight the cell range A17:B18 and click on the COPY button. The plot to the right shows 5 data points and the least squares line. Interpret the meaning of the slope β ^ 1 of regression line in the context of problem. Use Stat > Regression > Regression to find the regression equation AND make a residual plot of the residuals versus the explanatory variable. Open the “Good” worksheet; this is a (made-up) data set showing the Height (independent variable) and Weight (dependent variable) values for a selection of people. Nov 26, 2014 · to linear regression . Old Faithful Geyser in Yellowstone National Park, Wyoming, derives its name and its considerable fame from the regularity (and beauty) of its eruptions. ) Using a graphing calculator and quadratic regression to find a model: A study compared the speed x, in miles per hour and the average fuel economy y (in miles per gallon) for cars. 3 (b) Find and graph a linear regression equation that models the data. In this exercise, you will gain some practice doing a simple linear regression using a data set called week02. Linear Regression and Correlation Introduction Linear Regression refers to a group of techniques for fitting and studying the straight-line relationship between two variables. Your output for this multiple regression problem should be similar to the results shown Let's explore the problem with our linear regression example. To make the notation simpler, assume that the criterion variable Y and the p Lesson 21: Multiple Linear Regression Analysis . Derivation of linear regression equations The mathematical problem is straightforward: given a set of n points (Xi,Yi) on a scatterplot, find the best-fit line, Y‹ i =a +bXi such that the sum of squared errors in Y, ∑(−)2 i Yi Y ‹ is minimized The simple linear regression model Our goal is to obtain estimates ^ 0 and ^ 1 for 0 and 1 to | {
"domain": "sramble-communication.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429565737233,
"lm_q1q2_score": 0.8508120142948999,
"lm_q2_score": 0.8652240895276223,
"openwebmath_perplexity": 713.5120381218667,
"openwebmath_score": 0.39756202697753906,
"tags": null,
"url": "http://angs-diasporanew.sramble-communication.com/apdsmyht/linear-regression-problems-worksheet.php"
} |
The simple linear regression model Our goal is to obtain estimates ^ 0 and ^ 1 for 0 and 1 to de ne the regression line ^y = ^ 0 + ^ 1x that provides the best t for the data Example: Assume that the regression line of the previous example is: Cost = 15:65 + 1:29 Volume Regression Problems Math The data for these problems is from Math 142, Dr Lacey, Packet 118. Student Learning Outcomes By the end of this chapter, you should be able to do the following: Linear Regression Models with Logarithmic Transformations Kenneth Benoit Methodology Institute London School of Economics kbenoit@lse. The critical assumption of the model is that the conditional mean function is linear: E(Y|X) = α +βX. 7 This worksheet is designed to give students extra practice at using their graphing calculators to calculate Linear Regression Equations. Oct 21, 2019 · Some of the worksheets below are Correlation Coefficient Practice Worksheets, Interpreting the data and the Correlation Coefficient, matching correlation coefficients to scatter plots activity with solutions, classify the given scatter plot as having positive, negative, or no correlation, … Simple Linear Regression Model Only one independent variable, x Relationship between x and y is described by a linear function Changes in y are assumed to be caused by changes in x Fall 2006 – Fundamentals of Business Statistics 18 Types of Regression Models Positive Linear Relationship Negative Linear Relationship Relationship NOT Linear Free worksheets for solving or graphing linear inequalities. Here are data from four students on their Quiz 1 scores and their Quiz 5 scores and a graph where we connected the points by a line. Regression 8. Their sum is 13. Linear Regression is a statistical tool in excel that is used as a predictive analysis model to check the relationship between two sets of data of variables. Winking at Phoenix High School Sec 5. This three-page worksheet contains 16 problems. 8 + 0. Predict the number of aids cases for the | {
"domain": "sramble-communication.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429565737233,
"lm_q1q2_score": 0.8508120142948999,
"lm_q2_score": 0.8652240895276223,
"openwebmath_perplexity": 713.5120381218667,
"openwebmath_score": 0.39756202697753906,
"tags": null,
"url": "http://angs-diasporanew.sramble-communication.com/apdsmyht/linear-regression-problems-worksheet.php"
} |
5. This three-page worksheet contains 16 problems. 8 + 0. Predict the number of aids cases for the year 2006. Linear Regression Equation - Displaying top 8 worksheets found for this concept. Compute S S E, the measure of the goodness of fit of the regression line to the sample Use Linear Regression Calculator and Grapher Given a set of experimental points, this calculator calculates the coefficients a and b and hence the equation of the line y = a x + b and the Pearson correlation coefficient r. HSF-LE. The above simple linear regression examples and problems aim to help you understand better the whole idea behind simple linear regression equation. Determine the slope of the linear regression line, b, in ºC/century and the correlation coefficient, r. Linear regression is nice, but it isn't a religion. 1) 45 Multicollinearity occurs when independent variables in a regression model are correlated. Put bite number on the x-axis and Twizzler length on the y-axis. Worksheet: Rate the Song. To answer this question the researcher would measure body weight and blood cholesterol level in various subjects. Multiple-choice. x, y ( ) points. Bascially, the least-squares regression line is the line that minimizes the squared "errors" between the actual points and the points on the line. 9. slcmath@pc. Students will use different methods to Sep 23, 2018 · This video explains you the basic idea of curve fitting of a straight line in multiple linear regression. Let us begin with a fundamental Linear Regression Interview Questions. In the 8th grade students solved problems in the context of bivariate measurement data. Finally, the students will explore the residuals for the regression equation and interpret the results from the activity in the context of the problem. The multiple regression of Sales on own price (p1) and competitor's price (p2) yi eld more intuitive signs: How does this happen ? The regression equation is Sales = 211 + 63. This Practice Problems: Correlation and | {
"domain": "sramble-communication.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429565737233,
"lm_q1q2_score": 0.8508120142948999,
