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(To my mind this is neater than my argument but more rabbit-out-of-hat-ish.)
• Correct answer and reasonable proof, but there's a(nother) surprising fact that makes the proof even simpler. Jan 2, 2022 at 14:27
• As mentioned in the answer, I am currently looking for slicker proofs and have some plausible ideas of where to look. There is something of an analogy with a famous question about coins, but that pathway doesn't seem like it really produces something much simpler so far. Jan 2, 2022 at 14:29
• How about: ai is larger than i-1 elements (a1,...,a{i-1}), bi is larger than n-i elements (b{i+1},...,bn). The larger of ai and bi is larger than the other one and both these sets, that is 1 + i-1 + n-i elements. Jan 2, 2022 at 15:30
• ... and possibly other things too, but at least n elements, which means that all the right endpoints are >= n+1, and as there are n right endpoints that tells us what they all are? Yeah, that works. Jan 2, 2022 at 15:34
• The last spoilertag is the kind of argument I was thinking of. Nice example of a surprising non-obvious mathematical fact. Jan 2, 2022 at 17:48
This problem doesn't really have anything to do with the integers from 1 to 2n. Fix any set consisting of 2n real numbers (repetitions allowed); then split them into a_1 <= a_2 <= .... <= a_n and b_1 >= b_2 >= ... >= b_n. Then the sum |a_1 - b_1| + |a_2 - b_2| + ... + |a_n - b_n| is constant, that is, it doesn't depend on the partition. | {
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• Nice generalisation, and I suppose the same proof works in this case too, but can you edit your answer to include a quick proof of this fact? Jan 3, 2022 at 4:48
• @Randal'Thor Since the loopy walt's answer (up to the words "It follows that all the $m_i$ are smaller than all the $M_i$") does not use the fact that the numbers given are integers from 1 to 2n, if we replace $1\leqslant2\leqslant\dots\leqslant2n$ with any numbers $q_1\leqslant q_2\leqslant\dots\leqslant q_{2n}$, the assertion about the constant sum still holds (and it will be equal $\sum_{i=1}^n q_{n+i} - \sum_{i=1}^n q_i$). Jan 3, 2022 at 6:47
• @trolley813 I know how the proof would go, I'm just asking pmw to include it so that this answer is self-contained. Jan 3, 2022 at 8:23 | {
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# How many seating arrangements of $m$ people in a row of $n$ seats are there if $k$ of the people must sit together?
Suppose there are five people $$A, B, C, D, E$$ to be seated in a row of eight seats $$S_1, \ldots, S_8$$.
(1) How many possibilities are there if $$A$$ and $$B$$ are to sit next to each other?
(2) How many possibilities are there if $$A$$ and $$B$$ are not to sit next to each other?
My Attempt:
First Approach:
There are $$8$$ options of a seat for $$A$$. However, if $$A$$ is to be seated in $$S_1$$ or $$S_8$$, then $$B$$ can only be seated in $$S_2$$ or $$S_7$$, respectively. On the other hand, if $$A$$ is to be seated in any one of $$S_2$$ through $$S_7$$, then $$B$$ has two options of seat in each case.
Once $$A$$ and $$B$$ have been seated, there are $$6$$ seats left for $$C$$, and then $$5$$ seats left for $$D$$, and finally $$4$$ seats left for $$E$$.
In this way, the total number of seating arrangements are $$2 \times 1 \times 6 \times 5 \times 4 + 6 \times 2 \times 6 \times 5 \times 4 = 240 + 1440 = 1680.$$
Is this solution correct?
Second Approach:
Now let us treat $$A, B$$ as one entity. Then we have four entities to be accommodated in seven spots, for which there are $$7 \times 6 \times 5 \times 4 = 840$$ ways. And, each one of these $$840$$ arrangements, the members $$A$$ and $$B$$ of the block $$A, B$$ can be arranged within the block in two distinct ways. Therefore there are $$840 \times 2 = 1680$$ possible seating arrangements.
Is my answer correct? If so, then are both of my approaches also correct? If not, then where are the problems.
More generally, I have the following question:
Let $$k, m, n$$ be any natural numbers such that $$k \leq m \leq n$$. Then how many ways are there in total of seating $$m$$ people $$P_1 \ldots, P_m$$ in $$n$$ seats $$S_1, \ldots, S_n$$ such that some $$k$$ of these people insist on sitting next to each other?
My Attempt Using the Second Approach: | {
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My Attempt Using the Second Approach:
Let us treat that $$k$$ people as one block. Then there are
\begin{align} & (n-k+1) \underbrace{(n-k) (n-k-1) \ldots }_{(m-k) \mbox{ factors}} \\ &= (n-k+1) \big[ \ (n-k) (n-k-1) \ldots \big( (n-k) - (m-k-1) \big) \ \big] \\ &= (n-k+1)(n-k) \ldots (n-m+1) \\ &= ^{n-k+1}P_{n-m} . \end{align}
Finally, corresponding to each of the above arrangements with the $$k$$ people considered as one block, there are $$k!$$ ways of arranging the $$k$$ people within the block amongst themselves.
Hence there are a total of $$k! \ ^{n-k+1}P_{n-m}$$ ways of seating $$m$$ people in $$n$$ seats with $$k$$ people seated next to each other.
• The problem is not clearly stated until it is said which seats are next to each other. Among many possibilties two seem plausible: seats in a single row, or in a circle (in which case $S_1$ would be next to $S_8$). Nov 30, 2019 at 17:23
Both of your solutions to the first problem are correct. However, your general formula is not. Notice that in the first problem, $$n = 8$$, $$m = 5$$, and $$k = 2$$, so your formula gives $$2!P(8 - 2 + 1, 8 - 5) = 2!P(7, 3) = 2! \cdot 7 \cdot 6 \cdot 5 = 2 \cdot 210 = 420$$ Let's see what went wrong.
We wish to seat $$m$$ people, $$k$$ of whom must sit consecutively, in $$n$$ seats. Since the block takes up $$k$$ of the $$n$$ places, it must begin in one of the first $$n - (k - 1) = n - k + 1$$ positions. Once the block has been placed, there are $$n - k$$ seats left for the remaining $$m - k$$ people. They can be arranged in those seats in $$P(n - k, m - k)$$ ways. The people within the block can be arranged in $$k!$$ ways, which gives us the formula $$(n - k + 1)P(n - k, m - k)k!$$ As a sanity check, let's try our formula when $$n = 8$$, $$k = 2$$, and $$m = 5$$. It gives $$(8 - 2 + 1)P(8 - 2, 5 - 2)2! = 7 \cdot P(6, 3) \cdot 2! = 7 \cdot 6 \cdot 5 \cdot 4 \cdot 2 = 1680$$ which agrees with the answer you obtained in your example. | {
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Observe that \begin{align*} (n - k + 1)P(n - k, m - k) & = (n - k + 1) \cdot \frac{(n - k)!}{[(n - k) - (m - k)]!}\\ & = \frac{(n - k + 1)(n - k)!}{(n - m)!}\\ & = \frac{(n - k + 1)!}{(n - m)!}\\ & = \frac{(n - k + 1)!}{[(n - k + 1) - (m - k + 1)]!}\\ & = P(n - k + 1, m - k + 1) \end{align*} so we could write our formula in the form $$P(n - k + 1, m - k + 1)k!$$ As a sanity check, note that if $$n = 8$$, $$k = 2$$, and $$m = 5$$, then $$P(8 - 2 + 1, 5 - 2 + 1)2! = P(7, 4)2! = 7 \cdot 6 \cdot 5 \cdot 4 \cdot 2 = 1680$$
• thank you so much for such a beautiful answer! Nov 30, 2019 at 18:06
For part $$a.)$$, we can work on two cases: $$AB$$ ($$A$$ to the left of $$B$$) and $$BA$$. The possibilities for $$AB$$ are $$S_{1}S_{2}, S_{2}S_{3}, …, S_{7}S_{8}$$. The possibilities for $$BA$$ are also the same. So the answer is
$$7 \times \binom{6}{3} \times 3! + 7 \times \binom{6}{3} \times 3! = 1680$$
The $$\binom{6}{3}$$ is the number of ways we can choose 3 seats, and $$3!$$ is number of possible orders of $$C,D,E$$ in the 3 seats.
For part $$b.)$$ count all possible seating without any rule, and then subtract with $$1680$$. | {
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# Math Help - Counting-Permutations
1. ## Counting-Permutations
I have a question asking me to find:
The sum of all 4 digit numbers containing the digits 2,4,6,8 without repetitions
I realise that the digits can be chosen in (4)_4=4!=24 ways since repetition is not permitted and order does matter
The final answer is 133320
Any help would be great
2. Let the 24 numbers be $a_1,\dots,a_{24}$. Further, let the $i$th number $a_i$ be $1000b_{i1}+100b_{i2}+10b_{i3}+b_{i4}$ where all $b_{ij}$ are in {2,4,6,8}. Let's find $S_1=\sum_{i=1}^{24}b_{i1}$. Each of the four numbers is encountered 6 times, so the $S_1 = 6(2 + 4 + 6 + 8) = 120$. Therefore, the first digits of all numbers contribute 120 * 1000 to the whole sum. Similarly, the second digits contribute 120 * 100 and so on.
3. Originally Posted by qwerty10
I have a question asking me to find:
The sum of all 4 digit numbers containing the digits 2,4,6,8 without repetitions
The final answer is 133320
Think about it, each of those four numbers will be in each decimal place six times. The ones column adds up to $6(2+4+6+8)=120.$
Check out $\displaystyle120\left( {\sum\limits_{k = 0}^3 {10^k } } \right) = ~?$.
4. Hello, qwerty10!
Find the sum of all 4 digit numbers containing the digits 2,4,6,8 without repetitions.
The answer is: 133,320.
List the $4! = 24$ permutations of the four digits
,. . and consider their sum.
. . $\begin{array}{cc}
&2468 \\ &2486 \\ &2648 \\ &2684 \\ &2846 \\ &2864 \\ &\vdots \\
&8246 \\ &8264 \\ &8426 \\ &8462 \\ &8624 \\ + & 8642 \\ \hline
\end{array}$
We find that each column has: six 2's, six 4's, six 6's, six 8's.
The total of each column is: . $6\!\cdot\!2 + 6\!\cdot\!4 + 6\!\cdot\!6 + 6\!\cdot\!8 \:=\:120$
Hence, the addition has the form:
. . $\begin{array}{cccccc}
&&& 1 & 2 & 0 \\ && 1 & 2 & 0 \\ & 1 & 2 & 0 \\ 1 & 2 & 0 \\ \hline 1 & 3 & 3 & 3 & 2 & 0 \end{array}$
Therefore, the sum is: . $133,\!320$
Edit: Plato beat me to it . . . *sigh*
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The question is as follows:
I have the general solution of a differential equation as $P(t) = Ae^{kt}$ where $k < 0$ is a constant, and it describes the mass $P(t)$ of a radioactive isotope at future time $t$. I also have that the mass of the isotope is $10$ at time $0$ and it is $5$ at time $2$. I want to find the remainder of the isotope at time $3$.
So I can use that $P(0) = 10$ to obtain $A$ = $10$, and here now is where I get a bit confused. If I use $P(2) = 5$, I get $5 = 10e^{2k}$ and solving for $k$ I get $k = \frac{ln(0.5)}{2} = -0.34657$. And then substitute it all back in to get $P(3)$
However, when looking at other questions online, they talk a lot about half-life of the substance and we have that if we let $\tau$ denote the half-life of the substance, then $k = \frac{ln(2)}{\tau}$ (where they are using the general solution to be $Ae^{-kt}$ However, isn't $\tau$ in my case equal to $2$? So should I not be getting $k = \frac{ln(2)}{2}$ rather than $k = \frac{ln(0.5)}{2}$? Can someone please explain where I am going wrong?
Thank you
• You haven't don't anything wrong because $-\log{(0.5)}=\log{(2)}$. Which explains the negative sign, your solution is the same as theirs Mar 20 '18 at 12:51
• Ah ok I think I get where I am getting confused. So what I have written is fine, I just solved for $k$ where $k$ is negative whereas the one online solved for $k$ where $k$ is positive and when putting back into their formula they will take this into account and multiply it by negative $1$? Mar 20 '18 at 12:55
$Ae^{-kt}$ with $k=\frac{ln(2)}{\tau}$ is equivalent to $Ae^{kt}$ with $k=\frac{ln(0.5)}{\tau}$
• Thanks very much. One last thing, so using that then $P(3)$ is approx equal to $3.5$ yes? Mar 20 '18 at 12:58
• Yes, it makes sense as you can double check with half-life equal to two, which means that $P(4)=2.5$, and you know the exponential decay function is decreasing. Mar 20 '18 at 13:00 | {
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# 8.1.2 Insertion Sort
The list-sort-best-first procedure seems quite inefficient. For every output element, we are searching the whole remaining list to find the best element, but do nothing of value with all the comparisons that were done to find the best element.
An alternate approach is to build up a sorted list as we go through the elements. Insertion sort works by putting the first element in the list in the right place in the list that results from sorting the rest of the elements.
First, we define the list-insert-one procedure that takes three inputs: a comparison procedure, an element, and a List. The input List must be sorted according to the comparison function. As output, list-insert-one produces a List consisting of the elements of the input List, with the input element inserts in the right place according to the comparison function.
(define (list-insert-one cf el p) ; requires: p is sorted by cf
(if (null? p) (list el)
(if (cf el (car p)) (cons el p)
(cons (car p) (list-insert-one cf el (cdr p))))))
The running time for list-insert-one is in $\Theta(n)$ where $n$ is the number of elements in the input list. In the worst case, the input element belongs at the end of the list and it makes $n$ recursive applications of list-insert-one. Each application involves constant work so the overall running time of list-insert-one is in $\Theta(n)$.
To sort the whole list, we insert each element into the list that results from sorting the rest of the elements:
(define (list-sort-insert cf p)
(if (null? p) null
(list-insert-one cf (car p) (list-sort-insert cf (cdr p))))) | {
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Evaluating an application of list-sort-insert on a list of length $n$ involves $n$ recursive applications. The lengths of the input lists in the recursive applications are $n-1$, $n-2$, $\ldots$, 0. Each application involves an application of list-insert-one which has linear running time. The average length of the input list over all the applications is approximately $\frac{n}{2}$, so the average running time of the list-insert-one applications is in $\Theta(n)$. There are $n$ applications of \scheme|list-insert-one|, so the total running time is in $\Theta(n^2)$.
Exercise 8.5. We analyzed the worst case running time of list-sort-insert above. Analyze the best case running time. Your analysis should identify the inputs for which list-sort-insert runs fastest, and describe the asymptotic running time for the best case input.
Exercise 8.6. Both the list-sort-best-first-sort and list-sort-insert procedures have asymptotic running times in $\Theta(n^2)$. This tells us how their worst case running times grow with the size of the input, but isn't enough to know which procedure is faster for a particular input. For the questions below, use both analytical and empirical analysis to provide a convincing answer.
a. How do the actual running times of list-sort-best-first-sort and list-sort-insert on typical inputs compare?
b. Are there any inputs for which list-sort-best-first is faster than list-sort-insert?
c. For sorting a long list of $n$ random elements, how long does each procedure take? (See Exercise 8.2 for how to create a list of random elements.) | {
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# Some interesting observations on a sum of reciprocals
This recent question is the motivation for this post.
Consider the following equation $$\frac1{x-1}+\frac1{x-2}+\cdots+\frac1{x-k}=\frac1{x-k-1}$$ where $$k>1$$.
My claims:
1. There are $$k$$ solutions, all of which are real.
2. Let $$x_{\min}$$ be the minimum value of these $$k$$ solutions. Then as $$k\to\infty$$, $$x_{\min}$$ converges. (If it does, to what value does it converge?)
3. As $$k\to\infty$$, all of the solutions get closer and closer to an integer, which is bounded below. Furthermore, these integers will be $$1, 2, 3, \cdots, k-1, k+1$$.
To see these patterns, I provide the solutions of $$x$$ below. I used W|A for $$k\ge4$$. The values in $$\color{blue}{\text{blue}}$$ are those of $$x_{\min}$$.
$$\begin{array}{c|c}k&2&3&4&5&6\\\hline x&4.414&4.879&5.691&6.592&7.530\\&\color{blue}{1.585}&2.652&3.686&4.701&5.722\\&&\color{blue}{1.468}&2.545&3.588&4.615\\&&&\color{blue}{1.411}&2.487&3.531\\&&&&\color{blue}{1.376}&2.449\\&&&&&\color{blue}{1.352}\end{array}$$
Also, when $$k=2$$, the polynomial in question is $$x^2-6x+7$$, and when $$k=3$$, it is $$x^3-9x^2+24x-19$$.
The reason why I think $$x_{\min}$$ converges is because the difference between the current one and the previous gets smaller and smaller as $$k$$ increases.
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Are my claims true?
• It is easy to prove that it has at least $k-1$ distinct real soutions Aug 7, 2018 at 14:22
• and also it is true that $1<x_{min}<2$ for every $k$ and it converges to 1 Aug 7, 2018 at 14:24
• @Exodd: if you do that you know it has $k$ distinct solutions because of the limits of each side as $x \to \pm \infty$ Aug 7, 2018 at 14:27
• The method I used (removing the asymptotes) has been the basis of significant improvements in chemical engineering calculations (in particular for the so-called Rachford-Rice and Underwood equations). I have published quite a lor of papers for these. I had a lot of fun with your (may I confess that I cannot resist an equation ?). Cheers. Aug 8, 2018 at 8:05
• I updated my answer for some improvements. Cheers and thanks for the problem. Aug 10, 2018 at 8:05
## 3 Answers
Both your claims are true.
if you call $$f(x) = \frac1{x-1}+\frac1{x-2}+\cdots+\frac1{x-k}-\frac1{x-k-1}$$ then $f(1^+) = +\infty$, $f(2^-) = -\infty$ and $f$ is continuous in $(1,2)$, so it has a root in $(1,2)$. The same you can say about $(2,3)$, $(3,4), \cdots, (k-1,k)$, so there are at least $k-1$ real distinct roots. $f$ is also equivalent to a $k$-degree polynomial with the same root, but a $k$-degree polynomial with $k-1$ real roots has in reality $k$ real roots.
The last root lies in $(k+1,+\infty)$, since $f(k+1^+) = -\infty$ and $f(+\infty) = +\infty$.
The least root $x_{\min}$ must lie in $(1,2)$, since $f(x)<0$ for every $x<1$. Moreover, $$f(x) = 0\implies x = 1 + \frac{1}{\frac1{x-k-1}-\frac1{x-2}-\cdots-\frac1{x-k}}$$ and knowing $1<x<2$, we infer $\frac1{x-k-1}>\frac1{x-2}$ and $$1<x = 1 + \frac{1}{\frac1{x-k-1}-\frac1{x-2}-\cdots-\frac1{x-k}} < 1 - \frac{1}{\frac1{x-3}+\cdots+\frac1{x-k}}\to 1$$ so $x_{\min}$ converges to $1$ | {
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About the third claim, notice that you may repeat the same argument for any root except the biggest. Let us say that $x_r$ is the $r-th$ root, with $r<k$, and we know that $r<x_r<r+1$. $$f(x_r) = 0\implies x_r = r + \frac{1}{\frac1{x_r-k-1}-\frac1{x_r-1}-\cdots-\frac1{x_r-k}}$$ but $\frac1{x_r-k-1}>\frac1{x_r-1}$ holds, so $$r<x_r = r + \frac{1}{\frac1{x_r-k-1}-\frac1{x_r-1}-\cdots-\frac1{x_r-k}} < r - \frac{1}{\frac1{x_r-2}+\cdots+\frac1{x_r-k}}\to r$$ so $x_r$ converges to $r$.
