text stringlengths 1 2.12k | source dict |
|---|---|
$$\frac{x^4}{x^4+(x-x)^2}=1\xrightarrow[x=y\to0]{}1$$
Thus the limit depends on the path chosen $$\;\implies\;$$ the limit doesn't exist...and this is the relevant argument: going on different paths yields different limits.
You did right.
A possible improvement is to compute the limit along an arbitrary line: along $$y=mx$$ you have to compute $$\lim_{x\to0}\frac{x^2(mx)^2}{x^2(mx)^2+(x-mx)^2}= \lim_{x\to0}\frac{m^2x^4}{m^2x^4+x^2(1-m)^2}= \lim_{x\to0}\frac{m^2}{m^2+(1-m)^2x^{-2}}= \begin{cases} 0 & m\ne 1 \\[4px] 1 & m=1 \end{cases}$$ Doing this way may immediately show how to solve the business. Of course, if you find that all limits along lines are equal, you cannot conclude that the limit exists; instead, you should try along other curves, if you suspect the limit doesn't exist.
What you did is correct, but note that after proving that the limit is $$0$$ when you take $$y=0$$, the fact that it is also $$0$$ when you take $$x=0$$ is irrelevant. What matters here is that going to $$(0,0)$$ through two different directions leads you to two distinct limits. Therefore, there is no (global) limit at $$(0,0)$$. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9825575137315161,
"lm_q1q2_score": 0.8501324131535155,
"lm_q2_score": 0.865224072151174,
"openwebmath_perplexity": 183.92543095249806,
"openwebmath_score": 0.8499876856803894,
"tags": null,
"url": "https://math.stackexchange.com/questions/3102390/show-that-limit-does-not-exist-two-variables"
} |
# Is $\mathbb R^2$ a field?
I'm new to this very interesting world of mathematics, and I'm trying to learn some linear algebra from Khan academy.
In the world of vector spaces and fields, I keep coming across the definition of $\mathbb R^2$ as a vector space ontop of the field $\mathbb R$.
This makes me think, Why can't $\mathbb R^2$ be a field of its own? Would that make $\mathbb R^2$ a field and a vector space?
Thanks
-
A field has multiplication. How would you define multiplication on $\mathbb R^2$ so that it is a field? (There is a way to do so, but it isn't "obvious" until you realize that the resulting field is the complex numbers...) – Thomas Andrews Dec 4 '12 at 16:07
$\mathbb{R}^2$ can be a field but with multiplication defined as follows: $(a,b)(c,d) = (ac - bd, ad + bc)$. Indeed, this is one way of defining the complex numbers. – Rankeya Dec 4 '12 at 16:08
But, if you want to try to do this for $\mathbb{R}^n$, for $n \geq 3$ such that $\mathbb{R}$ is naturally embedded in $\mathbb{R}^n$ as a subfield, then it is not possible to do so, and this is a harder fact to prove. – Rankeya Dec 4 '12 at 16:11
ok this just answered my follow-up question. Is there any proof of that being true? – vondip Dec 4 '12 at 16:11
Vondip - Perhaps this is at a slight tangent, but a significant difference between R and C is that R is an ordered field and C is not. e.g. 5 is larger than 3, but which is "larger", 4 + 7i or 6 + 5i ? (Answer: well, defining how "large" or "the length" a complex number is is not as obvious as for the reals. In fact, there are many different ways of defining the length of a complex number). Just something to think about. – Adam Rubinson Dec 4 '12 at 16:37
If you define:
$$(a,b)+(x,y):=(a+x,b+y)$$
$$(a,b)\cdot (x,y):=(ax-by,ay+bx)$$
then the set $\,\Bbb R^2=\Bbb R\times\Bbb R\,$ turns into a field, and a rather well known and important one. Can you identify it? | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.982557512709997,
"lm_q1q2_score": 0.8501324088550003,
"lm_q2_score": 0.8652240686758841,
"openwebmath_perplexity": 279.8928370392323,
"openwebmath_score": 0.9212859272956848,
"tags": null,
"url": "http://math.stackexchange.com/questions/250768/is-mathbb-r2-a-field"
} |
-
Spoiler: take a look at one of the comments given. $\^\smile\^$ – FrenzY DT. Dec 4 '12 at 16:10
complex numbers indeed! fantastic how it all connects! Would that make R^2 a field and a vector space? – vondip Dec 4 '12 at 16:11
Yes, it would, because addition in $\mathbb{R}^2$, as @DonAntonio defines it, is component wise (the usual way). Remember, the vector space structure depends only on the underlying abelian group. – Rankeya Dec 4 '12 at 16:15
What do you mean "vector space"? Any field is a vector space over any subfield, so the field $\,\Bbb R^2\cong\Bbb C\,$ is a vector field ove an infinite number of subfields, say $\,\Bbb C\,,\,\Bbb R\,,\,\Bbb Q\,,\,\Bbb Q(i)\ldots\,$ , etc. – DonAntonio Dec 4 '12 at 16:15
So would that mean that I any vector space could be defined when F -being the field, as : F^n ? – vondip Dec 4 '12 at 16:17
It is important to understand that a set on its own has no algebraic structure. By defining operators on $\mathbb{R}^2$ you could turn it into (almost) anything you like.