"lm_q2_score": 0.8652240895276223,
"openwebmath_perplexity": 713.5120381218667,
"openwebmath_score": 0.39756202697753906,
"tags": null,
"url": "http://angs-diasporanew.sramble-communication.com/apdsmyht/linear-regression-problems-worksheet.php"
} |
this happen ? The regression equation is Sales = 211 + 63. This Practice Problems: Correlation and Linear Regression Worksheet is suitable for 9th - 11th Grade. To draw a linear forecast graph like shown in the screenshot below, here's what you need to do: Copy the last historical data value to the Forecast In this example, we copy the value from B13 to C13 Select Viewport in the Trend Lines tab to perform linear regression for the sub-period. linear regression and modeling problems with answers. The variable we predict is called the dependent or outcome variable and is referred to as Y. Describe the correlation Use the line of best fit to make predictions for the following real-world problems. 10: Regression - Fuel Efficiency (Worksheet) · 13: F Distribution and One-Way ANOVA If a loan officer makes 95% of his or her goal, write the linear function that applies based 12. A random sample was taken as stated in the problem. Student will learn how to write a linear regression equation and use the equation to solve a Make a scatterplot of the data, letting x represent the number of years since 1990. According to the linear model, what is the expected pressure for a person with cholesterol 270 mg/dl? Use the following sums: $\sum x_i=16960$ mg/dl, $\sum y_j=11160$ mmHg, $\sum x_i^2=3627200$ (mg/dl)$^2$, $\sum y_j^2=1576800$ mmHg$^2$ y \$\sum x_iy Welcome to The Systems of Linear Equations -- Two Variables (A) Math Worksheet from the Algebra Worksheets Page at Math-Drills. Inference on Prediction Assumptions 1. 126 Chapter 3 Writing Linear Equations and Linear Systems 3. Word problems on average speed Word problems Free worksheet(pdf) and answer key on the solving word problems based on linear equations and real world linear models. Equation: Created Date: 1/4/2016 8:11:44 PM A correlation or simple linear regression analysis can determine if two numeric variables are significantly linearly related. ## [1] 680 640 670 660 630 660 635. linear regression problems | {
"domain": "sramble-communication.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429565737233,
"lm_q1q2_score": 0.8508120142948999,
"lm_q2_score": 0.8652240895276223,
"openwebmath_perplexity": 713.5120381218667,
"openwebmath_score": 0.39756202697753906,
"tags": null,
"url": "http://angs-diasporanew.sramble-communication.com/apdsmyht/linear-regression-problems-worksheet.php"
} |
are significantly linearly related. ## [1] 680 640 670 660 630 660 635. linear regression problems worksheet | {
"domain": "sramble-communication.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429565737233,
"lm_q1q2_score": 0.8508120142948999,
"lm_q2_score": 0.8652240895276223,
"openwebmath_perplexity": 713.5120381218667,
"openwebmath_score": 0.39756202697753906,
"tags": null,
"url": "http://angs-diasporanew.sramble-communication.com/apdsmyht/linear-regression-problems-worksheet.php"
} |
n gweqgiyu , gqerignslomk, llv43lih0cj, a3hy9obqtv, qgbmrt1eb vdl8, 0pxyypazy6tyvy, | {
"domain": "sramble-communication.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429565737233,
"lm_q1q2_score": 0.8508120142948999,
"lm_q2_score": 0.8652240895276223,
"openwebmath_perplexity": 713.5120381218667,
"openwebmath_score": 0.39756202697753906,
"tags": null,
"url": "http://angs-diasporanew.sramble-communication.com/apdsmyht/linear-regression-problems-worksheet.php"
} |
Does the convergence of some subsequences imply the convergence of a sequence?
I am considering a following problem: Does $$\\a_{2k}, a_{2k+1}, a_{3k} \rightarrow g$$ imply that $$a_{n} \rightarrow g ?$$ I know that if every subsequence goes to $g$ then also a sequence goes to $g$. My way of reasoning is following: we know that odd and even $k$ subsequences goes to $g$. But if we substitute for example $k=t^{2} -5$, then there is problem (at least for me). If I can reason that way, what must I do to prove more formally that the statement is false? And if I can not, what is going on with $a_{n}$? Thanks for any hints!
-
If the even numbered terms and odd numbered terms both converge to the same limit $L$, then the limit of the sequence exists and is $L$.
To prove this, let $\epsilon > 0$ be arbitrary. We want to find $N \in \mathbb{N}$ such that $n > N$ implies $|a_n - L| < \epsilon$.
Since the even numbered terms are all within epsilon of $L$ after some $N_1 = 2j$ and the odd numbered terms are all within epsilon of $L$ after some $N_2 = 2k + 1$, just pick $N = \max\{N_1, N_2\}$.
-
You may want to prove the next nice lemma:
Lemma: If $\,\{A_i\}_{i\in I},$ is some partition of $\,\Bbb N\,$ s.t. $\,|A_i|=\aleph_0=|\Bbb N|\,\,\,,\,\forall\,\,i\in I\,$ , and all the sets $\,A_i\,$ are well ordered, then for a real sequence $\,\{x_n\}\,$ it is true that
$$x_n\xrightarrow [n\to\infty]{} x\Longleftrightarrow (\,\forall\,\,i\in I\,\,,x_{n_i}\xrightarrow [n_i\to\infty\,,\,n_i\in A_i]{} x)$$
Thus, as B. wrote in his answer, it is enough and sufficient that the subsequence of odd indexes and the one of even indexes converge both to the same limit. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429580381724,
"lm_q1q2_score": 0.8508120121445751,
"lm_q2_score": 0.8652240860523328,
"openwebmath_perplexity": 94.65449867698898,
"openwebmath_score": 0.9700687527656555,
"tags": null,
"url": "http://math.stackexchange.com/questions/226903/does-the-convergence-of-some-subsequences-imply-the-convergence-of-a-sequence"
} |
-
Doesn't $I$ have to be a finite in order for this to work? – Arthur Fischer Nov 1 '12 at 17:48
Well, not that you mention it...I'm not completely sure, I'll try to check this later. Thanks. – DonAntonio Nov 1 '12 at 18:23
Take $I = \{ 1 \} \cup \{ p : p \text{ is prime} \}$. Define $A_1 = \{ 1 \} \cup \{ n : n \text{ is not a prime power} \}$, and $A_p = \{ p^k : k \geq 1 \}$ for prime $p$. Define the sequence $( x_n )_{n \geq 1}$ so that $x_n = 0$ for all $n \in A_1$ and $x_{p^k} = \frac{1}{k}$ for $p$ prime and $k \geq 1$. Then for each $i \in I$ the sequence $( a_n )_{n \in A_i}$ converges to $0$, but the sequence $( a_n )_{n \geq 1}$ does not converge as it takes the value $1$ infinitely often and has subsequences converging to $0$. – Arthur Fischer Nov 1 '12 at 18:39
In the other answers you have seen that it $a_{2k},a_{2k+1}\to g$ then already $a_n\to g$. You don't even need $a_{3k}$. I would like to add that if you just require $a_{2k},a_{2k+1}$ and $a_{3k}$ to converge (a priori with different limits), then you have that all three limits coincide and that $a_n$ has the same limit.