For the biggest root, we know $k+1<x_k$ and $$f(x_k) = 0\implies k+1 < x_k = k+1 + \frac{1}{\frac1{x_k-1}+\cdots+\frac1{x_k-k}} \to k+1$$
• I would've argued that the LHS is continuous, monotone, and ranges from $-\infty$ to $+\infty$ between natural numbers, while the RHS is continuous and monotone, and since the LHS is asymptotically larger than the RHS as $x\to\infty$, there must be a solution $x>k+1$, where the LHS is less than the RHS. In any case, sound argument. Aug 7, 2018 at 14:52
• I think they're quite equivalent. Now I'm wondering if $x_{max}$ converges to $k+1$ Aug 7, 2018 at 14:57
• @Exodd Nice proofs. The third claim of converging to integers extends the claim $x_{\min}$ to all solutions of $x$. I strongly believe that it's true after plotting in Desmos, but I can't construct a rigorous proof for it. Aug 7, 2018 at 14:59
• Hard to exactly say what that's supposed to mean, $k+1$ is a moving target :P Aug 7, 2018 at 15:00
• Let's say $x_{max}(k)-k-1 = o_k(1)$ Aug 7, 2018 at 15:00
For the base case, $$\tag1f_2(x)=\frac1{x-1}+\frac1{x-2}-\frac1{x-3},$$ one readily verifies that there is a root in $(1,2)$ and a root $x^*$ in $(3,+\infty)$. | {
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If we multiply out the denominators of $$f_k(x)=\frac1{x-1}+\frac1{x-2}+\ldots+\frac1{x-k}-\frac1{x-k-1},$$ we obtain the equation $$\tag2(x-1)(x-2)\cdots(x-k-1)f_k(x)=0,$$ which is a polynomial of degree (at most) $k$, so we expect $k$ solutions, but some of these may be complex or repeated or happen to be among $\{1,2,\ldots, k+1\}$ and thus not allowed for the original equation. But $f_k(x)$ has simple poles with jumps from $-\infty$ to $+\infty$ at $1,2,3,\ldots, k$, and a simple pole with jump from $+\infty$ to $-\infty$ at $k+1$, and is continuous otherwise. It follows that there is (at least) one real root in $(1,2)$, at least one in in $(2,3)$, etc. up to $(k-1,k)$, so there are at least $k-1$ distinct real roots. Additionally, for $x>k+1$ and $k\ge2$, we have $$f_k(x)\ge f_2(x+k-2).$$ It follows that there is another real root between $k+1$ and $x^*+k-2$. So indeed, we have $k$ distinct real roots.
From the aboive, the smallest root is always in $(1,2)$. If follows from $f_{k+1}(x)>f_k(x)$ for $x\in(1,2)$ and the fact that all $f_k$ are strictly decreasing there, that $x_\min$ decreases with increasing $k$. As a decreasing bounded sequence, it does have a limit. | {
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Considering that you look for the first zero of function $$f(x)=\sum_{i=1}^k \frac 1{x-i}-\frac1 {x-k-1}$$ which can write, using harmonic numbers, $$f(x)=H_{-x}-H_{k-x}-\frac{1}{x-k-1}$$ remove the asymptotes using $$g(x)=(x-1)(x-2)f(x)=2x-3+(x-1)(x-2)\left(H_{2-x}-H_{k-x}-\frac{1}{x-k-1} \right)$$ You can approximate the solution using a Taylor expansion around $x=1$ and get $$g(x)=-1+(x-1) \left(-\frac{1}{k}+\psi ^{(0)}(k)+\gamma +1\right)+O\left((x-1)^2\right)$$ Ignoring the higher order terms, this gives as an approximation $$x_{est}=1+\frac{k}{k\left(\gamma +1+ \psi ^{(0)}(k)\right)-1}$$ which seems to be "decent" (and, for sure, confirms your claims). $$\left( \begin{array}{ccc} k & x_{est} & x_{sol} \\ 2 & 1.66667 & 1.58579 \\ 3 & 1.46154 & 1.46791 \\ 4 & 1.38710 & 1.41082 \\ 5 & 1.34682 & 1.37605 \\ 6 & 1.32086 & 1.35209 \\ 7 & 1.30238 & 1.33430 \\ 8 & 1.28836 & 1.32040 \\ 9 & 1.27726 & 1.30914 \\ 10 & 1.26817 & 1.29976 \\ 11 & 1.26055 & 1.29179 \\ 12 & 1.25403 & 1.28489 \\ 13 & 1.24837 & 1.27884 \\ 14 & 1.24339 & 1.27347 \\ 15 & 1.23895 & 1.26867 \\ 16 & 1.23498 & 1.26433 \\ 17 & 1.23138 & 1.26039 \\ 18 & 1.22810 & 1.25678 \\ 19 & 1.22510 & 1.25346 \\ 20 & 1.22233 & 1.25039 \end{array} \right)$$ For infinitely large values of $k$, the asymptotics of the estimate would be $$x_{est}=1+\frac{1}{\log \left({k}\right)+\gamma +1}$$
For $k=1000$, the exact solution is $1.12955$ while the first approximation gives $1.11788$ and the second $1.11786$.
Using such estimates would make Newton method converging quite fast (shown below for $k=1000$).
$$\left( \begin{array}{cc} n & x_n \\ 0 & 1.117855442 \\ 1 & 1.129429575 \\ 2 & 1.129545489 \\ 3 & 1.129545500 \end{array} \right)$$
Edit | {
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Edit
We can obtain much better approximations if, instead of using a Taylor expansion of $g(x)$ to $O\left((x-1)^2\right)$, we build the simplest $[1,1]$ Padé approximant (which is equivalent to an $O\left((x-1)^3\right)$ Taylor expansion). This would lead to $$x=1+ \frac{6 (k+k (\psi ^{(0)}(k)+\gamma )-1)}{\pi ^2 k+6 (k+\gamma (\gamma k+k-2)-1)-6 k \psi ^{(1)}(k)+6 \psi ^{(0)}(k) (2 \gamma k+k+k \psi ^{(0)}(k)-2)}$$ Repeating the same calculations as above, the results are $$\left( \begin{array}{ccc} k & x_{est} & x_{sol} \\ 2 & 1.60000 & 1.58579 \\ 3 & 1.46429 & 1.46791 \\ 4 & 1.40435 & 1.41082 \\ 5 & 1.36900 & 1.37605 \\ 6 & 1.34504 & 1.35209 \\ 7 & 1.32741 & 1.33430 \\ 8 & 1.31371 & 1.32040 \\ 9 & 1.30266 & 1.30914 \\ 10 & 1.29348 & 1.29976 \\ 11 & 1.28569 & 1.29179 \\ 12 & 1.27897 & 1.28489 \\ 13 & 1.27308 & 1.27884 \\ 14 & 1.26787 & 1.27347 \\ 15 & 1.26320 & 1.26867 \\ 16 & 1.25899 & 1.26433 \\ 17 & 1.25516 & 1.26039 \\ 18 & 1.25166 & 1.25678 \\ 19 & 1.24844 & 1.25346 \\ 20 & 1.24547 & 1.25039 \end{array} \right)$$
For $k=1000$, this would give as an estimate $1.12829$ for an exact value of $1.12955$.
For infinitely large values of $k$, the asymptotics of the estimate would be $$x_{est}=1+\frac{6 (\log (k)+\gamma +1)}{6 \log (k) (\log (k)+2 \gamma +1)+\pi ^2+6 \gamma (1+\gamma )+6}$$
• Interesting! What does that notation $\psi^{(0)}(k)$ mean by the way? Aug 8, 2018 at 8:10
• @TheSimpliFire. This is "just" the digamma function. Aug 8, 2018 at 8:18
• @TheSimpliFire. We use to write $\psi^{(n)}(k)$ for the polygamma function and, in usual notations,$\psi^{(0)}(k)=\psi(k)$ Aug 8, 2018 at 8:22 | {
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# Probability of selecting elements with replacement
If I'm selecting $N$ elements uniformly at random (with replacement) from $\{1, \dots, N\}$, what is the chance that a given value is selected at least once?
What is the name for the distribution for the more general case where I'm selecting $N$ elements from a domain with $M$ elements?
• by random do you mean equally-probable ? – user451844 Sep 6 '17 at 23:31
• @Roddy Uniformly at random means equally-probable. The key word is "uniformly". – Trevor Gunn Sep 6 '17 at 23:51
• @TrevorGunn yeah I'm stupid sorry. – user451844 Sep 6 '17 at 23:55
## 3 Answers
Your answer is a particular case of the binomial distribution. Let getting a $1$ be a Success. You have $N$ independent trials. In the general version, the success probability is $p = 1/M.$
If $X \sim \mathsf{Binom}(N, 1/M),$ then you seek $$P(X > 0) = 1 - (1-p)^N = 1 - (1 - 1/M)^N = 1 - \left(\frac{M-1}{M}\right)^N,$$ as in the Answer of @ConMan (+1), which I hope you will Accept. [On each trial, you might say that you are using a discrete uniform distribution on the integers $1$ through $M$.]
Example: What is the probability you get at least one $6$ in ten rolls of a fair die? Then $X = \mathsf{Binom}(10, 1/6)$ is the number of $6$'s and you seek $P(X > 0) = P(X \ge 1) = 0.8385,$ computed in R statistical software as:
1 - dbinom(0, 10, 1/6)
## 0.8384944
In the figure below, you want the total heights of the bars to the right of the dotted red line.
• All the answers here were excellent, but I accepted this one since I mentioned binomial distribution and included a graphic, and because you fixed up my notation in the question. – BeeOnRope Sep 7 '17 at 17:26
If you're drawing with equal probability and subsequent draws are independent of each other, then the probability that you do not draw, for example, "1" in a single draw from $M$ elements is $\frac{M-1}{M}$. | {
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The probability that you do not draw any "1"s in $N$ draws is the product of the individual probabilities, i.e. $\frac{M-1}{M}\cdot\frac{M-1}{M}\cdot\ldots\cdot\frac{M-1}{M}=\left(\frac{M-1}{M}\right)^N = \left(1 - \frac{1}{M}\right)^N$.
The probability that you draw at least 1 "1" in the $N$ draws is the complement of the probability that you draw none, i.e. it is $1 - \left(1 - \frac{1}{M}\right)^N$.
The distribution for drawing $M$ elements with replacement from a set of $N$ elements is the uniform distribution on all $M$-tuples from $\{1,\dots,N\}$. For example, with $M = 3$ and $N = 2$ we get 8 ($=2^3$) $3$-tuples:
$$(1,1,1), (1,1,2), (1,2,1), (1,2,2), (2,1,1), (2,1,2), (2,2,1), (2,2,2)$$
and each is equally likely.
Viewed this way, the question of how likely it is to see a given value at least once comes down to counting those sequences that contain the value somewhere. We can count this by counting the tuples that do not contain that value. We know that there are $N^N$ total $N$-tuples and $(N - 1)^N$ tuples not containing a specified value. Thus there are
$$N^N - (N - 1)^N$$
tuples that contain a specified value. Since the distribution is uniform, the probability of selecting such a tuple is
$$\frac{N^N - (N - 1)^N}{N^N} = 1 - \left( 1 - \frac{1}{N} \right)^N.$$
Now let's say that we have the elements $\{1,\dots,N\}$, we draw $M$ of them and we want at least $P$ of them to equal $1$ (or any other fixed element). Then to count the number of $M$-tuples with $P$ $1$'s, first we select the $P$ slots to put the $1$'s in, then from the $M - P$ remaining slots, we have $N - 1$ options each. This gives
$$\binom{M}{P} (N - 1)^{M - P}.$$
Now we may divide by the total number of $M$-tuples ($N^M$) to get a probability of
$$\binom{M}{P} \frac{(N - 1)^{M - P}}{N^{M - P}} \cdot \frac{1}{N^P} = \binom{M}{P} \left( 1 - \frac{1}{N} \right)^{M - P} \left( \frac1N \right)^{P}.$$ | {
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This is the probability mass function of the binomial distribution. Namely $\mathsf{Bin}(M, \frac{1}{N})$. | {
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# Cross product in higher dimensions
Suppose we have a vector $$(a,b)$$ in $$2$$-space. Then the vector $$(-b,a)$$ is orthogonal to the one we started with. Furthermore, the function $$(a,b) \mapsto (-b,a)$$ is linear.
Suppose instead we have two vectors $$x$$ and $$y$$ in $$3$$-space. Then the cross product gives us a new vector $$x \times y$$ that's orthogonal to the first two. Furthermore, cross products are bilinear.
Question. Can we do this in higher dimensions? For example, is there a way of turning three vectors in $$4$$-space into a fourth vector, orthogonal to the others, in a trilinear way?
• You might want to look at the Gram Schmitt method here :en.wikipedia.org/wiki/Gram%E2%80%93Schmidt_process – Furrane Jul 25 '17 at 10:06
• This construction has nothing to do with cross product, just happen to coincide in dimension 3. – Miguel Jul 25 '17 at 10:54
• A search for "generalized cross product" turns up a number of questions likely to be of interest, including Is the vector cross product only defined for 3D? and Generalized Cross Product. ;) (Not marking as a duplicate because you're better able to judge which question, if any, most nearly matches yours.) – Andrew D. Hwang Jul 25 '17 at 17:38
• You might be interested in the notion of the orthogonal complement. It can give you the vector orthogonal to a given set of $n-1$ independent vectors in $n$-space, like you're asking for $n=4$. But it can also give you $k$ independent vectors orthogonal to a given set of $n-k$ independent vectors in $n$-space. So you can take two vectors in 4-space and find two vectors perpendicular to them and to each other. – YawarRaza7349 Jul 25 '17 at 17:39
• – user57159 Jul 26 '17 at 3:18 | {
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Yes. It is just like in dimension $$3$$: if your vectors are $$(t_1,t_2,t_3,t_4)$$, $$(u_1,u_2,u_3,u_4)$$, and $$(v_1,v_2,v_3,v_4)$$, compute the formal determinant:$$\begin{vmatrix}t_1&t_2&t_3&t_4\\u_1&u_2&u_3&u_4\\v_1&v_2&v_3&v_4\\e_1&e_2&e_3&e_4\end{vmatrix}.$$ You then see $$(e_1,e_2,e_3,e_4)$$ as the canonical basis of $$\mathbb{R}^4$$. Then the previous determinant is $$(\alpha_1,\alpha_2,\alpha_3,\alpha_4)$$ with\begin{align*}\alpha_1&=t_4u_3v_2-t_3u_4v_2-t_4u_2v_3+t_2u_4v_3+t_3u_2v_4-t_2u_3v_4\\\alpha_2&=-t_4u_3v_1+t_3u_4v_1+t_4u_1v_3-t_1u_4v_3-t_3u_1v_4+t_1u_3v_4\\\alpha_3&=t_4u_2v_1-t_2u_4v_1-t_4u_1v_2+t_1u_4v_2+t_2u_1v_4-t_1u_2v_4\\\alpha_4&=-t_3u_2v_1+t_2u_3v_1+t_3u_1v_2-t_1u_3v_2-t_2u_1v_3+t_1u_2v_3\end{align*}It's a vector orthogonal to the other three.
I followed a suggestion taken from the comments on this answer: to put the entries $$e_1$$, $$e_2$$, $$e_3$$, and $$e_4$$ at the bottom. It makes no difference in odd dimension, but it produces the natural sign in even dimension.
Following another suggestion, I would like to add this remark:$$\alpha_1=-\begin{vmatrix}t_2&t_3&t_4\\u_2&u_3&u_4\\v_2&v_3&v_4\end{vmatrix}\text{, }\alpha_2=\begin{vmatrix}t_1&t_3&t_4\\u_1&u_3&u_4\\v_1&v_3&v_4\end{vmatrix}\text{, }\alpha_3=-\begin{vmatrix}t_1&t_2&t_4\\u_1&u_2&u_4\\v_1&v_2&v_4\end{vmatrix}\text{ and }\alpha_4=\begin{vmatrix}t_1&t_2&t_3\\u_1&u_2&u_3\\v_1&v_2&v_3\\\end{vmatrix}.$$ | {
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• Very lucid and appreciable answer.thanks – Arpit Yadav Jul 25 '17 at 10:18
• I'm pretty sure the row of basis vectors should be on the bottom to get the right-handedness correct; only in odd dimensions can this row be moved to the top without a change in sign whilst keeping the vectors in the same order. – user332714 Jul 25 '17 at 18:06
• (+1) This is the method I've used in a few answers. The formula is easy to remember! – robjohn Jul 25 '17 at 22:50
• @lastresort Nice remark. I shall edit my answer taking that into account. – José Carlos Santos Jul 25 '17 at 22:54
• Is $$\alpha_1 ={\rm det} \left| \matrix{ t_2 & t_3 & t_4 \\ u_2 & u_3 & u_4 \\ v_2 & v_3 & v_4} \right|$$ and so on? – John Alexiou Jul 26 '17 at 14:19
My answer is in addition to José's and Antinous's answers but maybe somewhat more abstract. In principle, their answers are using coordinates, whereas I'm trying to do it coordinate-free.
What you are looking for is the wedge or exterior product. The exterior power $$\bigwedge^k(V)$$ of some vector space $$V$$ is the quotient of the tensor product $$\bigotimes^k(V)$$ by the relation $$v\otimes v$$. To be somewhat more concrete and less abstract, this just means that for any vector $$v\in V$$ the wedge product $$v\wedge v=0\in\bigwedge^2(V)$$. Whenever you wedge vectors together, the result equals zero if at least two of the factors are linearly dependent. Think of what happens to the cross product in $$\mathbb{R}^3$$.
In fact, let $$e_1,e_2,\ldots,e_n$$ be a basis of an inner product space $$V$$. Then $$e_{i_1}\wedge e_{i_2}\wedge \ldots \wedge e_{i_k}$$ is a basis for $$\bigwedge^k(V)$$ where $$1\leq i_1 < i_2 < \ldots < i_k\leq n$$. | {
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If $$V=\mathbb{R}^3$$ then $$v \wedge w$$ equals $$v \times w$$ up to signs of the entries. This seems a bit obscure because technically $$v\wedge w$$ should be an element of $$\bigwedge^2(\mathbb{R}^3)$$. However, the latter vector space is isomorphic to $$\mathbb{R}^3$$. In fact, this relation is true for all exterior powers given an orientation on the vector space. The isomorphism is called the Hodge star operator. It says that there is an isomorphism $$\star\colon\bigwedge^{n-k}(V)\to\bigwedge^{k}(V)$$. This map operates on a $$(n-k)$$-wedge $$\beta$$ via the relation $$\alpha \wedge \beta = \langle \alpha,\star\beta \rangle \,\omega$$ where $$\alpha\in\bigwedge^{k}(V)$$, $$\omega\in\bigwedge^n(V)$$ is an orientation form on $$V$$ and $$\langle \cdot,\cdot \rangle$$ is the induced inner product on $$\bigwedge^{k}(V)$$ (see wiki). Notice that the wiki-page defines the relation the other way around. | {
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How does all this answer your question you ask? Well, let us take $$k=1$$ and $$V=\mathbb{R}^n$$. Then the Hodge star isomorphism identifies the spaces $$\bigwedge^{n-1}(\mathbb{R}^n)$$ and $$\bigwedge^{1}(\mathbb{R}^n)=\mathbb{R}^n$$. This is good because you originally wanted to say something about orthogonality between a set of $$n-1$$ linearly indepedent vectors $$v_1,v_2,\ldots,v_{n-1}$$ and their "cross product". Now let us exactly do that and set $$\beta :=v_1 \wedge v_2 \wedge \ldots \wedge v_{n-1}\in\bigwedge^{n-1}(\mathbb{R}^n)$$. Then the image $$\star\beta = \star(v_1 \wedge v_2 \wedge \ldots \wedge v_{n-1})$$ is a regular vector in $$\mathbb{R}^n$$ and the defining condition above implies for $$\alpha=v_i\in\mathbb{R}^n=\bigwedge^{1}(\mathbb{R}^n)$$ $$v_i \wedge (v_1 \wedge v_2 \wedge \ldots \wedge v_{n-1}) = \alpha \wedge \beta = \langle \alpha,\star\beta \rangle \,\omega = \langle v_i,\star\beta \rangle \,\omega.$$ However, the left hand side equals zero for $$i=1,2,\ldots,n-1$$, so that the vector $$\star\beta$$ is orthogonal to all vectors $$v_1,v_2,\ldots,v_{n-1}$$ which is what you asked for. So you might want to define the cross product of $$n-1$$ vectors as $$v_1 \times v_2 \times \ldots \times v_{n-1} := \star(v_1 \wedge v_2 \wedge \ldots \wedge v_{n-1})$$.