The natural operators on $\mathbb{R}^2$, namely $(x, y) + (a, b) \mapsto (x+a, y+b)$ and $(x, y) \cdot (a, b) \mapsto (x\cdot a, y\cdot b)$ do not define a field as $(0, 1)$ has no multiplicative inverse.
-
Usually in mathematics one defines these structures as tuples
A field is a triple $(K,+,\cdot)$ such that $K$ is a set and [...] and $\cdot:K \times K \rightarrow K$
A Vectorspace is a triple $(V,+,\cdot)$ such that $V$ is a set and [...] and $\cdot: K \times V \rightarrow V$
So your question is meaningless: A set (say $\mathbb R^2$) cannot be a field or a vectorspace or a group or anything - only if you add some additional structure (most of the time operations) you can ask this question. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.982557512709997,
"lm_q1q2_score": 0.8501324088550003,
"lm_q2_score": 0.8652240686758841,
"openwebmath_perplexity": 279.8928370392323,
"openwebmath_score": 0.9212859272956848,
"tags": null,
"url": "http://math.stackexchange.com/questions/250768/is-mathbb-r2-a-field"
} |
For example $\mathbb R^2$ can be the set used in the definition of a field, as well as the underlying set used in the definition of a vectorspace. And we are happy, the addition operation $$+:\mathbb R^2 \times \mathbb R^2 \rightarrow \mathbb R^2$$ is the same, and the multiplicaton for "the" vectorspace structure $$\mathbb R \times \mathbb R^2 \rightarrow \mathbb R^2$$ is "compatible" with the multiplication for "the" field structure $$\mathbb R^2 \times \mathbb R^2 \rightarrow \mathbb R^2$$
-
Adding to the above answer. With the usual exterior multiplication of the $\mathbb{R}-\{0\}$ as a ring with the natural addition and multiplication you can not make a field out of $\mathbb{R}^{2}-(0,0)$ \ But there may exist other products such as the one in the answers which can make a field out of ${\mathbb{R}\times\mathbb{R}}-\{ 0\}$ \ According to one of the theorems of Field theory every field is an Integral domain. So by considering : ${\mathbb{R}\times\mathbb{R}}-\{ 0\}$ With the following natural product: $(A,B)*(C,D)=(AB,CD)$ We see that $(1,0)*(0,1)=(0,0)$ Which means that $\mathbb{R}^{2}$is not an integral domain and hence not a field. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.982557512709997,
"lm_q1q2_score": 0.8501324088550003,
"lm_q2_score": 0.8652240686758841,
"openwebmath_perplexity": 279.8928370392323,
"openwebmath_score": 0.9212859272956848,
"tags": null,
"url": "http://math.stackexchange.com/questions/250768/is-mathbb-r2-a-field"
} |
-
Why are you considering $\mathbb{R}^*\times\mathbb{R}^*$? – Tobias Kildetoft Feb 23 '15 at 10:16
Bcoz he is asking that if $\mathbb{R}^{2}$is a field ... And I think he meant with the exterior multiplication and exterior addition. Otherwise it is obvious we can consider it as field aince it is isomorphism to $\mathbb{C}$ and hence a field – Romel Feb 23 '15 at 10:21
But what does that have to do with this? You are removing way more elements than $0$ when you consider this (in fact, it is clear that the usual multiplication does turn this into a group). – Tobias Kildetoft Feb 23 '15 at 10:22
And since the natural multiplication is defined on $\mathbb{R}$ We have to consider $\mathbb{R}^{*}$ as the set which the natural multiplication works on – Romel Feb 23 '15 at 10:24