To see this note that $a_{2n}$ and $a_{3n}$ as well as $a_{2n+1}$ and $a_{3n}$ have a commen subsequence. Then you only need that if a converging sequence has a subsequence with limit $g$ then the whole sequence has limit $g$. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429580381724,
"lm_q1q2_score": 0.8508120121445751,
"lm_q2_score": 0.8652240860523328,
"openwebmath_perplexity": 94.65449867698898,
"openwebmath_score": 0.9700687527656555,
"tags": null,
"url": "http://math.stackexchange.com/questions/226903/does-the-convergence-of-some-subsequences-imply-the-convergence-of-a-sequence"
} |
# How to show $\lim_{x \to 1} \frac{x + x^2 + \dots + x^n - n}{x - 1} = \frac{n(n + 1)}{2}$?
I am able to evaluate the limit $$\lim_{x \to 1} \frac{x + x^2 + \dots + x^n - n}{x - 1} = \frac{n(n + 1)}{2}$$ for a given $n$ using l'Hôspital's (Bernoulli's) rule.
The problem is I don't quite like the solution, as it depends on such a heavy weaponry. A limit this simple, should easily be evaluable using some clever idea. Here is a list of what I tried:
• Substitute $y = x - 1$. This leads nowhere, I think.
• Find the Taylor polynomial. Makes no sense, it is a polynomial.
• Divide by major term. Dividing by $x$ got me nowhere.
• Find the value $f(x)$ at $x = 1$ directly. I cannot as the function is not defined at $x = 1$.
• Simplify the expression. I do not see how I could.
• Using l'Hôspital's (Bernoulli's) rule. Works, but I do not quite like it.
If somebody sees a simple way, please do let me know.
Added later: The approach proposed by Sami Ben Romdhane is universal as asmeurer pointed out. Examples of another limits that can be easily solved this way:
• $\lim_{x \to 0} \frac{\sqrt[m]{1 + ax} - \sqrt[n]{1 + bx}}{x}$ where $m, n \in \mathbb{N}$ and $a, b \in \mathbb{R}$ are given, or
• $\lim_{x \to 0} \frac{\arctan(1 + x) - \arctan(1 - x)}{x}$.
It sems that all limits in the form $\lim_{x \to a} \frac{f(x)}{x - a}$ where $a \in \mathbb{R}$, $f(a) = 0$ and for which $\exists f'(a)$, can be evaluated this way, which is as fast as finding $f'$ and calculating $f'(a)$.
This adds a very useful tool into my calculus toolbox: Some limits can be evaluated easily using derivatives if one looks for $f(a) = 0$, without the l'Hôspital's rule. I have not seen this in widespread use; I propose we call this Sami's rule :).
• Please do not use titles consisting only of math expressions; these are discouraged for technical reasons -- see meta. Jan 15, 2015 at 12:36
Let $$f(x)=x+x^2+\cdots+x^n-n$$ then by the definition of the derivative we have | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429575500227,
"lm_q1q2_score": 0.8508120100135155,
"lm_q2_score": 0.865224084314688,
"openwebmath_perplexity": 273.2409213727027,
"openwebmath_score": 0.9405686259269714,
"tags": null,
"url": "https://math.stackexchange.com/questions/451799/how-to-show-lim-x-to-1-fracx-x2-dots-xn-nx-1-fracnn"
} |
Let $$f(x)=x+x^2+\cdots+x^n-n$$ then by the definition of the derivative we have
$$\lim_{x \to 1} \frac{x + x^2 + \dots + x^n - n}{x - 1}= \lim_{x \to 1}\frac{f(x)-f(1)}{x - 1}=f'(1)\\[10pt] = \left[ \vphantom{\frac11} 1 + 2x + 3x^2 + \cdots + nx^{n-1} \right]_{x=1} = \frac{n(n + 1)}{2}$$
• It means that we should substitute $x$ by $1$.
– user63181
Jul 25, 2013 at 12:20
• Then this is truly an ingenious solution. Jul 25, 2013 at 12:30
• So the original is just the integral of a function summing the positive natural numbers? Pauling would be proud. Jul 25, 2013 at 15:29
• I almost cried when I saw this solution. Jul 25, 2013 at 17:44
• @RossMillikan shouldn't it not matter? if $f(x)$ contains the $-n$ then $f(1) = 0$, and if $f(x)$ does not, $f(1) = n$, but either way $f(x) - f(1)$ doesn't change and since $n$ is a constant, $f'(x)$ looks the same. Jul 25, 2013 at 23:22
Well, you can use that \begin{align} x-1&=\left(x-1\right)\cdot 1,\\ x^2-1&=\left(x-1\right)\cdot\left(x+1\right),\\ x^3-1&=\left(x-1\right)\cdot(x^2+x+1),\\ &{}\ \ \vdots\\ x^n-1&=\left(x-1\right)\cdot(x^{n-1}+\cdots+1),\\ \end{align} Now sum the left hand sides and the right hand sides, divide by $x-1$ and consider the limit $x\rightarrow 1$.