Maybe keep in mind that the other two answers implicitly use the Hodge star operation (and also a basis) to compute the "cross product in higher dimension" through the formal determinant which is encoded in the use of the wedge product here. | {
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• So concretely, how do we actually know what the hodge star of a $k$-blade is? For example, work in $4$-space with the standard orientation. Suppose we want to know $\star(v_1 \wedge v_3).$ If I understand correctly, it's either $v_2 \wedge v_4$ or else $v_4 \wedge v_2$. How do we know which one? – goblin GONE Jul 25 '17 at 13:32
• It depends on the orientation that you choose for your vector space. Let's say $v_1,v_2,v_3,v_4$ form an oriented basis for $V$ (that is, $\omega = v_1\wedge v_2\wedge v_3\wedge v_4$) then $\star(v_1\wedge v_3)=v_4\wedge v_2$. This can be seen using the defining relation for $\alpha=v_i\wedge v_j$ cycling through all possible combinations $(i,j)$. This is what they say on the wiki-page linked above in the section "Computation of the Hodge star" albeit expressed a little bit complicated in my opinion. – Sven Pistre Jul 25 '17 at 14:25
• Of all combinations $(i,j)$ only $(2,4)$ and $(4,2)$ remain (because otherwise the left hand side equals zero). Then you assume $\star(v_1\wedge v_3)=v_k\wedge v_l$ and think about which combinations for $(k,l)$ remain on the right hand side of the def. relation. Then you will see that the only possible one is $(k,l)=(4,2)$. To see the last part, look at the definition of the induced scalar product on $\bigwedge^2(V)$. – Sven Pistre Jul 25 '17 at 14:28
• And also, as I forgot to mention this, $v_2\wedge v_4=-v_4\wedge v_2$. Changing the position of two vectors in a $k$-wedge just changes the sign. So really it only depends on the chosen orientation (or "right-handedness") of your vector space. – Sven Pistre Jul 25 '17 at 14:49
• @étale-cohomology The hodge star depends on a choice of inner product and orientation. So it is not canonical. I don't think that you can identify them canonically. – Sven Pistre Jul 25 '17 at 17:42
You can work out the cross product $p$ in $n$-dimensions using the following: | {
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You can work out the cross product $p$ in $n$-dimensions using the following:
$$p=\det\left(\begin{array}{lllll}e_1&x_1&y_1&\cdots&z_1\\e_2&x_2&y_2&\cdots&z_2\\\vdots&\vdots&\vdots&\ddots&\vdots\\e_n&x_n&y_n&\cdots&z_n\end{array}\right),$$ where $\det$ is the formal determinant of the matrix, the $e_i$ are the base vectors (e.g. $\hat{i},\hat{j},\hat{k}$, etc), and $x,y,\ldots,z$ are the $n-1$ vectors you wish to "cross".
You will find that $x\cdot p=y\cdot p=\cdots=z\cdot p=0$.
It's wonderful the determinant produces a vector with this property.
• Are there any requirements on the basis vectors $e_1, ..., e_n$? Like, do they need to form an orthonormal basis, or something? – étale-cohomology Jul 25 '17 at 17:01
• Yeah but do they have to be? Can't we take any other basis? – étale-cohomology Jul 25 '17 at 20:35
• By changing the basis to $\widetilde e_i$ you will have to change the vector entries to the coefficients $\widetilde x_i$ in the basis expansion for the new basis. Remember that the above is only a formal determinant as this is not actually a matrix (since the first column consists of entries that are vectors themselves). So it does not matter if the basis is orthonormal or not but you will have to adjust your formal determinant formula. – Sven Pistre Jul 25 '17 at 21:48
• (+1) This is the logical extension of José Carlos Santos' answer to $\mathbb{R}^n$ (at first, this is what I thought he had given, but now I see his only covers $\mathbb{R}^4$). – robjohn Jul 25 '17 at 22:53
• @robjohn I actually posted my answer 2 minutes before he did :-) – Pixel Jul 25 '17 at 23:00
Yes, and apart from other answers an interesting approach to think about it is using Clifford's algebra.
This can introduce you the basic concept in a nonrigorous but approachable manner. | {
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This can introduce you the basic concept in a nonrigorous but approachable manner.
• Thank you for your answer, however that article is extremely long and it's difficult to find a Clifford-style answer to my question by reading through it. Can I ask you to write up some details on how to compute an actual cross product using Clifford approach's? – goblin GONE Jul 26 '17 at 8:36 | {
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# Finding the interval of values of an unknown element such that a square matrix is positive semidefinite
I'm looking for a way to determine which values of $$c \in \mathbb{R}$$ make the matrix $$A$$ positive semidefinite:
$$A = \begin{bmatrix} 1 & c \\ c & 1 \end{bmatrix}$$
I'm looking for a way to determine which values of c∈R make the matrix A positive semidefinite:
By definition, A matrix is called positive semidefinite if it is symmetric and all its eigenvalues are non-negative. Hence
Clear["Global*"]
mat = {{1, c}, {c, 1}}
If[SymmetricMatrixQ[mat],
eigv = Eigenvalues[mat];
Reduce[eigv[[1]] >= 0 && eigv[[2]] >= 0, c]
]
So any value between -1 and 1 will do. The above code ofcourse assumes the matrix is 2 by 2, but it can easily be made more general.
• Congrats on 100K. Dec 6 '21 at 15:53
cp = CharacteristicPolynomial[{{1, c}, {c, 1}}, t]
(* Out[142]= 1 - c^2 - 2 t + t^2 *)
There are two important things to check. (i) Where do roots cross the y axis? (ii) Where might roots be complex-valued?
For (i) we just solve for c under the assumption thatt==0
Solve[cp == 0 && t == 0, {c, t}]
(* Out[143]= {{c -> -1, t -> 0}, {c -> 1, t -> 0}} *)
This splits the real line for c into three regions abd simply plugging in a value in each region will show that both solutions are nonnegative in the finite part.
For (ii) We check where the discriminant with respect to t vanishes. This will give conditions on c.
Discriminant[cp, t]
(* Out[144]= 4 c^2 *)
So it only vanishes at the origin. We already know that on neither side do we have complex valued roots in t. So there are no further restrictions.
An advantage to this approach is that it extends to higher dimensions.
If you compute the Cholesky decomposition of the matrix, and simplify it under the assumptions you gave:
Simplify[CholeskyDecomposition[{{1, c}, {c, 1}}], c ∈ Reals]
{{1, c}, {0, Sqrt[1 - c^2]}} | {
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Simplify[CholeskyDecomposition[{{1, c}, {c, 1}}], c ∈ Reals]
{{1, c}, {0, Sqrt[1 - c^2]}}
you can immediately read off the condition that Sqrt[1 - c^2] >= 0 for positive-semidefiniteness. Thus,
Reduce[%[[-1, -1]] >= 0, c, Reals]
-1 <= c <= 1
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# Floor function algebra question
I don't know how to put floor functions in but...
Solve $$\dfrac{19x + 16}{10} = \left \lfloor \dfrac{4x+7}{3}\right \rfloor$$
I have so far worked out that the RHS can either be $$(4x+7)/3 - 0.33$$, $$(4x+7)/3 - 0.67$$ or itself. When I solve for each of these three equations, I get $$x=12/17, 22/17, 2/17$$. From there, I subbed $$x$$ back into the equation to try and see which one works but none did. Can I have some help?
• You can get the floor functions with \lfloor and \rfloor, e.g., \lfloor \pi \rfloor = 3. If you need bigger ones, as around a fraction, for instance, use \left\lfloor and \right\rfloor, e.g., \left\lfloor\frac{4x+7}{3}\right\rfloor. May 21 '20 at 3:24
• You seem to be assuming that $4x+7$ is an integer. There's no reason to think it is. May 21 '20 at 3:37
• X must be of the form $$10{\alpha}+6;\alpha \inf \mathbb{Z}$$ May 21 '20 at 4:53
• @saulspatz so would it be safe to say that the RHS is in the range of $(4x+7)/3$ to $(4x+7)/3-0.99$ ?
– user377742
May 21 '20 at 8:52
• Look at my answer. May 21 '20 at 16:09
I'll get you started.
We know that $$x-1<\lfloor x\rfloor<=x$$, so we have $$\frac{4x+4}3<\frac{19x+16}{10}\leq\frac{4x+7}3\\ 40x+40<57x+48\leq40x+70\\ \frac{-8}{17} so that $$x=n+\varepsilon$$ where $$n\in\{-1,0,1\}$$ and $$0\leq\varepsilon<1$$.
Now we can test each of the three possibilities for $$n$$ separately. Suppose $$x=1+\varepsilon$$. Then $$\frac{19x+16}{10}=\frac{35+19\varepsilon}{10}$$ is an integer between $$3.5$$ and $$5.4$$ so there are only two possibilities for $$\varepsilon$$. Check these to see if $$x=1+\varepsilon$$ satisfies the equation. Repeat the process for $$n=0$$ and $$n=-1$$.
Since algebra with floor functions is rarely nice, I like to graph it if possible to visualize the solutions. In your case, this is the graph of $$\frac{19x+16}{10}-\Big\lfloor \frac{4x+7}{3}\Big\rfloor$$ | {
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(This is using Desmos by the way) Notice how any solutions would pass through the $$y=0$$, and there seem to be $$4$$. One seems to occur at $$x \approx -.3158$$ and by plugging this into the floor function we see it approaches $$1.912$$, the floor of which is $$1$$. So we are looking for $$x$$ such that $$\frac{19x+16}{10} = 1$$. The solution is $$x = \frac{-6}{19}$$ which, by substituting it back in, works. Similarly, we can show that we need $$x \approx .2105$$ such that $$\frac{19x+16}{10} = 2$$, giving the solution $$x = \frac{4}{19}$$. Then we need $$x \approx .7368$$ such that $$\frac{19x+16}{10} = 3$$, giving the solution $$x = \frac{14}{19}$$. Finally, we need $$x \approx 1.2632$$ such that $$\frac{19x+16}{10} = 4$$, giving the solution $$x = \frac{24}{19}$$. From here, you can use strict inequalities to prove that there are no more solutions.
$$\dfrac{19x + 16}{10} = \left \lfloor \dfrac{4x+7}{3}\right \rfloor$$
Let $$\dfrac{19x + 16}{10} = n \in \mathbb Z$$.
Then $$x = \dfrac{10n-16}{19}$$
and $$\dfrac{4x+7}{3} = \dfrac{40n+69}{57}$$.
So
$$n \le \dfrac{40n+69}{57} < n + 1$$
$$57n \le 40n+69 < 57n + 57$$
$$0 \le -17n+69 < 57$$
$$-69 \le -17n < -12$$
$$\dfrac{12}{17} < n \le 4\dfrac{1}{17}$$
So now you can find the values of $$n$$ and then the values of $$x$$.
First deal with the fact that $$\dfrac{19x + 16}{10}$$ is an integer $$k$$.
So $$19x = 10k -16$$ is and integer $$m$$ and $$x = \frac m{19}$$ for some integer $$m$$.
Now deal with $$\dfrac{19x + 16}{10}=\lfloor \dfrac{4x+7}{3} \rfloor$$ so
$$\dfrac{19x + 16}{10} \le \dfrac{4x+7}{3}< \dfrac{19x + 16}{10}+1=\dfrac{19x +26}{10}$$
Replace $$x$$ with $$\frac m{19}$$ and
$$\dfrac {m + 16}{10} \le \dfrac {\frac 4{19}m + 7}3 < \frac {m+26}{10}$$
$$3m + 48 \le \frac 40{19}m +70 < 3m + 78$$
$$-22 \le \frac {40}{19}m - 3m < 8$$
$$-22*19 \le 40m - 57m = -17*m < 8*19$$
$$\frac {-8*19}{17} < m \le \frac {22*19}{17}$$
$$-8 \frac {16}{17} < m \le 24 \frac {10}{17}$$
So $$-8 < m \le 24$$ | {
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$$-8 \frac {16}{17} < m \le 24 \frac {10}{17}$$
So $$-8 < m \le 24$$
But $$m = 10k-16$$ so $$m\equiv -16\equiv -6 \equiv 4 \pmod{10}$$ so so
$$m =-6,4,14,24$$ are acceptable values.
And $$x =-\frac 6{19},\frac 4{19}, \frac {14}{19}$$ and $$\frac {24}{29}$$ are acceptable answers. | {
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# How does ListInterpolation work?
ListInterpolation takes an array of data and interpolates between the entries. Let us interpolate between the elements of the identity matrix. For reference, this is what the identity matrix looks like:
MatrixPlot[IdentityMatrix[5]]
Now let's interpolate, and request first order interpolation:
if = ListInterpolation[IdentityMatrix[5], {{0, 1}, {0, 1}}, InterpolationOrder -> 1];
Plot[if[x, x], {x, 0, 1}]
The result is a piecewise function as expected, but not a piecewise linear function! Why? What does first order interpolation mean here?
The kind of linear interpolation I am familiar with is what ListDensityPlot does:
ListDensityPlot[IdentityMatrix[5], DataRange -> {{0, 1}, {0, 1}}, Mesh -> All]
Construct a Delaunay triangulation of the data and interpolate on each triangle.
This is not what ListInterpolation does (or even Interpolation when the data is strictly on a square grid). What does ListInterpolation do, then?
Inspired by this question: | {
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Inspired by this question:
• Well, it is doing bilinear interpolation in your first example, I would think. You definitely will get something with quadratic pieces if you sample across diagonals. – J. M. is in limbo Jun 15 '16 at 14:00
• @J.M. Well, we all have to learn at some point :-) Yes, you are right, and this is due to my ignorance of math. But I won't delete it. If you convert that to a one-line answer, I'll accept it. – Szabolcs Jun 15 '16 at 14:01
• Docs say the only methods available to ListInterpolation are Spline and Hermite – N.J.Evans Jun 15 '16 at 14:02
• Currently my comment is not that pedagogically useful; I'll try to come up with a sufficiently illuminating example first, unless somebody beats me to it. :) – J. M. is in limbo Jun 15 '16 at 14:04
• Szabolcs, please consider leaving the question up for a little longer. I was not aware of that behavior either, and I would find it instructive to see an interesting example, if @J.M. or others have the time / inclination. – MarcoB Jun 15 '16 at 14:43
I'll just treat the simplest case of interpolating across four noncoplanar points. The extension to general grids is straightforward (one just joins multiple pieces appropriately).
Consider the following bivariate interpolating polynomial:
ip[x_, y_] = InterpolatingPolynomial[{{{0, 0}, 1}, {{0, 1}, -1/2},
{{2, 0}, 1/2}, {{2, 1}, 0}}, {x, y}];
and the following InterpolatingFunction[]:
if = ListInterpolation[{{1, -1/2}, {1/2, 0}}, {{0, 2}, {0, 1}}, InterpolationOrder -> 1];
They are the same:
Plot3D[ip[x, y] - if[x, y] // Chop, {x, 0, 2}, {y, 0, 1}]
Let's have a look at what ip[] looks like:
Expand[ip[x, y]]
1 - x/4 - (3 y)/2 + (x y)/2
Huh. That last term certainly isn't linear. What gives? | {
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Huh. That last term certainly isn't linear. What gives?
In a bilinear interpolation such as this one, although the procedure is something along the lines of "linearly interpolate for each grid line in one direction, and then linearly interpolate those results", the result is inevitably quadratic. This should come as no surprise to people familiar with the hyperbolic paraboloid, which is one of the simplest examples of a ruled surface, or a surface formed by sweeping out a line in space. In fact, z == 1 - x/4 - (3 y)/2 + (x y)/2 is indeed a hyperbolic paraboloid.
{Plot3D[ip[x, y], {x, 0, 2}, {y, 0, 1}, MeshFunctions -> {#1 - #2 &}],
Plot[ip[x, x], {x, 0, 1}]} // GraphicsRow | {
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# How to find out the global minimum of the following expression
What is the global minimum of the expression \begin{align} |x-1| &+ |x-2|+|x-5|+|x-6|+|x-8|+|x-9|+|x- 10| \\&+ |x-11|+|x-12|+|x-17|+|x-24|+|x-31|+ |x-32|? \end{align}
I've solved questions of this sort before but there were only 3 terms. I solved those by expanding all the terms in the modulus and drawing a graph. This question came in a paper which requires the student to solve it within 5 minutes. What's a better method?
Unfortunately I needed some minutes to think about the problem before finding a solution that can be calculated very quickly:
Imagine the graph of the function $$f_a(x)=|x-a|$$. Having the graph in mind you see that the derivation $$f'(x)=-1$$ for $$x and $$f'(x)=1$$ for $$x>a$$.
For the intervals: $$(-\infty,1)$$, $$(1,2)$$, $$(2,5)$$, ..., $$(32,\infty)$$ we can now easily calculate the derivative $$f'(x)=f'_1(x)+f'_2(x)+f'_5(x)+...-f'_{32}(x)$$:
In the range $$(-\infty,1)$$ it is $$f'(x)=-1-1-1-...-1+1=-11$$.
In the range $$(1,2)$$ it is $$f'(x)=+1-1-1-...-1+1=-9$$.
In the range $$(2,5)$$ it is $$f'(x)=+1+1+1-...-1+1=-7$$.
...
In every step we simply have to invert one sign so "-1" becomes "+1". This means the derivative is changing by 2 at the points x=1,2,5,...
We start by calculating the derivative for $$x<1$$; it is -11.
Now we simply go through the ranges:
<1: -11
1..2: -9
2..5: -7
5..6: -5
6..8: -3
8..9: -1
9..10: +1
10..11: +3
11..12: +5
12..17: +7
17..24: +9
24..31: +11
31..32: +13
>32: +11
At $$x=32$$ the derivation decreases by 2 because of the minus sign before $$|x-32|$$; you could of course adapt this method for sums of elements of the form $$b|x-a|$$. | {
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We see that for $$x<9$$ the derivation is negative and for $$x>9$$ the derivation is positive. We also know that the function is continuous. (This is important because the derivation is not defined at x=1,2,5,...) This means that the function is strictly decreasing respectively increasing for $$x<9$$ and for $$x>9$$.
So we know that the global minimum must be at $$x=9$$.
• The right answer. Congratulations! Although I don't know if your method works when the number of terms is even, say $f(x) = |x-1|+ |x-2|$ or $f(x) = |x-11|+|x-12|+|x-17|+|x-24|$ (the minimum or maximum is an interval). Nov 25 '18 at 22:20
• @David Why should it be a problem if a whole interval represents the minimum or maximum? In the example of $|x-1|+|x-2|$ we would have $f'(x)=0$ in the whole interval 1...2. Nov 26 '18 at 6:28
• This answers the previous version of my typo question but the method is absolutely wonderful. Thanks!
– user619072
Nov 27 '18 at 17:36
You can in principle write out the function in a lot of intervals. But it would probably take too long. However, I will use this fact, without doing it explicitly. We know that if we do write this function, it is going to be linear on each interval (sum of linear functions is a linear function), and it's going to be continuous (sum of continuous functions is a continuous function). We also know that on a line you get minimum at one end, the other, or both (constant line). So all you need to do is to calculate your function at $$1,2,5,6,...$$ and find the minimum.
• I like how this answer asks us to imagine the graph of the function. I find this "imagine" technique useful, while still being intuitive. Similar examples includes "imagine the multiplication table of this finite group" (the group could be huge) or "imagine the truth table for this proposition" (the formula could be long). Nov 25 '18 at 22:02
The answer (minimizer) in this case is $$10$$, the median of the sequence $$(1,2,5,6,8,9,10,11,12,17,24,31,32).$$ | {
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You can plug in $$x=10$$ in the function and you would find that the minimum value is $$96$$. In general, the solution to the following minimization problem
$$\min\{|x-a_1| + |x-a_2| + \cdots + |x-a_n|\}$$ is the median of $$(a_1,\ldots,a_n)$$. To see why, consider first when $$n=2$$, and without loss of generality assume $$a_1. Then $$|x-a_1|+|x-a_2|$$ is the distance between $$x$$ and $$a_1$$ plus the distance between $$x$$ and $$a_2$$. It is easy to see that only when $$x$$ is in the middle of $$a_1$$ and $$a_2$$ should the sum of distances be minimal, which equals $$|a_2-a_1|$$ in this case. In this case the minimizer is not unique. Any points in $$[a_1,a_2]$$ is a minimizer.
When $$n=3$$, the function is $$|x-a_1|+|x-a_2|+|x-a_3|$$, and we order the parameters again so that $$a_1. When $$x$$ coincides with $$a_2$$, i.e. $$x=a_2$$, the value becomes $$|a_2-a_1|+|a_2-a_3|=|a_3-a_1|$$, the distance between $$a_3$$ and $$a_1$$. But when $$x\in[a_1,a_3], x\neq a_2$$, the value of the function is $$|x-a_2|+|x-a_1|+|x-a_3| = |a_3-a_1| + |x-a_2|,$$ which is largar than $$|a_3-a_1|$$, the distance between $$a_3$$ and $$a_1$$. Similarly the value would become larger when $$x$$ is outside $$[a_1,a_3]$$. So in this case, the minimizer is unique and is equal to $$a_2$$, the median of $$(a_1,a_2,a_3)$$.