No, to be a field we would need $(\mathbb{R}\times\mathbb{R})\setminus \{0\}$ to be a group, not $\mathbb{R}^*\times\mathbb{R}^*$. – Tobias Kildetoft Feb 23 '15 at 10:25 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.982557512709997,
"lm_q1q2_score": 0.8501324088550003,
"lm_q2_score": 0.8652240686758841,
"openwebmath_perplexity": 279.8928370392323,
"openwebmath_score": 0.9212859272956848,
"tags": null,
"url": "http://math.stackexchange.com/questions/250768/is-mathbb-r2-a-field"
} |
GMAT Question of the Day - Daily to your Mailbox; hard ones only
It is currently 20 Oct 2018, 06:49
### GMAT Club Daily Prep
#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
we will pick new questions that match your level based on your Timer History
Track
every week, we’ll send you an estimated GMAT score based on your performance
Practice
Pays
we will pick new questions that match your level based on your Timer History
# Each term of a certain sequence is calculated by adding a particular
Author Message
TAGS:
### Hide Tags
Math Expert
Joined: 02 Sep 2009
Posts: 50004
Each term of a certain sequence is calculated by adding a particular [#permalink]
### Show Tags
30 Jul 2018, 00:45
00:00
Difficulty:
15% (low)
Question Stats:
86% (01:48) correct 14% (01:40) wrong based on 42 sessions
### HideShow timer Statistics
Each term of a certain sequence is calculated by adding a particular constant to the previous term. The 2nd term of this sequence is 27 and the 5th term is 84. What is the 1st term of this sequence?
(A) 20
(B) 15
(C) 13
(D) 12
(E) 8
_________________
PS Forum Moderator
Joined: 25 Feb 2013
Posts: 1216
Location: India
GPA: 3.82
Each term of a certain sequence is calculated by adding a particular [#permalink]
### Show Tags
30 Jul 2018, 10:41
Bunuel wrote:
Each term of a certain sequence is calculated by adding a particular constant to the previous term. The 2nd term of this sequence is 27 and the 5th term is 84. What is the 1st term of this sequence?
(A) 20
(B) 15
(C) 13
(D) 12
(E) 8
let the constant be $$d$$ and the first term be $$a$$
so 2nd term $$= a+d=27$$ and 5th term will be $$a+4d=84$$. Subtract the two equations to get
$$3d=57 =>d=19$$
Hence $$a=27-19=8$$ | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9825575121992375,
"lm_q1q2_score": 0.8501324084130789,
"lm_q2_score": 0.8652240686758841,
"openwebmath_perplexity": 1686.1567387233056,
"openwebmath_score": 0.8939218521118164,
"tags": null,
"url": "https://gmatclub.com/forum/each-term-of-a-certain-sequence-is-calculated-by-adding-a-particular-271963.html"
} |
$$3d=57 =>d=19$$
Hence $$a=27-19=8$$
Option $$E$$
VP
Joined: 07 Dec 2014
Posts: 1104
Each term of a certain sequence is calculated by adding a particular [#permalink]
### Show Tags
Updated on: 31 Jul 2018, 17:11
Bunuel wrote:
Each term of a certain sequence is calculated by adding a particular constant to the previous term. The 2nd term of this sequence is 27 and the 5th term is 84. What is the 1st term of this sequence?
(A) 20
(B) 15
(C) 13
(D) 12
(E) 8
let d=difference between terms
84-27=57=3d
d=19
27-19=8=1st term
E
Originally posted by gracie on 30 Jul 2018, 18:15.
Last edited by gracie on 31 Jul 2018, 17:11, edited 1 time in total.