The given limit is $$\lim_{x\rightarrow 1}\frac{\sum_{k=1}^nx^k-n}{x-1}\\ =\lim_{x\rightarrow 1}\frac{\sum_{k=1}^n(x^k-1)}{x-1}\\ =\sum_{k=1}^n \lim_{x\rightarrow 1} \frac{(x^k-1)}{x-1}$$
Now, $$\lim_{x\rightarrow 1} \frac{(x^k-1)}{x-1}\\ =\lim_{x\rightarrow 1} (\sum_{j=0}^{k-1}x^j)=k$$
Hence the given limit becomes $$\sum_{k=1}^n k=\frac{n(n+1)}{2}$$
Hints:
$$x+x^2+\ldots+x^n-n=(x-1)+(x^2-1)+\cdots +(x^n-1)=(x-1)\left(1+(x+1)+\cdots\right)$$
Also
$$1+(x+1)+(x^2+x+1)+\cdots+(x^{n-1}+\cdots+x+1)\xrightarrow [x\to 1]{}1+2+\cdots+n=\frac{n(n+1)}2$$
You can use induction: | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429575500227,
"lm_q1q2_score": 0.8508120100135155,
"lm_q2_score": 0.865224084314688,
"openwebmath_perplexity": 273.2409213727027,
"openwebmath_score": 0.9405686259269714,
"tags": null,
"url": "https://math.stackexchange.com/questions/451799/how-to-show-lim-x-to-1-fracx-x2-dots-xn-nx-1-fracnn"
} |
You can use induction:
$$\frac{x + x^2 + \dots + x^n + x^{n+1} - (n+1)}{x - 1} =\\ \frac{x + x^2 + \dots + x^n - n}{x - 1} +\frac{x^{n+1}-1}{x-1}=\\ \frac{x + x^2 + \dots + x^n - n}{x - 1} +(1+x+x^2+\ldots+x^n)\xrightarrow[x\to 1]{} \frac{n(n + 1)}{2}+(n+1)=\frac{(n+1)(n + 2)}{2}.$$
• I am a little lost. How is that you are getting different result? Or is $\frac{(n + 1)(n + 2)}{2}$ only a part of the solution? Jul 25, 2013 at 12:28
• @DavidČepelík No, $\frac{(n+1)(n+2)}{2}$ is the limit for $\frac{x+x^2+\ldots+x^n\color{brown}{+x^{n+1}}-n\color{brown}{-1}}{x-1}$ and not for $\frac{x+x^2+\ldots+x^n-n}{x-1}$.
– P..
Jul 25, 2013 at 12:32
• Of course, that is the induction. Thank you! Jul 25, 2013 at 12:37
• So for $n = 1$ and $x \to 1$, we get $a_1 = \lim_{x \to 1} \frac{x - 1}{x - 1} = 1$ and $a_{n + 1} = a_n + (n + 1)$. The closed form for $a_n$ is then $\frac{n(n + 1)}{2}$. Correct? Jul 25, 2013 at 12:42
• @DavidČepelík: Yes that's right!
– P..
Jul 25, 2013 at 13:08
$$\frac{x + x^2 + \dots + x^n - n}{x - 1} =\frac{1+x + x^2 + \dots + x^n - n-1}{x - 1} =\frac{\frac{x^{n+1}-1}{x-1}-(n+1)}{x-1}$$
Putting $x-1=y,$
$$\lim_{x \to 1} \frac{x + x^2 + \dots + x^n - n}{x - 1} = \lim_{y\to0}\frac{\frac{(1+y)^{n+1}-1}y-(n+1)}y$$
$$=\lim_{y\to0}\frac{\frac{1+\binom {n+1}1y+\binom {n+1}2y^2+ O(y^3)-1}y-(n+1)}y\text{ (using Binomial Expansion)}$$
$$=\lim_{y\to0}\frac{(n+1)+\frac{n(n+1)}2y+ O(y^2)-(n+1)}y$$
$$=\frac{n(n+1)}2\text{ as }x\to1,y\to0\implies y\ne0$$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429575500227,
"lm_q1q2_score": 0.8508120100135155,
"lm_q2_score": 0.865224084314688,
"openwebmath_perplexity": 273.2409213727027,
"openwebmath_score": 0.9405686259269714,
"tags": null,
"url": "https://math.stackexchange.com/questions/451799/how-to-show-lim-x-to-1-fracx-x2-dots-xn-nx-1-fracnn"
} |
$$=\frac{n(n+1)}2\text{ as }x\to1,y\to0\implies y\ne0$$
If you already know l'Hopital's rule, Sami Ben Romdhane's answer isn't telling you anything you don't already know. When you have $$\lim_{x\to a}\frac{f(x)}{x - a}$$ and $f(a) = 0$ (otherwise, the limit is infinite on either side of $a$), then l'Hopital's rule says that the limit is the same as $$\lim_{x \to a}f'(x)= f'(a).$$ The only hard part then, in this case, is evaluating $f'(a)$. Here, $f(x) = x + x^2 + \cdots + x^n - n$, so $f'(x) = 1 + 2x + \cdots + nx^{n - 1}$, so $f'(1) = 1 + 2 + \cdots + n = \frac{n(n + 1)}{2}$ by a famous identity.
In fact, you can think of the definition of the derivative as just a special case of l'Hopital's rule (it isn't really, because l'Hopital's rule depends on the definition of the derivative, not the other way around, but it's useful to think of it this way).
• I do not think this is how he approached the problem; my guess is he noticed that the function and the derivative of the numerator are bound together by the definition of derivative of a function. I did not realize that he used the same principle Bernoulli's rule is built upon, thanks for pointing it out. Jul 25, 2013 at 16:23
Your objection to using l'Hospital's rule is on the basis that it feels like it's "too powerful" a tool for the problem, right?