In general, when $$n$$ is odd, there exists a unique minimizer, which is equal to the (unique) median of the parameters $$(a_1,\ldots,a_n)$$. When $$n$$ is even, the function is minimal and constant over the range $$[a_i,a_j]$$, where $$a_i$$ and $$a_j$$ are the two middle values. | {
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• That doesn't seem to be correct. Wolfram says the minimum is 51 at x=9. See here Nov 25 '18 at 20:50
• I see now that the confusion comes from the $−|x−32|$ in the problem statement @MartinRosenau. Note the minus sign. Fei Li solved it for a + sign. Nov 25 '18 at 21:18
• I plot the graph using wxmaxima and get the same result as @IlikeSerena. Note the minus sign at the end. Nov 25 '18 at 21:48
• Maybe the OP @CaptainQuestion made a typo. We need his or her classification, but I think the purpose of this question is exactly what I have demonstrated. Nov 25 '18 at 22:00
• @AlexVong So I hoped, but the next sentence "which equals $|a_2-a_1|$ in this case." excludes this interpretation. Nov 25 '18 at 22:39
TL;DR: Put the absolute values in ascending order, and look at the sum of the leading coefficients. One by one, change the signs in the sum from right to left. When the sum changes signs, you have passed a local extremum. When the sum equals zero, there is an extremum on an entire interval.
As an alternative to Andrei's excellent answer, or perhaps an extension, you could also look at the derivative. Clearly the function is continuous everywhere, and it is differentiable in all but finitely many points, call them $$a_1,\ldots,a_n$$ in ascending order. Then we want to minimize $$f(x)=\sum_{k=1}^nc_k|x-a_k|=\sum_{k=1}^n(-1)^{\delta_{x\leq a_k}}c_k(x-a_k),$$ where $$\delta$$ denotes the Kronecker delta, in this case defined as $$\delta_{x\leq a_k}:=\left\{\begin{array}{ll}1&\text{ if } x\leq a_k\\0&\text{ otherwise}\end{array}\right..$$ This has derivative (for all $$x$$ apart from the $$a_k$$) $$f'(x)=\sum_{k=1}^n(-1)^{\delta_{x\leq a_k}}c_k.$$ The expression has a local minimum at $$x$$ if either $$f'(x)=0$$, or if $$x=a_k$$ for some $$k$$ and $$f'(y)<0$$ for $$a_{k-1} and $$f'(y)>0$$ for $$a_k. | {
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This is all rather formal; in practice this means you put the $$c_k$$ in ascending order, so here $$n=13$$ and $$c_1=\cdots=c_{12}=1$$ and $$c_{13}=-1$$, and find all $$m$$ such that flipping the last $$m$$ signs in the sum $$c_1+c_2+c_3+c_4+c_5+c_6+c_7+c_8+c_9+c_{10}+c_{11}+c_{12}+c_{13}$$ makes the sum change signs compared to changing the last $$m-1$$ signs. Here a quick look gives $$c_1+c_2+c_3+c_4+c_5+c_6-c_7-c_8-c_9-c_{10}-c_{11}-c_{12}-c_{13}=1,$$ $$c_1+c_2+c_3+c_4+c_5-c_6-c_7-c_8-c_9-c_{10}-c_{11}-c_{12}-c_{13}=-1,$$ so $$f$$ has a local minimum at $$a_6=9$$, and it is not difficult to see that there is no other minimum. | {
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# A mistake on computing $\int \frac{dx}{\sqrt{x+1}+1}$
I have to find $$\int \frac{dx}{\sqrt{x+1}+1}$$. This was my attempt, which is wrong and I cannot find where exactly is the mistake.
First I write $$\frac{1}{\sqrt{x+1}+1}=\frac{\sqrt{x+1}-1}{x}=\frac{\sqrt{x+1}}{x}-\frac{1}{x}$$, therefore $$\int \frac{dx}{\sqrt{x+1}+1}=\int \frac{\sqrt{x+1}}{x}dx-\log\left (x\right )$$, so I only have to deal with $$\int \frac{\sqrt{x+1}}{x}dx$$.
Setting $$u=\sqrt{x+1}$$, we obtain $$du=\frac{1}{2\sqrt{x+1}}dx$$, therefore $$dx=2udu$$ and $$x=u^2-1$$, so
$$\int\frac{\sqrt{x+1}}{x}dx=\int\frac{2u^2}{u^2-1}du=\int \left(2+\frac{2}{u^2-1}\right)du=2\sqrt{x+1}+\int \left(\frac{1}{u-1}-\frac{1}{u+1}\right)du=$$
$$=2\sqrt{x+1}+\log \left (\sqrt{x+1}-1\right )-\log \left (\sqrt{x+1}+1\right )=2\sqrt{x+1}+\log\left ( \frac{\sqrt{x+1}-1}{\sqrt{x+1}+1} \right )$$
But then $$\int \frac{dx}{\sqrt{x+1}+1}=2\sqrt{x+1}+\log\left ( \frac{\sqrt{x+1}-1}{\sqrt{x+1}+1} \right )-\log\left (x\right )=2\sqrt{x+1}+\log\left ( \frac{x+2-2\sqrt{x+1}}{x^2} \right )$$, which is not what I am supposed to obtain. The actual answer is $$2\sqrt{x+1}-2\log\left ( 1+\sqrt{x+1} \right )$$.
Where is my mistake?
(I already know a correct way to solve it, I just want to know where I am committing a mistake).
• I believe that you went wrong in your first step. That “simplification” made everything a lot harder. Hint: Start with a u-substitution if u=x+1 . – Sina Babaei Zadeh Mar 2 '19 at 22:36
• The standard method suggestst to set $u=\sqrt{x+1}$, i.e. $x=u^2-1,\; u\ge 0$, from the very beginning. – Bernard Mar 2 '19 at 22:44
But you have been right all along!
Notice that $$\log\frac{\sqrt{x+1}-1}{\sqrt{x+1}+1} - \log x = \log\frac{\sqrt{x+1}-1}{x(\sqrt{x+1}+1)} = \log \frac{1}{(\sqrt{x+1}+1)^2} = -2\log(\sqrt{x+1}+1)$$ by your very first step.
Also, I would suggest that you use $$\int\frac{dx}{x} = \log|x|$$ (with the absolute value sign, to extend the domain). | {
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What I would also suggest is to use an online function grapher if you are not sure that your solution is correct. If two functions have identical graphs or if they differ by a constant value, then both are correct solutions of the integral.
• Oh, I see... My answer was correct, but it was not clear why. I wrote $\log\left ( \frac{\sqrt{x+1}-1}{\sqrt{x+1}+1} \right )=\log\left ( \frac{x+2-2\sqrt{x+1}}{x} \right )$ (i.e., i was rationalizing the denominator), but I should have rationalized the numerator, leading to $\log\left ( \frac{x}{\left ( \sqrt{x+1}+1 \right )^2} \right )$, which gives the solution. Thank you! – solomeo paredes Mar 2 '19 at 23:04
Note that $$\ln\frac{x+2-2\sqrt{x+1}}{x^2}=\ln\frac{(\sqrt{x+1}-1)^2}{x^2}=2\ln\frac{\sqrt{x+1}-1}{x}$$ if we further simplify, $$2\ln(\frac{\sqrt{x+1}-1}{x}\cdot\frac{\sqrt{x+1}+1}{\sqrt{x+1}+1}) =2\ln\frac{1}{\sqrt{x+1}+1}=-2\ln(\sqrt{x+1}+1)$$ You did nothing wrong.
Note that $$\frac{x+2-2\sqrt{x+1}}{x^2} =\frac {(x+2)^2-4(x+1)} {x+2+2\sqrt{x+1}} \cdot\frac 1{x^2}\ ,$$ and this has nothing to do with $$\frac{\sqrt{x+1}-1}{\sqrt{x+1}+1} \cdot \frac 1x = \frac{(\sqrt{x+1}-1)(\sqrt{x+1}+1)}{(\sqrt{x+1}+1)(\sqrt{x+1}+1)} \cdot \frac 1x = \frac 1{(\sqrt{x+1}+1)^2}\ ,$$ which leads to the solution. So the last step is the bad one.
You're fine.
$$\log\left(\frac{\sqrt{x+1}-1}{\sqrt{x+1}+1}\right) -\log x$$
$$= \log\left(\frac{\sqrt{x+1}-1}{\sqrt{x+1}+1}\frac{\sqrt{x+1}+1}{\sqrt{x+1}+1}\right) -\log x$$
$$=\log\left( \frac{x}{ (\sqrt{x+1}+1)^2}\right) -\log x$$
$$=\log x -2\log(\sqrt{x+1}+1)- \log x$$
$$=-2\log(\sqrt{x+1}+1).$$
No mistake! $$log((\frac{\sqrt{x+1}-1}{x})^2)=-2log(\frac{x}{\sqrt{x+1}-1})=-2log(\frac{x(\sqrt{x+1}+1)}{(\sqrt{x+1}-1)(\sqrt{x+1}+1)})=-2log\frac{x(\sqrt{x+1}+1)}{x}$$ | {
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## Graph Of Sine And Cosine Functions Real World Applications | {
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The cosine is known as an even function, and the sine is known as an odd function. Identities and equations do not include any half-angle relationships. Objectives: To graph the sine and cosine functions; to identify the graphs of the sine and cosine functions. The amplitude is 2. Are you looking to apply what you learn in the classroom to the real world? Trigonometry is a subject that has lots of practical applications. Markov processes are widely used in performance modelling of wireless and mobile communication systems. Frequency is the number of cycles in a given unit of time, so it is the reciprocal of the period of a function. Whenever you see an "oscilloscope," for example when you play music using certain programs on a computer, you're really seeing a whole bunch of sine waves added together. In that program you used the built-in sine and cosine functions. Frequency is the number of cycles in a given unit of time. Fortunately, we can analyze the problem by first representing it as a linear function and then interpreting the components of the function. 762-768) optional 2 optional 1 • Graph trigonometric functions. So the graph looks like a very simple wave. One method to write a sine or cosine function that models a sinusoid is to fi nd the values of a, b, h, and k for y = a sin b(x − h) + k or y = a cos b(x − h) + k where ∣ a ∣ is the amplitude, — 2π is the period. The inverses of sine, cosine and tangent functions are not functions unless the domains are limited. Request PDF | A Discrete Sine Cosine Algorithm for Community Detection | As an important research area in complex networks, community detection problems (CDPs) have aroused widespread concern. Find an expression for a volume. Evaluate Inverse Trigonometric Functions v. We often graph sine over time (so we don't write over ourselves) and sometimes the "thing" doing sine is also moving, but this is optional! A spring in one dimension is a perfectly happy sine wave. acute triangles using the sine law | {
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A spring in one dimension is a perfectly happy sine wave. acute triangles using the sine law and the cosine law, including problems arising from real-world applications; demonstrate an understanding of periodic relationships and the sine function, and make connections between the numeric, graphical, and algebraic representations of sine functions; identify and represent sine functions, and. Using your guidelines, you should be able to sketch in a smooth curve: see figure 3. FIND the coordinates of A & C. The graphs of sine, cosine, and tangent. ) like any other function with a domain or real numbers!. • Solve real world and mathematical problems using a system of two first-degree equations in two variables. First, take a look the Taylor series representation of exponential function, and trigonometric functions, sine, and cosine,. The vertical shift is 2. Find an equation for a sine function that has amplitude of 4 a period of π. 9) Graph sine and cosine functions (PC-N. THE COSINE FUNCTION You can graph the cosine function y = cos x. Trig Functions: The Ferris Wheel Applications of Trig Functions - Duration: 13:37. txt) or view presentation slides online. 25 scaffolded questions on equation graph involving amplitude and periodplus model problems explained step by step. Find Period and Amplitude 2. Whenever you see an "oscilloscope," for example when you play music using certain programs on a computer, you're really seeing a whole bunch of sine waves added together. Determine the values for angles between 0° and 360°. Trigonometry An In Depth Approach to Sine, Cosine, Tangent, Cotangent, Secant and Cosecant: Trigonometry Ratios and Their Graphs and Real World Applications. Recall that the sine and cosine functions relate real number values to the x– and y-coordinates of a point on the unit circle. Lesson 9-11: Graphing Trigonometric Functions and Applications in Real World Contexts Learning Goals #8, 9, 10: How do I use the critical values of the Sine and Cosine | {
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World Contexts Learning Goals #8, 9, 10: How do I use the critical values of the Sine and Cosine curve to assist in in graphing Sine and Cosine functions? How do I contextualize Trig Functions in real life situations? Warm-Up! WITH A PARTNER. I can identify the important features (amplitude, period, max, min, x-intercepts, and frequency) of the sine, cosine, and tangent functions. Sinusoidal functions graph wave forms. Graphs of the sine and cosine functions; and Periodic Behavior. The reality is that the functions of sine, cosine and tangent are embedded in the foundations of modern mathematics and, as you’ll discover, the world around us. Students use special triangles to geometrically determine the values of sine. KEY STANDARDS ADDRESSED: MA3A3. We start by revisiting the Ferris wheel. Write a rule in the form f(t) = A sin Bt that expresses this oscillation where t represents the number of seconds. Students are expected to be able to interpret functions in the context of a problem, write functions, and apply functions to periodic phenomenon. The hyperbolic sine and the hyperbolic cosine are. If we recall from the previous section we said that $$f\left( x \right)$$ is nothing more than a fancy way of writing $$y$$. Includes full solutions and score reporting. Line 1 just restates Euler's formula. So for example, if x = 2, the y-value will be y = 2 sin. Accelerated Mathematics III - Unit 5 Investigating Trigonometry Graphs Student Edition INTRODUCTION: In this unit students develop an understanding of the graphs of the six trigonometric functions: sine, cosine, tangent, cotangent, cosecant, and secant. Graphing Sine & Cosine w/out a Calculator Pt 2. Read: 'The sine of thirty degrees is zero point five. Trigonometry means the study of the triangle. Verify the size of a specified angle and hence that two triangles within the given diagram are similar. The resulting graph will be displayed as you change the values. ( Source: Wikipedia, try not to get hypnotized. Find | {
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will be displayed as you change the values. ( Source: Wikipedia, try not to get hypnotized. Find angle pi on a grid. 387 In-class/Homework: pg. Apply right-angle trigonometry to a scenario. Many compression algorithms like JPEG use fourier transforms that rely on sin and cos. Sine and Cosine Part 2 - given the graph of a sinusoidal function, determine an equation that defines it D2. We offer targeted baccalaureate programs of study, pre-baccalaureate programs of study for transfer, associate of arts and associate of science degrees, and serve as a portal to graduate education. By thinking of the sine and cosine values as coordinates of points on a unit circle, it becomes clear that the range of both functions must be the interval$\,\left[-1,1\right]. - He excelled in the trigonometric functions unit where he was able to sketch sinusoidal functions using the transformations of key points of the sine or. Different sounds create different waves. Use sinusoidal functions to solve problems. In this example, we are multiplying the sine of each x-value by the x-value. Transformations of Trigonometric Functions. " For example, "an oscilloscope is an electronic instrument used to display changing electrical signals. and “use sine and cosine functions to model real-life data,” iii. To be able to solve real-world problems using the Law of Sines and the Law of Cosines This tutorial reviews two real-world problems, one using the Law of Sines and one using the Law of Cosines. We do: Writing Sinusoidal Functions from a Graph. march 21, 2007. To find the cosine of angle pi, you need graph paper. Moreoever, the pattern repeats, so this is still a periodic function. can be used to assign a particular use of the plot function to a particular figure wi. Assessments Summative: Tests and benchmark. ppt), PDF File (. Markov processes are widely used in performance modelling of wireless and mobile communication systems. Graphing a Sinusoidal Function. ( Source: Wikipedia, try not to get | {
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mobile communication systems. Graphing a Sinusoidal Function. ( Source: Wikipedia, try not to get hypnotized. A periodic function is defined as a function that repeats its values in regular periods. Trig Functions: The Ferris Wheel Arnold Tutoring Solving Problems with the Sine and Cosine Functions - Lesson - Duration: Determining the Equation of a Sine and Cosine Graph - Duration:. For this, the phase shift will be 172. So what do they look like on a graph on a coordinate plane? Let’s start with the sine function. Sketch translations of graphs of sine and cosine functions; use sine and cosine functions to model real-life data. 2 Graph lines (including those of the form x = h and y = k) given appropriate information. I am confused as I am unsure whether time (e. As we can see in Figure 6, the sine function is symmetric about the origin. In fact, there is a zero derivative at this point here, which leads us to believe that maybe what we should have done was to have broken this curve down into the union of two one-to-one functions. y = 2 sin 2x 14. Amplitude and Period for Sine and Cosine Functions Worksheet Author: marthasweeney Last modified by: Leake, Sherry LHS-STAFF Created Date: 6/13/2016 5:56:00 PM. 2]Solve right and oblique triangles including application problems. Trigonometric formulae are useful for solving problems in two dimensions. Find all values of A and B such that. PERIODIC FUNCTIONS A. Four Function Scientific. Lesson 12: Ferris Wheels—Using Trigonometric Functions to Model Cyclical Behavior Student Outcomes Students review how changing the parameters 𝐴, 𝜔, ℎ, and 𝑘 in ( )=𝐴sin(𝜔( −ℎ))+𝑘 affects the graph of a sinusoidal function. 28rad has the same cosine as 5rad, they're inverse angles. php(143) : runtime-created function(1) : eval()'d code(156) : runtime-created. Solve real world applications problems using systems of equations and inequalities Graph rational functions with asymptotes Graph sine and cosine curves. How the values of A and B affect | {
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rational functions with asymptotes Graph sine and cosine curves. How the values of A and B affect the shape of the graph y = A sin(Bx). The table of values for the functions. The number π (pi)— or at least the closest approximation that fits in a JavaScript number—is available as Math. Graphing Sine & Cosine w/out a Calculator Pt 2. When they notice that these points are already on the graph, make a note about the cyclic nature of sine and cosine. List of trigonometric identities 2 Trigonometric functions The primary trigonometric functions are the sine and cosine of an angle. Correctness of a skewed cosine similarity graph I am currently implementing a word2vec model that uses the cosine similarity to determine the similarity between two vectors. A great discussion occurs when the students are asked to justify the function they want to use. The simplest one is y = sin(x). sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π–x, π+x, and 2π–x in terms of their values for x, where x is any real number. Topic:Real world application of sine graphs. Graph) 3 2 f x = ( ) 2sin(x. Graphing Trigonometric Functions Real-World Applications of Trigonometric Functions. In trig speak, you say something like this: If theta represents all the angles in the domain of the two functions. ) Circles are an example of two sine waves. The reality is that the functions of sine, cosine and tangent are embedded in the foundations of modern mathematics and, as you’ll discover, the world around us. In maths, you have real life applications on any thing that you study. For now, focus primarily on sine, cosine, and tangent. Trigonometric Functions, you will begin by learning about the inverses of quadratics and other functions. any anual average can be repeated year to year. Graphing Sine and Cosine Functions. Les't compare with. Graph a sine or cosine function with a given phase shift, period, and amplitude. When plotting | {
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with. Graph a sine or cosine function with a given phase shift, period, and amplitude. When plotting all the possible cosine similarities, I get the following. find the equation of a sinusoidal function given its graph or its properties. 7m, but I am unsure how to interval the axis. Of course, this was to be expected, but this isn't the real problem; the real problem is that if your function has some crucial points it must pass through (which is certainly true for trigonometry functions), the truncation will move the curve away from those points. So for example, if x = 2, the y-value will be y = 2 sin. Intro Tangent & Cotangent Graphs. Outcome 6: Set up and solve exponential and logarithmic equations; then identify and sketch graphs of the functions. 4 5 Graphing the Trigonometric Functions (Part 2) 2 The sine function Imagine a particle on the unit circle, starting at (1,0) and rotating counterclockwise around the origin. How to find and the slopes of lines to write and graph linear equations in two variables. Given an expression for the time rate of change of a quantity, find the initial rate. 762 Chapter 14 Trigonometric Graphs and Identities • Graph trigonometric functions. Sample Problem. 1 solve problems, including those that arise from real-world applications (e. graph the sine and cosine functions with varied amplitudes and periods. 14-1 Graphs of Sine and Cosine Example 3: Sound Application Use a sine function to graph a sound wave with a period of 0. Those functions were predefined for you, meaning that you didn't have to tell the computer how to compute the sine and cosine of an angle. Focus on creating a graph and formula from a story (starting from time 6:55) You Do Now: Warmup - Sketching a Cosine Wave. When the radian measure of angles was introduced in the early eighteenth century by Roger Cotes and the trigonometric ratios of angles in the plane began to be viewed as functions of a linearized angle, the undulating movement of the sine and cosine | {
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to be viewed as functions of a linearized angle, the undulating movement of the sine and cosine graphs fed the. Les't compare with. The hyperbolic functions may be defined in terms of the legs of a right triangle covering this sector. Trigonometric identities like finding. Graphing Secant & Cosecant. A real life example of the sine function could be a ferris wheel. Graphing Sine and Cosine Trig Functions With Transformations, Phase Shifts, Period - Domain & Range - Duration: 18:35. dilations, and horizontal and vertical translations of functions. This example shows how to use both CORDIC-based and lookup table-based algorithms provided by the Fixed-Point Designer™ to approximate the MATLAB® sine (SIN) and cosine (COS) functions. Every position of the particle corresponds with an angle, ?, where y sin ?. In line 3 we plug in -x into Euler's formula. Example: if your cutting down a tree and need to know where it will. Using f (x) = cos x as a guide, graph the function h (x) = __1 cos 2 3 x. SC-Common Core Precalculus Scope and Sequence Unit Topic Lesson Lesson Objectives Functions Functions and Graphs Graphing Linear Equations Find the slope of a line through two points Find the x- and y-intercepts of a line Find zeros of linear functions Graph linear equations Operations with Functions Find composite functions. Graph Translations 3. Plus ITEM TYPES: MC. Determine the characteristics of the graphs of the six basic trigonometric functions. I can graph the sine, cosine, and tangent functions. 3 make connections between the sine ratio and the sine function and between the cosine ratio and the cosine function by graphing the relationship between angles from 0º to 360º and the corresponding sine ratios or cosine ratios, defining this relationship as the function f(x) =sinx or f(x) =cosx, and explaining why the relationship is a function. ( Source: Wikipedia, try not to get hypnotized. Equation of Sine and Cosine from a Graph. Calculus is made up of Trigonometry and | {
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to get hypnotized. Equation of Sine and Cosine from a Graph. Calculus is made up of Trigonometry and Algebra. inverse MTH 112 Elementary Functions Chapter 5 The Trigonometric Functions -. You can use the graph for a trigonometry function to show average daily temperatures at a specific location. Most radio communication is based on the use of combinations of sines and cosine waves. To understand how the absolute value can be applied to the real world, we’ll look at two topics: What is considered normal with regard to the temperature of the human body and; Fuel economy of two vehicles: a Honda Odyssey and a Nissan Altima. Trig also has applications in fields as broad as financial analysis, music theory, biology, medical imaging, cryptology, game development, and seismology. Amplitude is first, then sine or cosine, then B, which you get by doing 2pi/10 (period), which reduces to pi/5, then it is the parenthesis with the transformation, and then last but not least is the sinusodial axis. Application: Sine and cosine functions can be used to model real-world phenomena, such as sound waves. In the chapter on Trigonometric Functions, we examined trigonometric functions such as the sine function. functions; increasing/decreasing functions; finding intercepts with axes. Evaluate trigonometric functions of real numbers. Modeling Using Sinusoidal Functions Sine and cosine functions model many real-world situations. The graph of the sine function is called the sine curve. Defining Sine and Cosine Functions. When they notice that these points are already on the graph, make a note about the cyclic nature of sine and cosine. Amplitude is first, then sine or cosine, then B, which you get by doing 2pi/10 (period), which reduces to pi/5, then it is the parenthesis with the transformation, and then last but not least is the sinusodial axis. Topic:Real world application of sine graphs. Example: if your cutting down a tree and need to know where it will. Find all values of A and B such | {
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Example: if your cutting down a tree and need to know where it will. Find all values of A and B such that. 7m, but I am unsure how to interval the axis. This shape is also called a sine. Evaluate Inverse Trigonometric Functions v. For large arguments we would expect these functions to be a quarter period out of phase, just like cosine and sine, since asymptotically J 1 is proportional to cos(z – 3π/4) / √z and Y 1 is proportional to sin(z – 3π/4) / √z. Graphing Sine & Cosine w/out a Calculator Pt1. The function is a model for the hours of daylight in Center City. Sum and Difference, Double Angle and Half Angle. The size of a hyperbolic angle is twice the area of its hyperbolic sector. Students will use the model to make a prediction about a missing piece of data. 2 cos 2 cos 2 cos 2 sin( ) cos Now we can clearly see that if we horizontally shift the cosine function to the right by π/2 we get the sine function. 1 works in a wide range of settings: any time we have an exponential equation of the form $$p \cdot q^t + r = s\text{,}$$ we can solve for $$t$$ by first isolating the exponential expression $$q^t$$ and then by taking the natural logarithm of both sides of the equation. In maths, you have real life applications on any thing that you study. The cosine is known as an even function, and the sine is known as an odd function. Cumulative Review. Along the way, youll get plenty of practice, from fully guided examples to independent end-of-chapter drills and test-like samples. Midline Amplitude And Period Review Article Khan Academy Solving Problems Using Sinusoidal Functions Real World Applications Ferris Wheel Trig Example Youtube Period Of Cosine Function Math Ladyprestige Club. Next, lets take the 5 from the front. 2 Define sine and cosine as functions of the radian measure of an angle in terms of the x- and y-coordinates of the point on the unit circle. But just as you could make the basic quadratic, y = x 2, more complicated, such as y = -(x + 5) 2 - 3, so also | {
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you could make the basic quadratic, y = x 2, more complicated, such as y = -(x + 5) 2 - 3, so also trig graphs can be made more complicated. I know that the amplitude of the function is 1. When I consider how to address the Precalculus objectives "to solve real-life problems involving harmonic motion" ii. A cosine function will start at its minimum or maximum value on the y-axis; however, the sine graph will start at its midline on the y-axis. Graph sine and cosine functions with various amplitudes and periods. 6 Modeling with Trigonometric Functions 507 Writing Trigonometric Functions Graphs of sine and cosine functions are called sinusoids. Determine the values for angles between 0° and 360°. can be used to assign a particular use of the plot function to a particular figure wi. A sine wave is what you get when you plot the sine function on a graph. Psst… don’t over-focus on a single diagram, thinking tangent is always smaller than 1. Defining Sine and Cosine Functions. Sketch the graphs of basic sine and cosine functions; use amplitude and period to help sketch the graphs of sine and cosine functions. Phase Shift = to the right Vertical Shift = 4 4. How to analyze graphs of functions. Sine, Cosine, Tangent Applications. My kids have seen this a bunch so this should only take a minute or two. 2]Solve right and oblique triangles including application problems. Analysis: I developed this lesson plan so that students would be introduced to graphs…. Aug 15, 2015 · C2 Sine and Cosine Rule - 1 - Basic Introduction (AS Maths A Level trigonometry) ukmathsteacher. In line 4 we use the properties of cosine (cos -x = cos x) and sine (sin -x = -sin x) to simplify the. Graphing the sine and cosine functions is a topic that students will carry on with them throughout the rest of their future science and mathematics courses. Graphs of tan, cot, sec and csc; 5. Real life word problems that involve the sine and cosine function can be used to keep the students engaged in the topic. | {
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that involve the sine and cosine function can be used to keep the students engaged in the topic. inverse MTH 112 Elementary Functions Chapter 5 The Trigonometric Functions -. The inverses of sine, cosine and tangent functions are not functions unless the domains are limited. Many compression algorithms like JPEG use fourier transforms that rely on sin and cos. Investigate Transformations of Sine and Cosine Functions Many real-world processes can be modelled with sinusoidal functions. functions for special angles using the unit circle and right triangle definitions. Evaluating Inverse. - She did an excellent job in the trigonometry unit where she was able to use the sine and cosine laws to solve problems on both her assignment and. We can prove this by using the cofunction identity and the negative angle identity for cosine. Lesson 6-6 Modeling Real-World Data with Sinusoidal Functions 389 Graphing Calculator Tip For keystroke instruction on how to find sine regression statistics, see page A25. Amplitude and Period for Sine and Cosine Functions Worksheet Author: marthasweeney Last modified by: Leake, Sherry LHS-STAFF Created Date: 6/13/2016 5:56:00 PM. The sine function “starts” at the midline, but the leading negative causes it to reflect across the midline. Course Objectives: * Students will be able to develop and understand transferable pre-calculus skills necessary for success in Precalculus and beyond * Students will be able to identify, evaluate, perform operations on, and find the domain, range, and inverse of functions * Students will be able to draw common graphs along with transformations and reflections * Students will be able to create. Use sinusoidal functions to solve problems. The heat equation is used to model how things get hot (electronics, spacecraft, ovens, etc). Find the amplitude, period, and phase shift of a sine or cosine functions from its equation. Sine and cosine functions can be used to model many real-life scenarios –radio waves, tides, | {
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Sine and cosine functions can be used to model many real-life scenarios –radio waves, tides, musical tones, electrical currents. Frequency is the number of cycles in a given. Analytic Trigonometry. Sine and cosine waves can be applied to all sinusoid problems of the real-world. Being a cofunction, means that complementary input angles leads to the same output , as shown in the following example: Problem 2. Students will investigate and use the graphs of the six trigonometric functions. 1 RTP Scenarios and Terminology 12. 2 RTP Packet Format 12. Understanding Basic Sine & Cosine Graphs. c) sketch the graph of the function by using transformations for at least a two-period interval; and d) investigate the effect of changing the parameters in a trigonometric function on the graph of the function. When teaching students the value of sinusoidal models and graphs, such as sine and cosine functions, students often feel like they must memorize formulas and apply them with little recollection or understanding. Graph y = sin x. Using f (x) = cos x as a guide, graph the function h (x) = __1 cos 2 3 x. In particular: The slope of a graph is related to how fast the quantity being graphed is changing. Graphing Sine and Cosine Functions. The Sine, Cosine and Tangent functions are often applied to real world scenarios. C8 demonstrate an understanding of real-world relationships by translating between graphs, tables, and written descriptions. Third, this is a sine graph with a leading negative. Hence, we start this lesson by recalling these functions and their corresponding graphs. functions, and graph them, with and without the use of digital technologies (ACMMG274) prove that the. Trigonometry can be used to roof a house, to make the roof inclined ( in the case of single individual bungalows) and the height of the roof in buildings etc. Figure 4 The cosine function Because we can evaluate the sine and cosine of any real number, both of these functions are defined for all real | {
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evaluate the sine and cosine of any real number, both of these functions are defined for all real numbers. Graphs of the Trigonometric Functions; 1. The Code of Federal Regulations is a codification of the general and permanent rules published in the Federal Register by the Executive departments and agencies of the Federal Government. Write a cosine function that describes the height of the nail above ground as a function of angular distance. Trigonometric functions (Level C) Radian measure (including what are radians, converting from degrees to radians, converting from radians to degrees, using radians measure in real world applications) Graphs of sine, cosine, and tangent functions. The sine function. A single note can be modeled on a sine curve, and a chord can be modeled with multiple sine curves used in conjunction with one another. Special triangles can be used to geometrically determine the values of the six trigonometric functions at ,, 6 4 3 , and their integral multiples. Duration: 2 Day(s) Learning Targets. Applications of Sine and Cosine Graphs Learning Task: Trigonometry functions are often used to model periodic data. Key Questions. SheLovesMath. Graphing Sine and Cosine Functions. Chapter Five – Trigonometric Functions What to Expect: A new formula designed for sin and cos graphs, details about frequencies, new vocabulary, explaining periodic trends, and the like. Notes Chapter 6 Section 6 ( Modeling Real-World Data with Sinusoidal Functions ) -pg. Section 3-5 : Graphing Functions. Trigonometric Functions By the end of this course, students will: - solve problems involving trigonometry in acute triangles using the sine law and the cosine law, including problems arising from real-world applications; - demonstrate an understanding of periodic relationships and the sine function, and make connections between the identify and. The cosine of an angle is the ratio of x r {\displaystyle {\tfrac {x}{r}}} , so in the unit circle, the cosine is the x | {
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is the ratio of x r {\displaystyle {\tfrac {x}{r}}} , so in the unit circle, the cosine is the x -coordinate of the point of rotation. - She did an excellent job in the trigonometry unit where she was able to use the sine and cosine laws to solve problems on both her assignment and. C2 create and analyse scatter plots of periodic data. Press the button marked "sin" to change to the cosine. 2-use technology, graphs, and tables to compare sine graphs. 3 Finding Maximum and Minimum Values and Zeros of Sine and Cosine 2. GCSE Maths Pythagoras and Trigonometry Level 8-9. These functions are most conveniently defined in terms of the exponential function, with sinh z = 1 / 2 (e z − e −z) and cosh z = 1 / 2 (e z + e. 391-392 #'s 7-16 all I will be able to: Model real-world data using sine and cosine functions. 8 Graphs, Inverses, and Applications of Trigonometric Functions. Find the amplitude and period of the function. Applications of Exponential Functions The best thing about exponential functions is that they are so useful in real world situations. If the set of input values is a finite set then the models are known as discrete models. When teaching students the value of sinusoidal models and graphs, such as sine and cosine functions, students often feel like they must memorize formulas and apply them with little recollection or understanding. 7 The student will identify the domain and range of the inverse trigonometric functions and recognize the graphs of these functions. So, temperature as a function of days. A real life example of the sine function could be a ferris wheel. Sample records for nino-southern oscillation eventsnino-southern oscillation events «. To be able to solve real-world problems using the Law of Sines and the Law of Cosines This tutorial reviews two real-world problems, one using the Law of Sines and one using the Law of Cosines. and "use sine and cosine functions to model real-life data," iii. Find the amplitude, period, and phase shift of a sine | {
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functions to model real-life data," iii. Find the amplitude, period, and phase shift of a sine or cosine functions from its equation. sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π–x, π+x, and 2π–x in terms of their values for x, where x is any real number. Write the sum and difference identities for sine, cosine, and tangent. Chapter 4 Sections 4. This means that bodies. On the ground, from the object you measure off a know distant in a straight line from the base. 2) Watch the video and pay attention to the key ideas listed. If you take a rope, fix the two ends, and let it hang under the force of gravity, it will naturally form a hyperbolic cosine curve. The sine and cosine of an angle are both circular functions, and they are the fundamental circular functions. Water Depth Word Problem Modeled with Cosine Sine Function. 5/175* share an equivalent sine. Namely, if we look at the graph 'y' equals hyperbolic cosine 'x', observe that whereas the function is single valued, it is not one-to-one. Find patterns in given array of numbers. The approach used in Example 3. 2 Define sine and cosine as functions of the radian measure of an angle in terms of the x- and y-coordinates of the point on the unit circle. Find the frequency in hertz for this sound wave. A single note can be modeled on a sine curve, and a chord can be modeled with multiple sine curves used in conjunction with one another. For large arguments we would expect these functions to be a quarter period out of phase, just like cosine and sine, since asymptotically J 1 is proportional to cos(z – 3π/4) / √z and Y 1 is proportional to sin(z – 3π/4) / √z. Frequency is the number of cycles in a given unit of time. Sineing on to the job Since we know that a triangle has 180 degrees, we can subtract 56 degrees and 91 degrees from it to find our missing angle Using the law of sines we can then set up this equation sin 91 degrees/ xft = sin | {
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our missing angle Using the law of sines we can then set up this equation sin 91 degrees/ xft = sin 33/6ft After crossmultipying and then dividing to. Now we must stretch out the amplitude of the sine graph. Les't compare with. So, temperature as a function of days. This is an example of how a Graph of a Cosine Function looks like:. Animated gifs showing relationships between functions and geometry eg trigonometric functions How the unit circle gives us the sine wave Teaching math with the Drum Loop Real sinusoid on a timebase, formed by a linear increment of complex argument in time. Therefore, the period of this graph is 2π/(1/3) which is 6π. To find the period of a sine graph, take the B value, which is 2, and divide 2π by it. Description: To be able to solve real-world problems using the Law of Sines and the Law of Cosines. Land surveying makes an extensive use of the sine and cosine law. The basic sine and cosine graphs C. Seymour Public Schools Curriculum Subject or course name 5 graph the basic sine and cosine functions. Do not forget this. state the properties for the Sine and Cosine graph identify the values of a, k, c and d and apply it to a sinusoidal function to graph the resulting transformation Lesson:. Graph Trigonometric Functions (1), cosine function with solution. ' Below is a listing of several popular input and output values for the three main trigonometry functions. 1) Graph sine functions (PC-N. Graphing Sine and Cosine Trig Functions With Transformations, Phase Shifts, Period - Domain & Range - Duration: 18:35. 1: Graphs of the Sine and Cosine Functions. Water Depth Word Problem Modeled with Cosine Sine Function. One method to write a sine or cosine function that models a sinusoid is to fi nd the values of a, b, h, and k for y = a sin b(x − h) + k or y = a cos b(x − h) + k where ∣ a ∣ is the amplitude, — 2π is the period. Master PHP 4 — the open source Web scripting breakthrough! Contains expert coverage of syntax, functions, design, and | {
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open source Web scripting breakthrough! Contains expert coverage of syntax, functions, design, and debugging! Leverage the amazing performance of the new Zend engine! 650+ real-world code examples! CD-ROM includes source code, plus everything you’ll need to run PHP 4 implementations on Windows and UNIX!. You can select the values of a and b within certain ranges. Sine wiggles in one dimension. We completed three cosine graphs together on the 13-5 vocab support page. Signal Processing. Real life word problems that involve the sine and cosine function can be used to keep the students engaged in the topic. ) Circles are an example of two sine waves. Find the amplitude and period of the function. Transformations of Trigonometric Functions. Graphs of the sine and cosine functions; and Periodic Behavior. Because we can evaluate the sine and cosine of any real number, both of these functions are defined for all real numbers. The student will graph sine or cosine graph affected by horizontal and vertical translations. Like What? [Filename: Notes 6. The Code of Federal Regulations is a codification of the general and permanent rules published in the Federal Register by the Executive departments and agencies of the Federal Government. It is given by parameter #a# in function #y = asinb(x - c) + d or y = acosb(x - c) + d# •The period of a graph is the distance on the x axis before the function repeats itself. • Use the cosine law to determine the interior angles of an acute triangle. The Sine Ratio Passy's World of Mathematics. Physical phenomenon such as tides, temperatures and amount of sunlight are all things that repeat themselves, and so are easily modeled by sine and cosine functions (collectively, they are called "sinusoidal functions"). Find an equation for a sine function that has amplitude of 4 a period of π. To understand how the absolute value can be applied to the real world, we'll look at two topics: What is considered normal with regard to the temperature of the | {
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world, we'll look at two topics: What is considered normal with regard to the temperature of the human body and Fuel economy of two vehicles: a Honda Odyssey and a Nissan Altima. 8 L5 Trig Applications Part 1 - solve problems that arise from real world applications involving periodic phenomena D3. Lynn Weiss. Sketch the graphs of basic sine and cosine functions; use amplitude and period to help sketch the graphs of sine and cosine functions. Given an expression for the time rate of change of a quantity, find the initial rate. ppt), PDF File (. The maximum value appears to be 72 and the minimum value is 11. Goal: Students will use technology to discover how to graph trigonometric functions. Apr 15, 2018 - Discovering the graphs of Sine & Cosine using the measurements of the Unit Circle (Activity from Rice APSI) Stay safe and healthy. PreCalculus Honors - Curriculum Pacing Guide - 2017 - 2018 First Half of Semester Anderson School District Five Page 3 2017-2018 Unit 2 - Application of Trigonometry to Triangles (7 days) PC. solve problems involving trigonometry in acute triangles using the sine law and the cosine law, including problems arising from real-world applications; demonstrate an understanding of periodic relationships and the sine function, and make connections between the numeric, graphical, and algebraic representations of sine functions;. A sine wave is what you get when you plot the sine function on a graph. MCF 3M - Functions and Applications (Mixed) Spring 2014 Page history last edited by Peter Kim 5 years, 9 months ago This course introduces basic features of the function by extending students' experiences with quadratic relations. The product of two cosine functions is a sum of two other cosine functions. Trigonometric functions allow us to specify the shapes and proportions of objects independent of exact dimensions. Free practice questions for Precalculus - Graph the Sine or Cosine Function. Real world uses of hyperbolic trigonometric functions. | {
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- Graph the Sine or Cosine Function. Real world uses of hyperbolic trigonometric functions. Specific Expectations. Perform a sine regression using the Ti83/84 Activity #2 - Steamboat problem From given information: 1. Amplitude is first, then sine or cosine, then B, which you get by doing 2pi/10 (period), which reduces to pi/5, then it is the parenthesis with the transformation, and then last but not least is the sinusodial axis. 1) Find properties of sine functions (PC-N. Evaluating Inverse Trigonometric. Graphing Sine and Cosine Functions. \begingroup Basically, the answer is: yes, there are many other periodic functions, and the reason you typically see harmonics (like \sin,e^{i\omega t}) used is because in most simple applications of interest which are easily understandable, either the behavior itself is harmonic, or the behavior is most easily understood in terms of harmonics. First, you must understand that each of these functions has its own graph. In the chapter on Trigonometric Functions, we examined trigonometric functions such as the sine function. First of all, the graph is no longer a sine curve, but there's definitely a pattern to it. It exposes the student to the basic concepts of algebra and trigonometry with an emphasis in practical applications. But just saying "the sine function" doesn't tell you anything about the probability measure in the domain, which is what gets you probabilistic information, including joint distributions. the sine or cosine function and use the equation as a mathematical model to make predictions and interpretations about the real world. Using trigonometric functions to model climate. In ICSA, an elitism strategy is used to select the global solution, and a new updating mechanism for the new solution was proposed. You cannot graph an exponential functions without finding its horizontal asymptote. In addition, how do I know if this the graph of sine or cosine?. functions; increasing/decreasing functions; finding intercepts | {
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if this the graph of sine or cosine?. functions; increasing/decreasing functions; finding intercepts with axes. real world applications of sinusoids Sinusoids are applicable to the real world. To graph the sine function, we mark the angle along the horizontal x axis, and for each angle, we put the sine of that angle on the vertical y-axis. y-2π -π π 2π 0. The rise and fall of tides can have great impact on the communities and ecosystems that depend upon them. Every angle shares the same sine with another angle (and -sine with two others). • Find the amplitude and period of variation of the sine, cosine, and tangent functions. Sine and cosine functions are periodic functions. 4 COMPLEX NUMBERS Add and subtract complex numbers Multiply and divide complex numbers Simplify powers of i 2. The amplitude is 2. These graphs act as a reference every time you use a trigonometric function. Also for every n π/2 radians (where "n" is an interger) the cosine function passes through y=0. 21 from both sides: Divide both sides by 1. Outcome 5: Set up, solve, and graph equations from problems that require use of trigonometric functions, tangent, sine, and cosine and the Pythagorean Theorem. 3 make connections between the sine ratio and the sine function and between the cosine ratio and the cosine function by graphing the relationship between angles from 0º to 360º and the corresponding sine ratios or cosine ratios, defining this relationship as the function f(x) =sinx or f(x) =cosx, and explaining why the relationship is a function. The sine and cosine of an angle are both circular functions, and they are the fundamental circular functions. Full Range Fourier Series - various forms of the Fourier Series 3. Focus on creating a graph and formula from a story (starting from time 6:55) You Do Now: Warmup - Sketching a Cosine Wave. Learn how to construct trigonometric functions from their graphs or other features. Physical phenomenon such as tides, temperatures and amount of sunlight are | {