Lets go all the way to the other end, then, and prove the limit using just the definition.
$$\lim_{x\to a} f(x) = L \iff \forall \varepsilon \gt 0 \ \exists \delta \gt 0 \ni \left| x-a \right| \le \delta \Rightarrow \left| f(x)-L \right| \le \varepsilon$$
Now, all we need to do is figure out a way that we can always pick a $\delta$ small enough to keep the function within $\varepsilon$ of $L$.
The particular limit in question: $$\lim_{x \to 1} \frac{x + x^2 + \dots + x^n - n}{x - 1} = \frac{n(n + 1)}{2}$$
From that, we can take: \begin{align*} a&=1\\ f(x)&=\frac{x + x^2 + \dots + x^n - n}{x - 1}\\ L&=\frac{n(n + 1)}{2} \end{align*} | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429575500227,
"lm_q1q2_score": 0.8508120100135155,
"lm_q2_score": 0.865224084314688,
"openwebmath_perplexity": 273.2409213727027,
"openwebmath_score": 0.9405686259269714,
"tags": null,
"url": "https://math.stackexchange.com/questions/451799/how-to-show-lim-x-to-1-fracx-x2-dots-xn-nx-1-fracnn"
} |
So we need to solve $$\left| \frac{x + x^2 + \dots + x^n - n}{x - 1} - \frac{n(n + 1)}{2}\right| < \varepsilon$$ for $\left|x-1\right|$.
I really don't feel like doing any of that crunchwork solving that equation. If someone out there does feel like it, please do so, and edit it into my answer. For now, though, I am going to skip ahead bunch of steps, assume we solved it, and have our solution of $$\left|x-1\right| \ge g(\varepsilon)$$
Now, we know from the definition that $\left| x-1 \right| \le \delta$, so we can conclude that if we pick $\delta = g(\varepsilon)$, we ensure that the value of $f(x)$ is within $\varepsilon$ of $L$, satisfying our definition of the limit, and proving that $$\lim_{x \to 1} \frac{x + x^2 + \dots + x^n - n}{x - 1} = \frac{n(n + 1)}{2}$$
• Thanks for your answer. You already know the limit exists and you know what it equals to ($\frac{n(n+1)}{2}$ was not my first guess). Finding what a limit equals to and proving the solution is correct by checking it satisfies the definition of the limit are two completely different exercises. (Similar to the difference between the P and NP classes in the theory of computational complexity.) I am aware I gave correct solution along with the limit, maybe that is why we did not understand each other. Sorry about it. Jul 26, 2013 at 20:01
• @DavidČepelík Personally, I think that if you prove the limit without using your super-powered tool, it doesn't matter if you used it to find the value of the limit, since the validity of your answer then does not depend on the validity of the method you applied. Jul 29, 2013 at 12:57 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429575500227,
"lm_q1q2_score": 0.8508120100135155,
"lm_q2_score": 0.865224084314688,
"openwebmath_perplexity": 273.2409213727027,
"openwebmath_score": 0.9405686259269714,
"tags": null,
"url": "https://math.stackexchange.com/questions/451799/how-to-show-lim-x-to-1-fracx-x2-dots-xn-nx-1-fracnn"
} |
### 2 Do these iterative methods always work?
No. It is not difficult to invent examples where the iteration fails to approach the solution of $AX=B$ . The key point is related to matrix norms seen in the preceding Section.
The two iterative methods we encountered above are both special cases of the general form
$\phantom{\rule{2em}{0ex}}{X}^{\left(k+1\right)}=M{X}^{\left(k\right)}+N.$
1. For the Jacobi method we choose $M=-{D}^{-1}\left(L+U\right)$ and $N={D}^{-1}B$ .
2. For the Gauss-Seidel method we choose $M=-{\left(D+L\right)}^{-1}U$ and $N={\left(D+L\right)}^{-1}B$ .
The following Key Point gives the main result.
##### Key Point 13
For the iterative process ${X}^{\left(k+1\right)}=M{X}^{\left(k\right)}+N$ the iteration will converge to a solution if the norm of $M$ is less than 1 .
Care is required in understanding what Key Point 13 says. Remember that there are lots of different ways of defining the norm of a matrix (we saw three of them). If you can find a norm ( any norm ) such that the norm of $M$ is less than 1, then the iteration will converge. It doesn’t matter if there are other norms which give a value greater than 1, all that matters is that there is one norm that is less than 1.
Key Point 13 above makes no reference to the starting “guess" ${X}^{\left(0\right)}$ . The convergence of the iteration is independent of where you start! (Of course, if we start with a really bad initial guess then we can expect to need lots of iterations.)
Show that the Jacobi iteration used to approximate the solution of
$\phantom{\rule{2em}{0ex}}\left[\begin{array}{ccc}\hfill 4\hfill & \hfill -1\hfill & \hfill -1\hfill \\ \hfill 1\hfill & \hfill -5\hfill & \hfill -2\hfill \\ \hfill -1\hfill & \hfill 0\hfill & \hfill 2\hfill \end{array}\right]\left[\begin{array}{c}\hfill {x}_{1}\hfill \\ \hfill {x}_{2}\hfill \\ \hfill {x}_{3}\hfill \end{array}\right]=\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 2\hfill \\ \hfill 3\hfill \end{array}\right]$ | {
"domain": "github.io",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429629196683,
"lm_q1q2_score": 0.8508120095333598,
"lm_q2_score": 0.8652240791017536,
"openwebmath_perplexity": 231.83796962038983,
"openwebmath_score": 0.8845521211624146,
"tags": null,
"url": "https://bathmash.github.io/HELM/30_5_itrtv_mthds_systms_eqns-web/30_5_itrtv_mthds_systms_eqns-webse2.html"
} |
is certain to converge. (Hint: calculate the norm of $-{D}^{-1}\left(L+U\right)$ .)