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graphs or other features. Physical phenomenon such as tides, temperatures and amount of sunlight are all things that repeat themselves, and so are easily modeled by sine and cosine functions (collectively, they are called "sinusoidal functions"). 2-use technology, graphs, and tables to compare sine graphs. The property represented here is based on the right triangle and the two acute or complementary angles in a right triangle. The Code is divided into 50 titles which represent broad areas subject to Federal regulation. Sinusoidal Modeling - Real World Application Project. Then we looked at identifying the amplitude, frequency and period from a given cosine (or sine) function. Time-saving lesson video on Sine and Cosine Functions with clear explanations and tons of step-by-step examples. One way to distinguish sounds is to measure frequency. sinA sin B sin(AB) cosAcosB cos(AB) 59. 9) Graph sine and cosine functions (PC-N. 534), where x is the day of the year starting with January 1 as Day 1. Graph functions, plot data, evaluate equations, explore transformations, and much more - for free! Start Graphing Four Function and Scientific Check out the newest additions to the Desmos calculator family. Recall that the sine and cosine functions relate real number values to the x– and y-coordinates of a point on the unit circle. Day 7 HW F 15-Dec Graphing Cosine Functions NC. Determine the values for angles between 0° and 360°. Chapter Test. Evaluate and graph logistic growth functions. Find the amplitude and period of the function. Applications of Trigonometric Graphs; 6. Khan Academy is a 501(c)(3) nonprofit organization. Graph one period of s(x) = –cos(3x) The "minus" sign tells me that the graph is upside down. There are many uses of sin,cos,tan in real life. Trigonometric Functions • Understand some real-world situations that demonstrate periodic behavior. Write a cosine function that describes the height of the nail above ground as a function of angular distance. | {
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function that describes the height of the nail above ground as a function of angular distance. Trigonometry Graphing Trigonometric Functions Applications of Radian Measure. Discovery of Euler's Equation. Example 15 6. The new 3rd edition of Precalculus: Algebra and Trigonometry prepares students for further study in mathematics, the sciences, engineering, the social sciences, and other disciplines. By correctly labeling the coordinates of points A, B, and C, you will get the graph of the function given. 'transform') each of the three trigonometric functions discussed thus far: sine, cosine, and tangent. 25 scaffolded questions on equation graph involving amplitude and periodplus model problems explained step by step. Amplitude and Period of Sine and Cosine Functions BOATING A signal buoy between the coast of Hilton Head Island, South Carolina, and Savannah, Georgia, bobs up and down in a minor squall. Graphs of y = a sin x and y = a cos x 2. Yep, sine and cosine are practically twins. In maths, you have real life applications on any thing that you study. Lesson 7: Applications with Acute Triangles • Identify when to apply the sine and cosine laws given incomplete information about the • Solve a multistep problem that involves o two or more applications of the sine or cosine laws,. Sine Law and Cosine Law Find each measurement indicated. Free trigonometry worksheets, in PDF format, with solutions to download. Khan Academy is a 501(c)(3) nonprofit organization. stock market values can be formed into a sine, cosine graph (given a constant average) 5. Precalculus. Example: if your cutting down a tree and need to know where it will. I'm not exactly sure what level of mathematics you've studied so far, but sine and cosine are used heavily in math (calculus and functions) and physics (kinematics and dynamics), as well as having some real world applications. 2 The Inverse Trigonometric Functions. The trigonometric functions sine, cosine and tangent calculate the ratio of | {
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Trigonometric Functions. The trigonometric functions sine, cosine and tangent calculate the ratio of two sides in a right triangle when given an angle in that triangle. Although you. 4 The Real-Time Transport Protocol 12. the graph of a sine function. C2 create and analyse scatter plots of periodic data. Of course, this was to be expected, but this isn't the real problem; the real problem is that if your function has some crucial points it must pass through (which is certainly true for trigonometry functions), the truncation will move the curve away from those points. Standard 9: Find values of trigonometric functions using the unit circle. The unit circle and values of sine and cosine on the unit circle B. The fruit fly optimization algorithm (FOA) is a well-regarded algorithm for searching the global optimal solution by simulating the foraging behavior …. which means that theta can be any angle in degrees or radians — any real number. 6 u00ad Modeling Real World Data with Sinusoidal Functions Section 6. The unit circle can be used to express the values of sine, cosine, and tangent for π – x,. Students graph transformations of trigonometric functions (sine and cosine), including af(x), f(x) + d, f(x – c), and f(bx) for specific values of a, b, c, and d, in mathematical and real-world problem situations. Graphing Secant & Cosecant. Investigate Transformations of Sine and Cosine Functions Many real-world processes can be modelled with sinusoidal functions. The identities that arise from the triangle are called the cofunctionidentities. Sine and Cosine can be used in everyday life in various ways. Topic:Real world application of sine graphs. Press the button marked "sin" to change to the cosine. ratios, ie. Intro Tangent & Cotangent Graphs. To graph the sine function, we mark the angle along the horizontal x axis, and for each angle, we put the sine of that angle on the vertical y-axis. McGraw-Hill Education Trigonometry Review and Workbook, 1st Edition by William | {
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the vertical y-axis. McGraw-Hill Education Trigonometry Review and Workbook, 1st Edition by William Clark and Sandra Luna McCune (9781260128925) Preview the textbook, purchase or get a FREE instructor-only desk copy. 3 make connections between the sine ratio and the sine function and between the cosine ratio and the cosine function by graphing the relationship between angles from 0º to 360º and the corresponding sine ratios or cosine ratios, defining this relationship as the function f(x) =sinx or f(x) =cosx, and explaining why the relationship is a function. If the set of input values is a finite set then the models are known as discrete models. org (pre-calculus), and Khan Academy (SAT. This builds into learning about graphing and interpreting logarithmic functions and models. Les't compare with. These graphs act as a reference every time you use a trigonometric function. 2) A car's tire has a diameter of 32 inches. The caternary curve (a dangling string/chain) is really just cosh – crasic Oct 30 '10 at 23:48. state the properties for the Sine and Cosine graph identify the values of a, k, c and d and apply it to a sinusoidal function to graph the resulting transformation Lesson:. 14-1 Graphs of Sine and Cosine Example 3: Sound Application Use a sine function to graph a sound wave with a period of 0. pdf), Text File (. A single note can be modeled on a sine curve, and a chord can be modeled with multiple sine curves used in conjunction with one another. Graphing Sine and Cosine Trig Functions With Transformations, Phase Shifts, Period - Domain & Range - Duration: 18:35. In the 36 intensively illustrated lectures of Mathematics Describing the Real World: Precalculus and Trigonometry, he takes you through all the major topics of a typical precalculus course taught in high school or college. The graphs of the sine and cosine functions illustrate a property that exists for several pairings of the different trig functions. Then you will learn about modeling trigonometric | {
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several pairings of the different trig functions. Then you will learn about modeling trigonometric functions by graphing the sine and cosine functions. Recall from The Other Trigonometric Functions that we determined from the unit circle that the sine function is an odd function because [latex]sin(−x)=−sinx$. Sample records for nino-southern oscillation eventsnino-southern oscillation events «. The inverses of sine, cosine and tangent functions are not functions unless the domains are limited. Chapter Five – Trigonometric Functions What to Expect: A new formula designed for sin and cos graphs, details about frequencies, new vocabulary, explaining periodic trends, and the like. 2]Solve right and oblique triangles including application problems. Outcome 6: Set up and solve exponential and logarithmic equations; then identify and sketch graphs of the functions. Precalculus students are required to understand how sine and cosine functions model the real world. A lot of waves actually follow a sine graph, so we can prove that sinusoidal motion is a real thing in nature. We completed three cosine graphs together on the 13-5 vocab support page. f(x)= Calculus: Early Transcendental Functions Use Exercise 25 to find the moment of inertia of a circular disk of radius a with constant density about a. 0333 170 -170 t e 0. Have them graph these points and note whether or not they are on the same graph. Use a horizontal scale where one unit represents 0. With the sine law. Students will find equations for transformed sine curves and will graph transformed sine functions. Sketch translations of graphs of sine and cosine functions; use sine and cosine functions to model real-life data. Other circular functions can all be derived from the sine and cosine of an angle. 002 s and an amplitude of 3 cm. Examples: • Square wave function • Saw tooth functions Split Functions Much of the behaviour of current, charge and voltage in an AC circuit can be described. The inverses of sine, cosine | {
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of current, charge and voltage in an AC circuit can be described. The inverses of sine, cosine and tangent functions are not functions unless the domains are limited. You will be observing the graph of r = a + bsin() or r = a + bcos(), where a and b are non-zero real numbers. The period of sine and cosine functions is 2pi/B. graph the sine and cosine functions with varied amplitudes and periods. Now we must stretch out the amplitude of the sine graph. Sketch the graphs of the sine and cosine functions and state their domain and range. Applications of Sine and Cosine Graphs Learning Task: Trigonometry functions are often used to model periodic data. The sum formula for tangent states that the tangent of the sum of two angles equals the sum of the tangents of the angles divided by 1 minus the product of the tangents of the angles. Graphs the 6 Trigonometric Functions. Rational functions: asymptotes and excluded values (PC-E. It is given by parameter #a# in function #y = asinb(x - c) + d or y = acosb(x - c) + d# •The period of a graph is the distance on the x axis before the function repeats itself. But just as you could make the basic quadratic, y = x 2, more complicated, such as y = -(x + 5) 2 - 3, so also trig graphs can be made more complicated. Trigonometry plays a major role in musical theory and production Real world examples of cosine functions. We typically use degree measures when measuring angles, however we can use radian angle measure as an alternate way of measuring angles in advanced math courses. KEY STANDARDS ADDRESSED: MA3A3. Many compression algorithms like JPEG use fourier transforms that rely on sin and cos. Trigonometric functions can be used to model real world data that is periodic in nature. Since the sine function y = sin t begins at the origin (when t = 0), sine is the more convenient of the two for this purpose. First, find two easily solvable angles that add up to 75. 1 The Inverse Sine, Cosine, and Tangent Functions. Objectives: To graph | {
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angles that add up to 75. 1 The Inverse Sine, Cosine, and Tangent Functions. Objectives: To graph the sine and cosine functions; to identify the graphs of the sine and cosine functions. ' Or: sin(30 °) = 0. Domain and Range of Sine and Cosine The domain of sine and cosine is all real numbers, The range of sine and cosine is the interval [-1, 1] x or (, ) Both these graphs are considered sinusoidal graphs. Find Period and Amplitude 2. Not only is this a real-world visual representation of both graphs, but it also demonstrates that a cosine graph is simply a sine graph at a displacement of 90°. The table of values for the functions. Find the frequency in hertz for this sound wave. The graph of the sine function is called the sine curve. The property represented here is based on the right triangle and the two acute or complementary angles in a right triangle. 7 sketch graphs of y = af (k(x - d)) + c by applying one or more transformations to the graphs of f(x) =sinx and f(x) =cosx, and state the domain and range of the transformed functions 2. It is the reciprocal of the. Hyperbolic functions, also called hyperbolic trigonometric functions, the hyperbolic sine of z (written sinh z); the hyperbolic cosine of z (cosh z); the hyperbolic tangent of z (tanh z); and the hyperbolic cosecant, secant, and cotangent of z. Each of the six trig functions is equal to its co-function evaluated at the complementary angle. However, if we stretch the sine graph and change the amplitude, it just might work. In problems 12 & 13, the graphs of the sine and cosine functions are waveforms like the figure below. Since the multiplier out front is an "understood" –1, the amplitude is unchanged. If we change the number of cycles the wave completes every second -- in other words, if we change the period of the sine wave -- then we change the sound. Draw three right triangles and label the angle and two sides that apply to the sine, cosine and tangent functions respectively. Four Function | {
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angle and two sides that apply to the sine, cosine and tangent functions respectively. Four Function Scientific. The hyperbolic functions take a real argument called a hyperbolic angle. Discovery of Euler's Equation. Cumulative Review. Lesson 9-11: Graphing Trigonometric Functions and Applications in Real World Contexts Learning Goals #8, 9, 10: How do I use the critical values of the Sine and Cosine curve to assist in in graphing Sine and Cosine functions? How do I contextualize Trig Functions in real life situations? Warm-Up! WITH A PARTNER. Water Depth Word Problem Modeled with Cosine Sine Function. One example of applying tangent functions to solve a real world problem is:- Find the gradient and the actual length of a path represented as x cm (a known measurable quantity) in the 1 : n (a known given quantity) scaled contour map. Sound waves travel in a repeating wave pattern, which can be represented graphically by sine and cosine functions. The SOH means that the sine of an angle equals the side opposite the angle over the triangle's hypotenuse. One way to distinguish sounds is to measure frequency. Animated gifs showing relationships between functions and geometry eg trigonometric functions How the unit circle gives us the sine wave Teaching math with the Drum Loop Real sinusoid on a timebase, formed by a linear increment of complex argument in time. Let’s look at a few examples of real-world situations that can best be modeled using trigonometric functions. This is an example of how a Graph of a Cosine Function looks like:. Use sinusoidal functions to solve problems. which means that theta can be any angle in degrees or radians — any real number. To understand how the absolute value can be applied to the real world, we’ll look at two topics: What is considered normal with regard to the temperature of the human body and; Fuel economy of two vehicles: a Honda Odyssey and a Nissan Altima. Periodicity of trig functions. Trigonometric identities like finding. | {
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Odyssey and a Nissan Altima. Periodicity of trig functions. Trigonometric identities like finding. Bookmark File PDF Functions Modeling Change Answers Functions Modeling Change Answers Math Help Fast (from someone who can actually explain it) See the real life story of how a cartoon dude got the better of math Modeling with Functions Part 1 We model real life scenarios of sales and volume of a box with functions. 4 Solving Word Problems Involving Sine or Cosine Functions. By finding a few key points or aspects of the graph, any of the real-life problems we have today can be explained mathematically and much of the vibrations surrounding us can be better understood. active oldest votes. 3π 2 = = Period Amplitude. Fortunately, we can analyze the problem by first representing it as a linear function and then interpreting the components of the function. Read: 'Zero point five is the sine of thirty degrees. Sound waves travel in a repeating wave pattern, which can be represented graphically by sine and cosine functions. Academic Precalculus. functions to model real world problems A cosine graph appears. Topic:Real world application of sine graphs. Trigonometric functions can be used to solve right triangles. Overview of Fourier Series - the definition of Fourier Series and how it is an example of a trigonometric infinite series 2. Maths Applications: Graph sketching. Euler's equation (formula) shows a deep relationship between the trigonometric function and complex exponential function. | {
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# Sequences_and_Series - 11 Sequences and Series Consider the...
• 42
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Chapter 8 / Exercise 14
Essential Calculus: Early Transcendentals
Stewart
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11 Sequences and Series Consider the following sum: 1 2 + 1 4 + 1 8 + 1 16 + · · · + 1 2 i + · · · The dots at the end indicate that the sum goes on forever. Does this make sense? Can we assign a numerical value to an infinite sum? While at first it may seem di cult or impossible, we have certainly done something similar when we talked about one quantity getting “closer and closer” to a fixed quantity. Here we could ask whether, as we add more and more terms, the sum gets closer and closer to some fixed value. That is, look at 1 2 = 1 2 3 4 = 1 2 + 1 4 7 8 = 1 2 + 1 4 + 1 8 15 16 = 1 2 + 1 4 + 1 8 + 1 16 and so on, and ask whether these values have a limit. It seems pretty clear that they do, namely 1. In fact, as we will see, it’s not hard to show that 1 2 + 1 4 + 1 8 + 1 16 + · · · + 1 2 i = 2 i 1 2 i = 1 1 2 i 255
##### We have textbook solutions for you!
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Chapter 8 / Exercise 14
Essential Calculus: Early Transcendentals
Stewart
Expert Verified | {
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Chapter 8 / Exercise 14
Essential Calculus: Early Transcendentals
Stewart
Expert Verified
256 Chapter 11 Sequences and Series and then lim i →∞ 1 1 2 i = 1 0 = 1 . There is one place that you have long accepted this notion of infinite sum without really thinking of it as a sum: 0 . 3333 ¯ 3 = 3 10 + 3 100 + 3 1000 + 3 10000 + · · · = 1 3 , for example, or 3 . 14159 . . . = 3 + 1 10 + 4 100 + 1 1000 + 5 10000 + 9 100000 + · · · = π . Our first task, then, to investigate infinite sums, called series , is to investigate limits of sequences of numbers. That is, we o cially call i =1 1 2 i = 1 2 + 1 4 + 1 8 + 1 16 + · · · + 1 2 i + · · · a series, while 1 2 , 3 4 , 7 8 , 15 16 , . . ., 2 i 1 2 i , . . . is a sequence, and i =1 1 2 i = lim i →∞ 2 i 1 2 i , that is, the value of a series is the limit of a particular sequence. While the idea of a sequence of numbers, a 1 , a 2 , a 3 , . . . is straightforward, it is useful to think of a sequence as a function. We have up until now dealt with functions whose domains are the real numbers, or a subset of the real numbers, like f ( x ) = sin x . A sequence is a function with domain the natural numbers N = { 1 , 2 , 3 , . . . } or the non-negative integers, Z 0 = { 0 , 1 , 2 , 3 , . . . } . The range of the function is still allowed to be the real numbers; in symbols, we say that a sequence is a function f : N R . Sequences are written in a few di ff erent ways, all equivalent; these all mean the same thing: a 1 , a 2 , a 3 , . . . { a n } n =1 { f ( n ) } n =1 As with functions on the real numbers, we will most often encounter sequences that can be expressed by a formula. We have already seen the sequence a i = f ( i ) = 1 1 / 2 i , | {
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11.1 Sequences 257 and others are easy to come by: f ( i ) = i i + 1 f ( n ) = 1 2 n f ( n ) = sin( n π / 6) f ( i ) = ( i 1)( i + 2) 2 i Frequently these formulas will make sense if thought of either as functions with domain R or N , though occasionally one will make sense only for integer values. Faced with a sequence we are interested in the limit lim i →∞ f ( i ) = lim i →∞ a i . We already understand lim x →∞ f ( x ) when x is a real valued variable; now we simply want to restrict the “input” values to be integers. No real di ff erence is required in the definition of limit, except that we specify, per- haps implicitly, that the variable is an integer. Compare this definition to definition 4.10.4. | {
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# Finding the two planes that contain a given line and form the same angle with two other lines
I am asked the following question:
There are two planes $\pi_1$ and $\pi_2$ such that each of them contain the line $t \{ (-1,4,0) + \lambda (1,2,3)$ and form the same angle with the lines $r \{ (0,0,0) + \beta (1,1,1)$ and $s \{ (-1,2,3) + \alpha (1,1,-1)$. Find the angle formed between those planes
My solution:
Let's say the normal vector of $\pi$ is $\vec{n} = (a,b,c)$. Since the angle between the plane(s) and the lines are equal,
\begin{align*} \measuredangle \left( \pi, r \right) &= \measuredangle \left( \pi, s \right)\\ \frac{\left( a,b,c \right) \cdot (1,1,1)}{\sqrt{a^2+b^2+c^2} \ \sqrt{3}} &= \frac{\left( a,b,c \right) \cdot (1,1,-1)}{\sqrt{a^2+b^2+c^2} \ \sqrt{3}}\\ a+b+c &= a+b-c\\ c &= 0 \end{align*}
Also, since $t$ is contained in the plane, the dot product between its vector and the normal vector of the plane is zero:
\begin{align*} (a,b,c) \cdot (1,2,3) &= 0\\ a+2b+3c&=0\\ a &= -2b \end{align*}
Assuming that the normal vector has length one,
$$a^2 + b^2 + c^2 = 1$$
So we have three equations with three variables and the solution for those is
$$a = \mp \frac{2}{\sqrt{5}} \quad a = \pm \frac{1}{\sqrt{5}} \quad c = 0$$
so the normal vectors of the two planes found are
$$\left( \frac{2}{\sqrt{5}} , - \frac{1}{\sqrt{5}} , 0 \right) \quad \left( - \frac{2}{\sqrt{5}} , \frac{1}{\sqrt{5}} , 0 \right)$$
and the angle between them is
$$\theta = \arccos \left(\frac{\left \vert - \frac{2}{\sqrt{5}} \frac{2}{\sqrt{5}} - \frac{1}{\sqrt{5}} \frac{1}{\sqrt{5}} + 0 \right \vert}{1} \right) = \arccos(1) = 0^\circ$$
My answer does not agree with the textbook's solution.
Textbook's solution:
$$\theta = \arccos \left( \frac{9}{\sqrt{95}} \right)$$
Did I make a mistake somewhere?
Thank you.
$$|(a,b,c) \cdot (1,1,1)| = |(a,b,c) \cdot (1,1,-1)| \\ |a+b+c| = |a+b-c|$$ yields
$$a+b+c = a+b-c \quad \text{ and } \quad a+b+c = -a-b+c$$ | {
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# A bag has only r red marbles and b blue marbles. If five red marbles
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A bag has only r red marbles and b blue marbles. If five red marbles [#permalink]
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25 Jul 2018, 23:16
00:00
Difficulty:
5% (low)
Question Stats:
87% (00:39) correct 13% (01:00) wrong based on 54 sessions
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### HideShow timer Statistics
A bag has only r red marbles and b blue marbles. If five red marbles and one blue marble are added to the bag, and if one marble is then selected at random from the bag, then the probability that the marble selected will be blue is represented by
A. b/r
B. b/(b + r)
C. (b + 1)/(r + 5)
D. (b + 1)/(r + b + 5)
E. (b + 1)/(r + b + 6)
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Re: A bag has only r red marbles and b blue marbles. If five red marbles [#permalink]
### Show Tags
25 Jul 2018, 23:20
Number of red marbles = r
Number of blue marbles = b
five red marbles and one blue marble are added to the bag
Number of red marbles = r + 5
Number of blue marbles = b + 1
Total number of marbles = r + 5 + b + 1 = r + b + 6
Probability of selecting a blue marbles = $$\frac{b + 1}{r + b + 6}$$
Hence option E
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Re: A bag has only r red marbles and b blue marbles. If five red marbles [#permalink]
### Show Tags
26 Jul 2018, 01:02
Bunuel wrote:
A bag has only r red marbles and b blue marbles. If five red marbles and one blue marble are added to the bag, and if one marble is then selected at random from the bag, then the probability that the marble selected will be blue is represented by
A. b/r
B. b/(b + r)
C. (b + 1)/(r + 5)
D. (b + 1)/(r + b + 5)
E. (b + 1)/(r + b + 6)
Red marbles = r
Blue marbles = b
Changing Situation :
Red = r + 5
Blue = b + 1
Total = r + s + 6
we are asked to pick up a blue marbles = $$\frac{( b + 1)}{r + s + 6}$$ | {
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Total = r + s + 6
we are asked to pick up a blue marbles = $$\frac{( b + 1)}{r + s + 6}$$
Thus the best answer is E.
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Re: A bag has only r red marbles and b blue marbles. If five red marbles [#permalink]
### Show Tags
30 Jul 2018, 10:51
Bunuel wrote:
A bag has only r red marbles and b blue marbles. If five red marbles and one blue marble are added to the bag, and if one marble is then selected at random from the bag, then the probability that the marble selected will be blue is represented by
A. b/r
B. b/(b + r)
C. (b + 1)/(r + 5)
D. (b + 1)/(r + b + 5)
E. (b + 1)/(r + b + 6)
After 5 red marbles and 1 blue marble are added to the bag, the bag has b + 1 blue marbles and r + b + 5 + 1 = r + b + 6 total marbles. So the probability of selecting a blue marble is (b + 1)/(b + r + 6).
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Re: A bag has only r red marbles and b blue marbles. If five red marbles &nbs [#permalink] 30 Jul 2018, 10:51
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## Probability Puzzles (Part 3)
Here’s a puzzle based on something interesting that I learned from Greg Egan. I’ve dramatized it a bit.
Traditional Tom and Liberal Lisa are dating. They discuss their plans for having children:
Tom: I plan to keep having kids until I get two sons in a row.
Lisa: What?! That’s absurd. Why?
Tom: I want two to run my store when I get old.
Lisa: Even ignoring your insulting assumption that only boys can manage your shop, why in the world do you need two in a row?
Tom: From my own childhood, I’ve learned there’s a special bond between sons who are next to each other in age. They play together, they grow up together… they can run my shop together.
Lisa: Hmm. Well, then maybe I should have children until I have a girl followed directly by a boy!
Tom: What?!
Lisa: Well, I’ve observed that something special happens when a boy has an older sister, with no intervening siblings. They play together, they grow up together… and maybe he learns not to be such a sexist pig!
They decide they are incompatible, so they split up and each one separately tries to find a mate who will go along with their reproductive plan.
Now for some puzzles. In these puzzles, assume that each time someone has a child, they have a 50% chance of having either a daughter or a son. Also assume each event is independent: that is, the gender of any children so far has no effect on that of later ones. Also ignore twins and other tricky issues.
Puzzle 1. If Tom carries out his plan of having children until he has two consecutive sons, and then stops, what is the expected number of children he will have?
Puzzle 2. If Lisa carries out her plan of having children until she has a daughter followed directly by a son, and then stops, what is the expected number of children she will have?
Puzzle 3: Which is greater, Tom’s expected number of children or Lisa’s? Or are they equal? | {
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Puzzle 3: Which is greater, Tom’s expected number of children or Lisa’s? Or are they equal?
For maximum benefit, try to answer Puzzle 3 before doing the calculations required to answer Puzzles 1 or 2.
### 64 Responses to Probability Puzzles (Part 3)
1. It’s late here right now so for now only preliminary quick guesses.
If you wanna have a more formal go, maybe don’t look at this until you want a hint or a distraction, depending on whether my guesses are right.
Puzzle 3 guess:
Since any particular two-birth-sequence is equally likely (1/4 each), I’d guess that both scenarios are equally likely.
Puzzle 1:
Possible sequences:
SS (1/4)
DSS (1/8)
DDSS (1/16)
SDSS (1/16)
DDDSS (1/32)
DSDSS (1/32)
SDDSS (1/32)
DDDDSS (1/64)
DDSDSS (1/64)
DSDDSS (1/64)
SDDDSS (1/64)
SDSDSS (1/64)
DDDDDSS (1/128)
DDDSDSS (1/128)
DDSDDSS (1/128)
DSDDDSS (1/128)
SDDDDSS (1/128)
DSDSDSS (1/128)
SDDSDSS (1/128)
SDSDDSS (1/128)
This looks suspiciously like $\sum_{n=1}^\infty \frac{\text{fib}\left(n\right)}{2^{n+1}}$
There is probably some nice combinatorial argument to make to prove this.
This sum – I didn’t prove but just quickly looked up – sums up to 1, so that’s good.
The expectation value version of that sum just adds an n:
$\sum_{n=1}^\infty n \frac{\text{fib}\left(n\right)}{2^{n+1}}$
And that appears to sum to 5.
Puzzle 2:
Possible sequences:
DS (1/4)
SDS (1/8)
DDS (1/8)
SSDS (1/16)
SDDS (1/16)
DDDS (1/16)
SSSDS (1/32)
SSDDS (1/32)
SDDDS (1/32)
DDDDS (1/32)
SSSSDS (1/64)
SSSDDS (1/64)
SSDDDS (1/64)
SDDDDS (1/64)
DDDDDS (1/64)
The series appears to be
$\displaystyle{ \sum_{n=1}^\infty \frac{n}{2^{n+1}} }$
which also sums to 1.
And the expectation value
$\displaystyle{ \sum_{n=1}^\infty \frac{n^2}{2^{n+1}} }$
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Furthermore, if I did this right, Lisa can expect her value to hover around 3 by 2, whereas for Tom that uncertainty sits around 4.69.
So in a fairly pessimistic (assuming you want to get there asap) outcome would be 5 trials for Lisa but 10 for Tom.
(I used that $\sigma^2 = \langle-^2\rangle$ but I’m tired and so possibly error-prone.)
• Michael Nelson says:
Here’s how fibonacci shows up:
You can count the number of ways of Tom winning via a string of n turns by breaking it up into two cases:
…….DSS
….DSDSS
The first case you counted already in the previous string of length n-1. And the second case you counted already from the string before that of length n-2.
• This is how I also ended up solving it, getting the Fibonnaci sequence immediately by using a construction approach to compute the number of binary strings (1 == boy, 0 == girl) that had the constraint of ending in ’11’ but with no other ’11’ in them. The construction as you show has the two ways to generate N+1: extend all the length N cases with a ‘0’ in the front and extend the N-1 cases adding ’10’ to the front. This means the number of ways to generate an appropriate string of length N+1 is Fib(N) since Count(N+1) = Count(N -1) + Count(N). And the total strings of length N+1 is $2^{N+1}$. Dividing gives you the probability of each string length (number of kids) and the expectation is as was shown above as well. And of course phi (the Golden Mean) shows up in the solution of the characteristic of the Fibonnaci recurrence so ultimately I ended up using it to solve this. Pretty cool!
2. It’s much quicker, Kram, to model each problem as a state machine.
Let $T_0$ and $T_1$ be Tom’s expectation of future children after zero and one son respectively. We can say immediately that $T_0 = \frac{1}{2}(s+T_1) + \frac{1}{2}(d+T_0)$ and $T_1 = \frac{1}{2}(s) + \frac{1}{2}(d+T_0)$. The solution, if I haven’t blundered, is $T_0 = 3d+3s$. | {
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For Lisa: $L_0$ is her expectation before her first daughter; $L_1$ is her expectation after any daughter. $L_0 = \frac{1}{2}(s+L_0) + \frac{1}{2}(d+L_1)$ and $L_1 = \frac{1}{2}(d+L_1) + \frac{1}{2}(s)$. Solution: $L_0 = 2d+2s$.
Lisa can expect two sons and two daughters; Tom can expect three daughters and three sons.
The qualitative difference is that for Tom a daughter after a son puts him back a step, while for Lisa a child of the sex not hoped for leaves her in the same position.
• John Baez says:
This looks impressively efficient, Anton! But what does an equation like
$T_0 = \frac{1}{2}(s+T_1) + \frac{1}{2}(d+T_0)$
actually mean? I guess I’m lacking the necessary familiarity with state machines to know what type of entities these variables represent, and what addition means. I know what a state machine is; I’m just not familiar with this kind of equation.
I can guess that $T_n$ is allowed to be any expression of the form $a s + b d$ where $a,b \ge 0$, and
$T_n = a s + b d$
means “after having had $n$ sons, Tom expects in the future to have a total of $a$ sons and $b$ daughters”.
But then I don’t know why
$T_0 = \frac{1}{2}(s+T_1) + \frac{1}{2}(d+T_0)$
should be true. It seems to say “after having no sons, Tom expects in the future to have the same thing as if he had a 50% chance of having one more son than what he expected after having one son, and a 50% chance of having one more daughter than what he expected after having no sons”.
The horrible quoted phrase here certainly shows why a more concise notation is a good thing! It reminds me of what happens when you try to state the quadratic equation in ‘plain English’ without variables.
But the problem is, I don’t see why it should be true.
• My subscripts mean only that state 0 must occur before state 1 can occur. I could also say explicitly $T_2 = L_2 = 0$; does that make anything clearer? | {
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My equations are the invariants of the tree of futures. The one that you query can be read as: “From state 0, the branches of Tom’s future are: with probability 1/2, he has a son and moves to state 1; and with probability 1/2, he has a daughter and remains in state 0.” As long as his sequence of daughters continues, his expectation of future children remains the same, so the definition of T0 is recursive. From state 1 (one son) Tom cannot remain in state 1; he must either terminate or go back to 0. After a daughter, previous sons are irrelevant to Tom’s future tree, which is why there are not more states in this model.
• Chris Upshaw says:
Shouldn’t that be s * T1 then, if its a parallel notation to sum of products datatype notation? Its certainly not the same operation as the middle ‘+’.
• That’s cos I don’t know nothin bout notating no datatypes.
• John Baez says:
I found Anton’s trick very hard to understand at first, mainly because the variables and operations in his equations had not been explained (and I didn’t already know this trick). Like Greg, I found XJY’s comment here helpful. Even then, I had to hit myself over the head with a rock a few times before I really understood what was going on.
I’ve spent a lot of time thinking about other aspects of Markov processes, but I never thought about this particular trick. It’s very nice, but I think I need to digest it and blend it with other things I know, like generating functions and—yes—recursively defined data types.
• Anton,
This is a really cool solution! Not only do you compute Lisa’s expected # of kids, you compute how many sons and daughters she can expect. I have never seen this concept of expressing the invariants of the tree of futures before. Do you have a good reference for learning more about this? Is it standard CS?
Thanks for posting this.
• Rob, sorry, I have no idea where I got the, er, idea. | {
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Thanks for posting this.
• Rob, sorry, I have no idea where I got the, er, idea.
“Invariant” may be a misnomer; thinking about it further, I reckon the analogy with the loop invariants of CS is not so strong.
I may have misunderstood the problem but it seems to me that as soon as Tom or Lisa reach their objective they STOP. Then Tom can have 2 sons in a row and it is over. On the other hand if they have a daughter Lisa can get a son and they stop, Tom can never get two sons.
• John Baez says:
Wow, you assumed that Tom and Lisa are married, and that their plans affect each other’s activities!
I never thought that: I was imagining two people at a bar, who obviously hate each other, each discussing their own separate plans. But now I see that there’s an ambiguity in the phrase “their plans for having children”. I’ll fix that!
• John Baez says:
Upon rereading what I wrote, I see there’s no real ambiguity in the puzzles as stated. One asks what will happen if Tom carries out his plan. The other asks what will happen if Lisa carries out her plan. There’s nothing about what will happen if both of them try to carry out their conflicting plans.
Nonetheless I added some stuff to make it crystal clear that Tom and Lisa are not going to be having children together.
4. jamievicary says:
I deliberately haven’t worked this out formally, because it’s more fun to see if intuition leads to a right answer. Write their progeny as a string in D and S. It’s ‘easy’ for a string of length n to lack a SS substring, as it could have no Ss, or just one S, or just two Ss spaced at least one apart, etc. But to lack a DS substring is relatively ‘hard’: you have to have all the sons before all the daughters. So, holding out for two sons in a row will take longer than waiting for a daughter followed by a son. | {
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I guess the reason it’s a bit paradoxical is that for any two adjacent children, SS is just as likely as DS. I suppose the reason is that combinatorially, like-gendered children are indistinguishable. In reality this is not the case. They should just ask for “two children adjacent in age, both with the temperament to run a shop”!
(By the way, I found this post cognitively difficult to write, since on parenting forums DD and DS often meen ‘darling son’ and ‘darling daughter’.)
• John Baez says:
Jamie wrote:
I guess the reason it’s a bit paradoxical is that for any two adjacent children, SS is just as likely as DS.
Right, that was supposed to make the problem cognitively difficult.
(By the way, I found this post cognitively difficult to write, since on parenting forums DD and DS often meen ‘darling son’ and ‘darling daughter’.)
I never knew! I hadn’t expected that to cause cognitive difficulties. Do SD and SS mean ‘stupid daughter’ and ‘stupid son’?
(I would probably be banned from the forum for that joke.)
• Tamfang says:
or Silly or Sweet
• Charles Jackson says:
Well, you did make the problem cognitively difficult for me.
I did #3 first. I thought P(SS) = P(DS) => Expected births to first SS is same as for DS. WRONG—but
I did 2 relatively quickly with something like Anton’s approach above.
Then I looked at the answers in the comments. Oh, oh. I thought about it for awhile and I think I see an easy explanation for the answer.
Consider the outcomes with up to three children. Make a little truth table-like figure with all 8 possible sequences of combinations of boy and girl babies. See below. Notice that there are 4 occurrences of the sequence DS in the table—each of which is the first occurrence of DS in the sequence containing it. (capitalized in table below)
Birth Order
1 2 3
d d d
d D S
D S d
D S s
s d d
s D S
s s d
s s s
Now, search for SS, but only the 3 that are the first occurence in time.
d d d
d d s
d s d
d S S
s d d
s d s
S S d
S S s | {
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d d d
d d s
d s d
d S S
s d d
s d s
S S d
S S s
The difference (only 3 SS versus 4 DS) arises from the second SS that is a subsequence of the SSS sequence
d d d
d d s
d s d
d s s
s d d
s d s
s s d
s S S
So, if the parties were restricted to 3 or fewer babies, Lisa would have a 50% chance of getting DS, but Tom would only have a 3/8 chance of getting SS.
Of course, if they both had three babies, Tom would have a 1/8 chance of getting SS twice, but Lisa would have zero chance of getting DS twice.
At the birth of the second baby they each have a 25% chance of getting their desired outcome. On the third baby, Lisa would also have a 25% chance but Tom would only have a 12.5% chance.
Lisa’s probability of terminating the process at 3, 4, or 5 is higher than the probability for Tom. This pushes up the expected number for him relative to her.
Chuck
PS Thanks for your blog. I only read some of the posts but always enjoy those I complete. No doubt if I had more time and patience I get even more enjoyment from it.
• John Baez says:
Nice. I’m glad you’re enjoying this blog!
• Jenny Meyer says:
The DD and DS tags are just a convention for anonymizing your family members. The D can be heavily ironic in context.
(The picture of the Duggars was enough to make this too cognitively difficult for me.)
5. Michael Nelson says:
I think I understand. If we think of this as a game, where Lisa needs to get the sequence DS and Tom needs to get the sequence SS, then Lisa will certainly have the advantage. As soon as a D shows up, Tom will lose. On the other hand, if a S shows up, Lisa can still win. So Lisa has a greater chance of winning this game.
• John Baez says:
I’m not assuming that Tom and Lisa are having children with each other: they are two separate people, each pursuing their own separate agenda with their own spouse, each assuming their spouse will knuckle under and go along with their insane plan. | {
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Nonetheless you can still think of it as a game, where we imagine a random stream of babies being produced and both Tom and Lisa saying when they, personally, would prefer the stream to stop.
• Michael Nelson says:
Yes, this is what I meant. You phrased it better for me. I think I have a better way to think about this now. The number of ways that Lisa can win at the nth spot is just n. For example:
DDDDDS
SDDDDS
SSDDDS Lisa wins at 5th spot
SSSDDS
SSSSDS
DDDDDDS
SDDDDDS
SSDDDDS
SSSDDDS Lisa wins at 6th spot
SSSSDDS
SSSSSDS
The number of ways that Tom can win at the nth spot involves the fibonacci numbers though. For example:
DDDDSS
SDDDSS
DSDDSS Tom wins at 5th spot
DDSDSS
SDSDSS
DDDDDSS
SDDDDSS
DSDDDSS
DDSDDSS Tom wins at 6th spot
SDSDDSS
SDDSDSS
DSDSDSS
DDDDDDSS
SDDDDDSS
DSDDDDSS
DDSDDDSS Add D just before SS in 6th spot wins
SDSDDDSS
SDDSDDSS
DSDSDDSS
plus Tom wins at 7th spot
SDSDSDSS
DDSDSDSS
DSDDSDSS Replace the last S in 5th spot wins with DSS
SDDDSDSS
DDDDSDSS
That’s pretty cool that the fibonacci numbers are involved.