The Jacobi iteration matrix is
$\begin{array}{rcll}-{D}^{-1}\left(L+U\right)& =& {\left[\begin{array}{ccc}\hfill 4\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -5\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 2\hfill \end{array}\right]}^{-1}\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill -1\hfill & \hfill 0\hfill & \hfill 2\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill 0.25\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -0.2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0.5\hfill \end{array}\right]\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill -1\hfill & \hfill 0\hfill & \hfill 2\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]& \text{}\\ & =& \left[\begin{array}{ccc}\hfill 0\hfill & \hfill 0.25\hfill & \hfill 0.25\hfill \\ \hfill -0.2\hfill & \hfill 0\hfill & \hfill 0.4\hfill \\ \hfill 0.5\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]& \text{}\end{array}$
and the infinity norm of this matrix is the maximum of $0.25+0.25$ , $0.2+0.4$ and $0.5$ , that is
$\phantom{\rule{2em}{0ex}}∥-{D}^{-1}\left(L+U\right){∥}_{\infty }=0.6$
which is less than 1 and therefore the iteration will converge.
#### 2.1 Guaranteed convergence
If the matrix has the property that it is strictly diagonally dominant , which means that the diagonal entry is larger in magnitude than the absolute sum of the other entries on that row, then both Jacobi and Gauss-Seidel are guaranteed to converge. The reason for this is that if $A$ is strictly diagonally dominant then the iteration matrix $M$ will have an infinity norm that is less than 1. | {
"domain": "github.io",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429629196683,
"lm_q1q2_score": 0.8508120095333598,
"lm_q2_score": 0.8652240791017536,
"openwebmath_perplexity": 231.83796962038983,
"openwebmath_score": 0.8845521211624146,
"tags": null,
"url": "https://bathmash.github.io/HELM/30_5_itrtv_mthds_systms_eqns-web/30_5_itrtv_mthds_systms_eqns-webse2.html"
} |
A small system is the subject of Example 20 below. A large system with slow convergence is the subject of Engineering Example 1 on page 62.
##### Example 20
Show that $A=\left[\begin{array}{ccc}\hfill 4\hfill & \hfill -1\hfill & \hfill -1\hfill \\ \hfill 1\hfill & \hfill -5\hfill & \hfill -2\hfill \\ \hfill -1\hfill & \hfill 0\hfill & \hfill 2\hfill \end{array}\right]$ is strictly diagonally dominant.
##### Solution
Looking at the diagonal entry of each row in turn we see that
$\begin{array}{rcll}4& >& |-1|+|-1|=2\phantom{\rule{1em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}& \text{}\\ \left|-5\right|& >& 1+\left|-2\right|=3& \text{}\\ 2& >& \left|-1\right|+0=1& \text{}\end{array}$
and this means that the matrix is strictly diagonally dominant.
Given that $A$ above is strictly diagonally dominant it is certain that both Jacobi and Gauss-Seidel will converge.
#### 2.2 What’s so special about strict diagonal dominance?
In many applications we can be certain that the coefficient matrix $A$ will be strictly diagonally dominant. We will see examples of this in HELM booklet 32 and HELM booklet 33 when we consider approximating solutions of differential equations.
##### Exercises
1. Consider the system
$\phantom{\rule{2em}{0ex}}\left[\begin{array}{cc}\hfill 2\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 2\hfill \end{array}\right]\left[\begin{array}{c}\hfill {x}_{1}\hfill \\ \hfill {x}_{2}\hfill \end{array}\right]=\left[\begin{array}{c}\hfill 2\hfill \\ \hfill -5\hfill \end{array}\right]$ | {
"domain": "github.io",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429629196683,
"lm_q1q2_score": 0.8508120095333598,
"lm_q2_score": 0.8652240791017536,
"openwebmath_perplexity": 231.83796962038983,
"openwebmath_score": 0.8845521211624146,
"tags": null,
"url": "https://bathmash.github.io/HELM/30_5_itrtv_mthds_systms_eqns-web/30_5_itrtv_mthds_systms_eqns-webse2.html"
} |
1. Use the starting guess ${X}^{\left(0\right)}=\left[\begin{array}{c}\hfill 1\hfill \\ \hfill -1\hfill \end{array}\right]$ in an implementation of the Jacobi method to show that ${X}^{\left(1\right)}=\left[\begin{array}{c}\hfill 1.5\hfill \\ \hfill -3\hfill \end{array}\right]$ . Find ${X}^{\left(2\right)}$ and ${X}^{\left(3\right)}$ .
2. Use the starting guess ${X}^{\left(0\right)}=\left[\begin{array}{c}\hfill 1\hfill \\ \hfill -1\hfill \end{array}\right]$ in an implementation of the Gauss-Seidel method to show that ${X}^{\left(1\right)}=\left[\begin{array}{c}\hfill 1.5\hfill \\ \hfill -3.25\hfill \end{array}\right]$ . Find ${X}^{\left(2\right)}$ and ${X}^{\left(3\right)}$ .
(Hint: it might help you to know that the exact solution is $\left[\begin{array}{c}\hfill {x}_{1}\hfill \\ \hfill {x}_{2}\hfill \end{array}\right]=\left[\begin{array}{c}\hfill 3\hfill \\ \hfill -4\hfill \end{array}\right]$ .)