6. Dave Doty says:
I worked it out like this. I am nagged by the possibility that there is a more elegant simple way to do it.
Each is picking a string from the alphabet $\{ s,d \}$. Both have a list of “allowed” strings of length $n$ that indicate they must keep going.
Lisa omits from this list all strings with the substring $sd$, and Tom omits all strings with the substring $ss$. Already we can see that Lisa has fewer allowed strings than Tom, and will therefore wait a shorter time before stopping, because to remain in an allowed string, she can only transition once (from $s$ to $d$), whereas Tom can transition between $s$ and $d$ an arbitrary number of times.
Lisa’s allowed strings match the regular expression $s^*d^*$; e.g., for length 5, her 6 allowed strings are
$sssss, ssssd, sssdd, ssddd, sdddd, ddddd$ | {
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$sssss, ssssd, sssdd, ssddd, sdddd, ddddd$
In general, Lisa has $n+1$ allowed strings of length $n$, so probability $\frac{n+1}{2^n}$ that a given length $n$ string will be allowed. If the string is $s^*$, then either of $s$ or $d$ is allowed next, whereas if it ends in a $d$, only a $d$ is allowed next. So the probability that Lisa ends on length $n$ is the probability that she does not end before (i.e., the length $n-1$ prefix is allowed) times the probability that the whole length $n$ string is disallowed, conditioned on length $n-1$ prefix being allowed. This is
$\displaystyle{ \frac{1}{2^{n-1}} \cdot 0}$
(the probability the string is $s^{n-1}$, times probability 0 since she will not stop after $s^{n-1}$), plus
$\displaystyle{ \frac{n}{2^{n-1}} \cdot \frac{1}{2} }$
(the probability the string is $s$‘s followed by at least one $d$, times the probability that the next symbol is $s$).
So Lisa’s probability of stopping after $n$ children is $\frac{n}{2^n}$.
Thus her expected stopping time is
$\displaystyle{ \sum_{n=2}^\infty n \cdot \frac{n}{2^n} = \sum_{n=2}^\infty \frac{n^2}{2^n} = \frac{11}{2} }$
Tom’s allowed strings omit the substring $ss$. We can count recursively how many strings of length $n$ satisfy this, denoted by $T_n$. If the first symbol is $d$, then all allowed strings of length $n-1$ could follow, of which there are $T_{n-1}$. If the first symbol is $s$, then the next symbol must be $d$, and any allowed string of length $n-2$ could follow, of which there are $T_{n-2}$. So $T_n = T_{n-1} + T_{n-2}$, the $n$‘th Fibonacci number, which is
$\displaystyle{ \frac{(1+\sqrt{5})^n - (1-\sqrt{5})^n}{2^n \sqrt{5}} }$
So Tom’s probability of stopping after $n$ children is the probability that the last three symbols are $dss$ (if it were $sss$ he would have stopped at the second $s$) the prefix of length $n-3$ is allowed, which is
$\displaystyle{ \frac{T_n}{2^n} = \frac{(1+\sqrt{5})^n - (1-\sqrt{5})^n}{2^{2n} \sqrt{5}} }$ | {
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$\displaystyle{ \frac{T_n}{2^n} = \frac{(1+\sqrt{5})^n - (1-\sqrt{5})^n}{2^{2n} \sqrt{5}} }$
Thus his expected stopping time is
$\displaystyle{ \sum_{n=2}^\infty n \cdot \frac{(1+\sqrt{5})^n - (1-\sqrt{5})^n}{2^{2n} \sqrt{5}} = \frac{19}{2} }$
• John Baez says:
Wow, what a tour de force! You are not getting the officially approved correct answers, but the officially approved correct answers do involve Fibonacci numbers, at least in the most direct approach.
7. Bruce Smith says:
I carefully solved each puzzle before reading beyond it (in case of spoilers), so puzzle 3 was not very interesting! (I had thought of it while starting to work on puzzle 2, but didn’t have the benefit of your stating it was worthwhile to work on it first, so I didn’t do so.)
Another mistake — I initially assumed Tom and Lisa were married and were discussing their joint plan for having kids! This leads to puzzle 4 — suppose they do marry (before having any kids) and agree to have kids until both their criteria are satisfied…
• John Baez says:
I think it’s too late for me to make Puzzle 3 into Puzzle 1 without causing massive confusion.
I have added verbiage to the puzzle to make it clear that Tom and Lisa are not married and never will be.
Nonetheless, Puzzle 4 is interesting, and will doubtless lead to an even larger expected family.
• For that case I get seven expected children, again equally divided.
Next question: For each scenario, what are the variances? Do the variance in daughters and the variance in sons add up to the variance in total children? I don’t know how to figure these. | {
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• Unless I made an error, I already did the variances – or, well, the standard deviations which are just the square roots of the variances – in my reply (first one). However, all the other contributions here are much more fleshed out and nice. My approach was really just trying a couple and then guessing. It seems like the results were right, however it’s nice to read in the others’ replies why they are right.
To get the variance, if you have all the sub-probabilities, you just do:
Probability: $P=\sum_{n \in \mathbb{N}} p\left(n\right) = 1$
Expectation: $E = \sum_{n \in \mathbb{N}} n p\left(n\right)$
Variance: $\sigma^2 = \sum_{n \in \mathbb{N}} n^2 p\left(n\right) - E^2$
Standard Deviation: $\sigma = \sqrt{\sigma^2}$
For some reason, in my original reply, the $\LaTeX$ didn’t work right. Looks like the fixed version still isn’t right though. Hopefully the above works.
• John Baez says:
The LaTeX at the very end of this comment of yours didn’t work, and it was pretty cryptic, so I guessed you were trying to say this:
$\sigma^2 = \langle-^2\rangle$
This means ‘the square of the standard deviation is the expected value of the square of the quantity in question’. (The little dash is a standard symbol for ‘the quantity in question’: a blank to be filled in.) But this is only true for quantities whose mean is zero, which isn’t the case in this problem. So, I must have guessed your meaning incorrectly. I should have written
$\sigma^2 = \langle-^2\rangle - \langle - \rangle^2$
But probably you weren’t trying to use the ‘little dash’ notation, which gets a bit scary in formulas that also contain minus signs!
8. Ilya Surdin says:
Puzzle 1:
suppose d is the expectation value we’re looking for.
then d=0.5(d+1)+0.5(0.5*2+0.5(d+2)) [if a daughter is born first, we’re back to point 1 again, if a son is born first then if another son is born the value is 2, and if a daughter is born second we’re back to square 1].
so we get d = 0.75d +1.5 -> d=6 | {
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Puzzle 2:
suppose e is the expectation value we’re looking for.
e = 0.5(e+1) + 0.5f,
where summand 1 is if a son is born, and summand 2 is if a daughter is born, we only need to get a son (ds, dds, ddds, all options are good)
f = 2 [ f=0.51+0.5(f+1) ]
so e = 0.5e + 0.5 +1 -> e=3.
Puzzle 3:
At first sight, I guessed they will be equal.
9. Graham says:
Here’s a solution for Tom’s case using generating functions. Let $T$ be a random variable representing the number of children Tom has. Let $f(s) = \sum p_i s^i$ be the generating function for T, where $p_i = \Pr(T=i)$. The first few $p_i$ are
$p_0 = p_1 = 0, p_2=1/4, p_3=1/8.$
For $i \ge 4$, the sequence must end DSS, and there must be no SS occurring in the first $i-3$. So
$p_4 = (1/8)(1-p_1),$
$p_5 = (1/8)(1-p_1-p_2),$
and so on.
From this it follows that
$p_i = p_{i-1} - (1/8)p_{i-3}$
for $i \ge 4$.
Then by comparing $f(s), sf(s), s^3f(s)$, it is possible to calculate that
$\displaystyle{ f(s) = \frac{-s^2}{s^2+2s-4}. }$
Then
$E(X) = f'(1) = 6$
and
$E(X(X-1)) = f''(1) = 52$
from which $\mathrm{var}(X) = 52 + 6 -6^2 = 22$.
10. “In one word he told me secret of success in mathematics: Generalize!” What if the sex ratio at birth is not 1:1? I’ll get right on that — later.
• John Baez says:
Puzzle 5. For what sex ratio at birth would Tom and Lisa have an equal expected number of children?
(Obviously it needs to be more sons than daughters.)
• Greg Egan says:
John wrote:
Puzzle 5. For what sex ratio at birth would Tom and Lisa have an equal expected number of children?
$\displaystyle{ p_{son} = \frac{\sqrt{5}-1}{2} }$
or:
$\displaystyle{\frac{p_{son}}{p_{daughter}} = \frac{\sqrt{5}+1}{2} }$ | {
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or:
$\displaystyle{\frac{p_{son}}{p_{daughter}} = \frac{\sqrt{5}+1}{2} }$
To prove this (stealing from and generalising an answer by “XJY”, which is a slightly more coarse-grained version of Anton Sherwood’s approach), we can write the expectation values for the number of additional children that Tom and Lisa will each have after their latest child was a daughter or a son (who didn’t yet complete their target sequence) as:
$T_d = p_{son} (1+T_s) + (1-p_{son}) (1+T_d)$
$T_s = p_{son} +(1-p_{son}) (1+T_d)$
$L_d = p_{son} + (1-p_{son}) (1+L_d)$
$L_s = p_{son} (1+L_s) + (1-p_{son})(1+L_d)$
These are solved by:
$\displaystyle{T_d = \frac{1+p_{son}}{p_{son}^2} }$
$\displaystyle{T_s = \frac{1}{p_{son}^2} }$
$\displaystyle{L_d = \frac{1}{p_{son}} }$
$\displaystyle{L_s = \frac{1}{p_{son}(1-p_{son})} }$
But $T_d$ and $L_s$ are also the expectation values for the number of children each will have in total, because they have 0 out of 2 elements of their target sequence.
So if we solve:
$L_s = T_d$
the positive solution for $p_{son}$ is:
$\displaystyle{ p_{son} = \frac{\sqrt{5}-1}{2} }$
and
$L_s = T_d = 2+\sqrt{5}$
11. Bruce Smith says:
It turns out these puzzles show up (for numbers 1 and 3 sought by Alice and Bob) in this (otherwise mostly unrelated) blog post about primes:
https://rjlipton.wordpress.com/2016/03/26/bias-in-the-primes/
• John Baez says:
• Erica Klarreich, Mathematicians discover prime conspiracy, Quanta, 13 March 2016. | {
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• Erica Klarreich, Mathematicians discover prime conspiracy, Quanta, 13 March 2016.
Soundararajan was drawn to study consecutive primes after hearing a lecture at Stanford by the mathematician Tadashi Tokieda, of the University of Cambridge, in which he mentioned a counterintuitive property of coin-tossing: If Alice tosses a coin until she sees a head followed by a tail, and Bob tosses a coin until he sees two heads in a row, then on average, Alice will require four tosses while Bob will require six tosses (try this at home!), even though head-tail and head-head have an equal chance of appearing after two coin tosses.
I decided to dramatize this using children instead of coin tosses.
By the way, I wrote about the ‘prime conspiracy’ here:
• John Baez, Weirdness in the primes, _The n-Category Café 15 March 2016.
There’s a bit of math here that’s not in the Quanta article.
12. Rob says:
Nice Puzzle.
from “Mathematical Plums” edited by Ross Honsberger, chap 5 some surprises in probability, “Suppose a fair coin is tossed repeatedly and an indefinitely long sequence of H’s and T’s is generated…. what a surprise to learn that it is twice as likely that the triple TTH will be encountered ahead of the triple THT…” I love surprises in probability, and I find that most things about probability surprise me! The chapter also talks about something else you may like if you haven’t seen it, “Efron’s Dice”.
By the way, have you read this paper, http://www.emis.ams.org/journals/TAC/volumes/26/4/26-04.pdf, Commutative Monads as a Theory of Distributions. I gather Prof. Kock is developing the theory to study probability theory. I can’t really understand it, but you may find it interesting.
• John Baez says:
Probability theory is often counterintuitive to humans, which must have something to do with why humans mess up so often. | {
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I haven’t read that paper. I’m pretty good friends with Anders Koch’s son Joachim, who also does category theory related work. But this paper is by Anders. I find it pretty hard going, at first because I’ve never understood strengths as much as I’d like, even though I did help come up with the more conceptual definition of them given on the link. (The more common definition, which involves writing down some commutative diagrams, always bothered me.)
• Rob says:
Say the first child is a G. Then we are guarateed that GB will occur prior to to BB. (cause for BB to happen, we must first have had GB)
Now say the first child is a B. Then either the second child is a B, in which case BB happens first, or it is a G, in which case, again, GB will happen before BB.
So we see that GB is 3x more likely to occur in any given familiy than BB. Thus I would expect that the expected value of the second random varable is less than the expected value of the first. Hah! well probability is so amazing, I expect that I am wrong! Now I will read through the comments where I expect to find the right answer.
Also, if you don’t understand that paper, I never will! But I think it is really interesting so I will keep reading it. BTW I think your notes on Cat Theory are great. Thanks to all your students for putting that together.
13. arch1 says:
Puzzle 3: Represent all possible birth sequences as an infinite binary tree (at the risk of scaring you we’ll pretend that childbearing doesn’t stop when the goal is reached). Yes, at each level of the tree starting with the 2nd, ¼ of the arcs are DS arcs and ¼ are SS arcs. But the symmetry breaks when we look at the correlation between adjacent levels: While an SS arc can immediately follow an SS arc, a DS arc can’t immediately follow a DS arc (as that would require the shared middle node to be both an S and a D). | {
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Upshot: Half of the SS arcs “hide behind” an immediate SS parent, but DS-DS repulsion prevents DS arcs from ever doing this. This makes a DS arc likelier than an SS arc to be the first arc of its kind on a path from the root (put differently, it causes DS arcs to more efficiently “cover” the tree as viewed from the root). So the expected number of levels before encountering a DS arc is less than for an SS arc.
Greg and John, I find these probability paradoxes very instructive and great for my humility. I hope you keep them coming.
• John Baez says:
Nice answer! Yes, probability puzzles are great for ones humility—maybe the right verb is “humiliating”?
There should be a book of probability puzzles, similar to Mark Levi’s excellent physics puzzle book Why Cats Land on Their Feet: And 76 Other Physical Paradoxes and Puzzles. But I don’t know it!
If anyone here hasn’t tried Probability Puzzles (Part 1) and Probability Puzzles (Part 2), now is the time! They’re all about boys and girls, oddly enough. The first one is particularly devastating, or infuriating.
• arch1 says:
(chuckle) I actually tend to find them more invigorating than humiliating, partly for noble reasons but mostly I think because I can’t wait to spring them on others.
Ironically the closest I’ve come to humiliation recently was last week when I, er, “judged” a high school science fair project which explained how cats can land on their feet using gauge theory.
• cclingen says:
High school?! Cats landing their feet?! Gauge theory?! Where can I learn more?
• John Baez says:
Hi, Charlie! It’s probably good to start here:
• Wikipedia, Falling cat problem.
but then go to this paper by a friend of mine who is an expert on classical mechanics at U. C. Santa Cruz:
• Richard Montgomery, Gauge theory of the falling cat, in M. J. Enos, Dynamics and Control of Mechanical Systems, American Mathematical Society, pp. 193–218.
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• arch1 says:
Nowhere as far as I know (certainly not me as I gleaned only a scant understanding). Maybe a paper will come out later; meanwhile I’d prefer to let the student pursue things in his own way.
• Re Prob Puzzle books. I just bought “Fifty Challenging Problems in Probability”. It looks like it has some really cool stuff in it. And best of all it was cheap!
• Greg Egan says:
The counterintuitive aspects of real statistical phenomena might have put an innocent man in prison for 38 years:
An Unusual Pattern
Like the statisticians who discuss the case, I’m not claiming any certainty that Geen is innocent, but on the face of it the statistical basis of the prosecution’s argument appears alarmingly wrong-headed.
• That is creepy. As teachers we need to do a better job at explaining statistics. Examples like the above and the OP help a lot.
14. pauljurczak says:
I hate to be a party pooper, but using infinite series in the top answer is yet another reason why lies and statistics are often mentioned in the same sentence. Human female can only have a limited number of children, much smaller than 100, which is much smaller than infinity.
This is in general a problem of using mathematics to model physical “reality”, which always requires certain assumptions resulting in departure from “reality”. In this case, the assumption is immortality or more specifically, unlimited childbearing capability.
P.S. Don’t take this post too seriously ;-)
• John Baez says:
I won’t take it too seriously—just more seriously than you expected.
Puzzle 7: How do the answers for Tom’s and Lisa’s expected number of children change if we impose a ‘cutoff’, saying that a woman can have no more than 10 children?
• …Or if each pregnancy has probability q of leaving the mother unable to bear further children?
• Charles Jackson says:
Note that that there is about a 1% that Tom will have to have 24 or more children before the desired SS outcome occurs.
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15. Edon says:
These processes can be seen as absorbing Markov chains. Let $T$ be the block in the transition matrix that holds the transitions from transient states to other transient states. It is a basic property that $(I - T)$ is invertible and that the $(i,j)$ entry of $M=(I - T)^{-1}$ gives the expected number of times the chain is in state $j$ before it is absorbed, given that we start from state $i$. Summing the entries in the appropriate row yields the answers to the puzzles.
16. For some reason there isn’t a reply button below that latest reply of you, John, but yeah, that was what I was going for, although indeed not with the ‘little dash’ notation, but that works too. With just one minus sign it’s not that bad. Thanks for the fix. There must have been more wrong with it than just $\langle \cdot \rangle$ if it was so cryptic. I’m not sure what I typed anymore but while typing it that was the only part I didn’t know whether it’d work. The rest I was pretty sure of. But oh well. I thought \left{<} would work for \langle but you guessed that part right.
• John Baez says:
Kram wrote:
For some reason there isn’t a reply button below that latest reply of you…
Unfortunately if you tell this blog that you don’t want comments to get indented more than 4 times (because they become absurdly skinny), it implements this by eliminating the ‘Reply’ button after a comment that’s been indented 4 times.
What you should do in this case is reply not to the comment but to the comment it’s commenting on! In fact it’s often wise to do this. This keeps the comments from getting too skinny. I can tell who is a really expert blog-commenter by whether they do this.
I thought \left{<} would work for \langle but you guessed that part right.
I’ve never seen \left{<} in LaTeX; I think LaTeX is pretty careful about the difference between the less than symbol $<$ and the left angle bracket $\langle$. | {
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But there’s an extra problem with LaTeX in most blogs! In HTML, < is an escape symbol, so you to get a less than symbol in LaTeX you have to use the HTML symbol < rather than <. It’s very bizarre to be using HTML commands inside LaTeX equations, but that’s how it works! That’s true not only for this rather brain-dead WordPress blog but also some others I use.
17. Tom says:
Puzzle 6: Why did John use the name of his wife for the female character in this drama?
• John Baez says:
Answer: because it was the first woman’s name beginning with L that I thought of.
18. David Cinabro says:
This one got in my head, and I had to try to solve it.
First Puzzle 3. Two children SS, SD, DS, DD. Tom and Lisa both stop 1 out of 4. Three Children for Tom SDS, SDD, DSS, DSD, DDS, DDD. Tom stops 1 out of 6. Three Children for Lisa SSS, SSD, SDS, SDD, DDS, DDD. Lisa stops 2 out of 6. Clearly the expected number of children is different and Lisa will have a smaller number than Tom.
For Tom’s expected number of children, he has 1/4 chance stopping at 2, 1/6 stopping at 3, 2/10 stopping at 4, 3/16 stopping at 5, 5/26 stopping at 6, etc. The stopping chance numerators are Fibonacci seeded by 1 and 1, and the denominators are Fibonacci seeded by 4 and 6. The expected number of children is then 2(1/4) + 3(1-1/4)1/6 + 4(1-1/4)(1-1/6)3/16+…. ie he stops at 2 with a probability of 1/4, at 3 with a probability of (1-1/4)*1/6, etc. Perhaps there is a clever way to write that, but I wrote a little program to tell me that it goes to 6.
Lisa has 1/4 chance stopping at 2, 2/6 stopping at 3, 3/8 stopping at 4, etc. The stopping chance numerators are 1,2,3,etc. and the denominators at 4,6,8, etc. The same sort of calculation gives the expected number of children 21/4 + 3(1-1/4)2/6+4(1-1/4)(1-2/6)*3/8+… This one is certainly even simpler to work out in closed from , but again the program tells me that it sums to 4.
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