1. Show that the Jacobi iteration applied to the system
$\phantom{\rule{2em}{0ex}}\left[\begin{array}{cccc}\hfill 5\hfill & \hfill -1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -1\hfill & \hfill 5\hfill & \hfill -1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -1\hfill & \hfill 5\hfill & \hfill -1\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -1\hfill & \hfill 5\hfill \end{array}\right]\left[\begin{array}{c}\hfill {x}_{1}\hfill \\ \hfill {x}_{2}\hfill \\ \hfill {x}_{3}\hfill \\ \hfill {x}_{4}\hfill \end{array}\right]=\left[\begin{array}{c}\hfill 7\hfill \\ \hfill -10\hfill \\ \hfill -6\hfill \\ \hfill 16\hfill \end{array}\right]$
can be written | {
"domain": "github.io",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429629196683,
"lm_q1q2_score": 0.8508120095333598,
"lm_q2_score": 0.8652240791017536,
"openwebmath_perplexity": 231.83796962038983,
"openwebmath_score": 0.8845521211624146,
"tags": null,
"url": "https://bathmash.github.io/HELM/30_5_itrtv_mthds_systms_eqns-web/30_5_itrtv_mthds_systms_eqns-webse2.html"
} |
can be written
$\phantom{\rule{2em}{0ex}}{X}^{\left(k+1\right)}=\left[\begin{array}{cccc}\hfill 0\hfill & \hfill 0.2\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0.2\hfill & \hfill 0\hfill & \hfill 0.2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0.2\hfill & \hfill 0\hfill & \hfill 0.2\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0.2\hfill & \hfill 0\hfill \end{array}\right]{X}^{\left(k\right)}+\left[\begin{array}{c}\hfill 1.4\hfill \\ \hfill -2\hfill \\ \hfill -1.2\hfill \\ \hfill 3.2\hfill \end{array}\right].$
2. Show that the method is certain to converge and calculate the first three iterations using
zero starting values.
(Hint: the exact solution to the stated problem is $\left[\begin{array}{c}\hfill 1\hfill \\ \hfill -2\hfill \\ \hfill 1\hfill \\ \hfill 3\hfill \end{array}\right]$ .)
1. $\phantom{\rule{2em}{0ex}}2{x}_{1}^{\left(1\right)}=2-1{x}_{2}^{\left(0\right)}=2$
and therefore ${x}_{1}^{\left(1\right)}=1.5$
$\phantom{\rule{2em}{0ex}}2{x}_{2}^{\left(1\right)}=-5-1{x}_{1}^{\left(0\right)}=-6$
which implies that ${x}_{2}^{\left(1\right)}=-3$ . These two values give the required entries in ${X}^{\left(1\right)}$ . A second and third iteration follow in a similar way to give
$\phantom{\rule{2em}{0ex}}{X}^{\left(2\right)}=\left[\begin{array}{c}\hfill 2.5\\ \hfill -3.25\end{array}\right]\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{X}^{\left(3\right)}=\left[\begin{array}{c}\hfill 2.625\\ \hfill -3.75\end{array}\right]$
2. $\phantom{\rule{2em}{0ex}}2{x}_{1}^{\left(1\right)}=2-1{x}_{2}^{\left(0\right)}=3$
and therefore ${x}_{1}^{\left(1\right)}=1.5$ . This new approximation to ${x}_{1}$ is used straight away when finding a new approximation to ${x}_{2}^{\left(1\right)}$ .
$\phantom{\rule{2em}{0ex}}2{x}_{2}^{\left(1\right)}=-5-1{x}_{1}^{\left(1\right)}=-6.5$ | {
"domain": "github.io",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429629196683,
"lm_q1q2_score": 0.8508120095333598,
"lm_q2_score": 0.8652240791017536,
"openwebmath_perplexity": 231.83796962038983,
"openwebmath_score": 0.8845521211624146,
"tags": null,
"url": "https://bathmash.github.io/HELM/30_5_itrtv_mthds_systms_eqns-web/30_5_itrtv_mthds_systms_eqns-webse2.html"
} |
$\phantom{\rule{2em}{0ex}}2{x}_{2}^{\left(1\right)}=-5-1{x}_{1}^{\left(1\right)}=-6.5$
which implies that ${x}_{2}^{\left(1\right)}=-3.25$ . These two values give the required entries in ${X}^{\left(1\right)}$ . A second and third iteration follow in a similar way to give
$\phantom{\rule{2em}{0ex}}{X}^{\left(2\right)}=\left[\begin{array}{c}\hfill 2.625\\ \hfill -3.8125\end{array}\right]\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}{X}^{\left(3\right)}=\left[\begin{array}{c}\hfill 2.906250\\ \hfill -3.953125\end{array}\right]$
where ${X}^{\left(3\right)}$ is given to 6 decimal places
1. In this case $D=\left[\begin{array}{cccc}\hfill 5\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 5\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 5\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 5\hfill \end{array}\right]$ and therefore ${D}^{-1}=\left[\begin{array}{cccc}\hfill 0.2\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0.2\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0.2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0.2\hfill \end{array}\right]$ . | {
"domain": "github.io",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429629196683,
"lm_q1q2_score": 0.8508120095333598,
"lm_q2_score": 0.8652240791017536,
"openwebmath_perplexity": 231.83796962038983,
"openwebmath_score": 0.8845521211624146,
"tags": null,
"url": "https://bathmash.github.io/HELM/30_5_itrtv_mthds_systms_eqns-web/30_5_itrtv_mthds_systms_eqns-webse2.html"
} |
So the iteration matrix $M\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}{D}^{-1}\left[\begin{array}{cccc}\hfill 0\hfill & \hfill -1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill -1\hfill & \hfill 0\hfill & \hfill -1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -1\hfill & \hfill 0\hfill & \hfill -1\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill -1\hfill & \hfill 0\hfill \end{array}\right]\phantom{\rule{0.3em}{0ex}}=\phantom{\rule{0.3em}{0ex}}\left[\phantom{\rule{0.3em}{0ex}}\begin{array}{cccc}\hfill 0\hfill & \hfill 0.2\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0.2\hfill & \hfill 0\hfill & \hfill 0.2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0.2\hfill & \hfill 0\hfill & \hfill 0.2\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0.2\hfill & \hfill 0\hfill \end{array}\phantom{\rule{0.3em}{0ex}}\right]$
and that the Jacobi iteration takes the form
$\phantom{\rule{2em}{0ex}}{X}^{\left(k+1\right)}=M{X}^{\left(k\right)}+{M}^{-1}\left[\begin{array}{c}\hfill 7\hfill \\ \hfill -10\hfill \\ \hfill -6\hfill \\ \hfill 16\hfill \end{array}\right]=\left[\begin{array}{cccc}\hfill 0\hfill & \hfill 0.2\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0.2\hfill & \hfill 0\hfill & \hfill 0.2\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0.2\hfill & \hfill 0\hfill & \hfill 0.2\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0.2\hfill & \hfill 0\hfill \end{array}\right]{X}^{\left(k\right)}+\left[\begin{array}{c}\hfill 1.4\hfill \\ \hfill -2\hfill \\ \hfill -1.2\hfill \\ \hfill 3.2\hfill \end{array}\right]$
as required. | {
"domain": "github.io",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429629196683,
"lm_q1q2_score": 0.8508120095333598,
"lm_q2_score": 0.8652240791017536,
"openwebmath_perplexity": 231.83796962038983,
"openwebmath_score": 0.8845521211624146,
"tags": null,
"url": "https://bathmash.github.io/HELM/30_5_itrtv_mthds_systms_eqns-web/30_5_itrtv_mthds_systms_eqns-webse2.html"
} |
as required.
2. Using the starting values ${x}_{1}^{\left(0\right)}={x}_{2}^{\left(0\right)}={x}_{3}^{\left(0\right)}={x}_{4}^{\left(0\right)}=0$ , the first iteration of the Jacobi method gives $\begin{array}{rcll}{x}_{1}^{1}& =& 0.2{x}_{2}^{0}+1.4=1.4& \text{}\\ {x}_{2}^{1}& =& 0.2\left({x}_{1}^{0}+{x}_{3}^{0}\right)-2=-2& \text{}\\ {x}_{3}^{1}& =& 0.2\left({x}_{2}^{0}+{x}_{4}^{0}\right)-1.2=-1.2\phantom{\rule{1em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}& \text{}\\ {x}_{4}^{1}& =& 0.2{x}_{3}^{0}+3.2=3.2& \text{}\end{array}$
The second iteration is
$\begin{array}{rcll}{x}_{1}^{2}& =& 0.2{x}_{2}^{1}+1.4=1& \text{}\\ {x}_{2}^{2}& =& 0.2\left({x}_{1}^{1}+{x}_{3}^{1}\right)-2=-1.96& \text{}\\ {x}_{3}^{2}& =& 0.2\left({x}_{2}^{1}+{x}_{4}^{1}\right)-1.2=-0.96\phantom{\rule{1em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}& \text{}\\ {x}_{4}^{2}& =& 0.2{x}_{3}^{1}+3.2=2.96& \text{}\end{array}$
And the third iteration is
$\begin{array}{rcll}{x}_{1}^{3}& =& 0.2{x}_{2}^{2}+1.4=1.008& \text{}\\ {x}_{2}^{3}& =& 0.2\left({x}_{1}^{2}+{x}_{3}^{2}\right)-2=-1.992\phantom{\rule{1em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}& \text{}\\ {x}_{3}^{3}& =& 0.2\left({x}_{2}^{2}+{x}_{4}^{2}\right)-1.2=-1& \text{}\\ {x}_{4}^{3}& =& 0.2{x}_{3}^{2}+3.2=3.008& \text{}\end{array}$ | {
"domain": "github.io",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429629196683,
"lm_q1q2_score": 0.8508120095333598,
"lm_q2_score": 0.8652240791017536,
"openwebmath_perplexity": 231.83796962038983,
"openwebmath_score": 0.8845521211624146,
"tags": null,
"url": "https://bathmash.github.io/HELM/30_5_itrtv_mthds_systms_eqns-web/30_5_itrtv_mthds_systms_eqns-webse2.html"
} |
# cumtrapz
Cumulative trapezoidal numerical integration
## Syntax
``Q = cumtrapz(Y)``
``Q = cumtrapz(X,Y)``
``Q = cumtrapz(___,dim)``
## Description
example
````Q = cumtrapz(Y)` computes the approximate cumulative integral of `Y` via the trapezoidal method with unit spacing. The size of `Y` determines the dimension to integrate along:If `Y` is a vector, then `cumtrapz(Y)` is the cumulative integral of `Y`.If `Y` is a matrix, then `cumtrapz(Y)` is the cumulative integral over each column.If `Y` is a multidimensional array, then `cumtrapz(Y)` integrates over the first dimension whose size does not equal 1.```
example
````Q = cumtrapz(X,Y)` integrates `Y` with respect to the coordinates or scalar spacing specified by `X`. If `X` is a vector of coordinates, then `length(X)` must be equal to the size of the first dimension of `Y` whose size does not equal 1.If `X` is a scalar spacing, then `cumtrapz(X,Y)` is equivalent to `X*cumtrapz(Y)`. ```
example
````Q = cumtrapz(___,dim)` integrates along the dimension `dim` using any of the previous syntaxes. You must specify `Y`, and optionally can specify `X`. If you specify `X`, then it can be a scalar or a vector with length equal to `size(Y,dim)`. For example, if `Y` is a matrix, then `cumtrapz(X,Y,2)` cumulatively integrates each row of `Y`.```
## Examples
collapse all
Calculate the cumulative integral of a vector where the spacing between data points is 1.
Create a numeric vector of data.
`Y = [1 4 9 16 25];`
`Y` contains function values for $\mathit{f}\left(\mathit{x}\right)={\mathit{x}}^{2}$ in the domain `[1 5]`.
Use `cumtrapz` to integrate the data with unit spacing.
`Q = cumtrapz(Y)`
```Q = 1×5 0 2.5000 9.0000 21.5000 42.0000 ```
This approximate integration yields a final value of 42. In this case, the exact answer is a little less, $41\frac{1}{3}$. The `cumtrapz` function overestimates the value of the integral because f(x) is concave up. | {
"domain": "mathworks.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9833429604789206,
"lm_q1q2_score": 0.8508120091302669,
"lm_q2_score": 0.8652240808393984,
"openwebmath_perplexity": 781.9710638136748,
"openwebmath_score": 0.8963515162467957,
"tags": null,
"url": "https://au.mathworks.com/help/matlab/ref/cumtrapz.html"
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.