Manager
Joined: 11 Mar 2018
Posts: 80
Re: Each term of a certain sequence is calculated by adding a particular [#permalink]
### Show Tags
30 Jul 2018, 19:34
Bunuel wrote:
Each term of a certain sequence is calculated by adding a particular constant to the previous term. The 2nd term of this sequence is 27 and the 5th term is 84. What is the 1st term of this sequence?
(A) 20
(B) 15
(C) 13
(D) 12
(E) 8
Let the first term be 'a' and the constant being added be d.
So, 2nd term becomes -
$$a_2$$ = a + d = 27 ------ (1)
and the 5th term becomes -
$$a_5$$ = a + d + d + d + d = 84
Hence,
$$a_5$$ = a + 4d = 84 ------ (2)
Now multiplying equation (1) by 4 to eliminate d
hence (1) becomes -
$$a_2$$ * 4 =>
4a + 4d = 108 ---- (3)
Subtracting equation (2) from (3)
4a + 4d - a - 4d = 108 - 84
3a = 24
a = 8
_________________
Regards
---------------------------------
A Kudos is one more question and its answer understood by somebody !!!
Target Test Prep Representative
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
Posts: 3896
Location: United States (CA)
Re: Each term of a certain sequence is calculated by adding a particular [#permalink]
### Show Tags | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9825575121992375,
"lm_q1q2_score": 0.8501324084130789,
"lm_q2_score": 0.8652240686758841,
"openwebmath_perplexity": 1686.1567387233056,
"openwebmath_score": 0.8939218521118164,
"tags": null,
"url": "https://gmatclub.com/forum/each-term-of-a-certain-sequence-is-calculated-by-adding-a-particular-271963.html"
} |
### Show Tags
01 Aug 2018, 16:36
Bunuel wrote:
Each term of a certain sequence is calculated by adding a particular constant to the previous term. The 2nd term of this sequence is 27 and the 5th term is 84. What is the 1st term of this sequence?
(A) 20
(B) 15
(C) 13
(D) 12
(E) 8
Let n = the constant added to each term to get the next term.
We have:
2nd term = 27
3rd term = 27 + n
4th term = 27 + 2n
5th term = 27 + 3n
Since we are given that the 5th term is 84, we have:
27 + 3n = 84
3n = 57
n = 19
Since the second term was given as 27, we know the first term is 27 - n. So the first term is 27 - 19 = 8.
_________________
Scott Woodbury-Stewart
Founder and CEO
GMAT Quant Self-Study Course
500+ lessons 3000+ practice problems 800+ HD solutions
Re: Each term of a certain sequence is calculated by adding a particular &nbs [#permalink] 01 Aug 2018, 16:36
Display posts from previous: Sort by | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9825575121992375,
"lm_q1q2_score": 0.8501324084130789,
"lm_q2_score": 0.8652240686758841,
"openwebmath_perplexity": 1686.1567387233056,
"openwebmath_score": 0.8939218521118164,
"tags": null,
"url": "https://gmatclub.com/forum/each-term-of-a-certain-sequence-is-calculated-by-adding-a-particular-271963.html"
} |
# Iterated Integrals - setting up limits of integration
1. Sep 30, 2008
### mirandasatterley
1. The problem statement, all variables and given/known data
Find the volume of the region under the graph of f(x,y) = x+y and above the region y2≤x, 0≤x≤9
3. The attempt at a solution
From these equations, x will be integrated from 0-9, but i'm not sure about y.
My thinking is that y will be intgrated from 0-3 because y2≤x and the smallest value of x is 0, and the square root of 0 is 0, so that is the smallest y, and the largest x value possible is 9 and the positive quare root of 9 is 3, so this is the largest value of y,
So I would integrate:
∫0-9∫0-3 (x+y)dydx.
Is this correct or am I missing something, where the limits of integration also involve using f(x,y)?
2. Sep 30, 2008
### HallsofIvy
Staff Emeritus
No, that is not correct. The region $a\le x\le b$, $c\le y\le d$, with a, b, c, d numbers is always a rectangle and the figure here is not a rectangle. But the bounds do NOT involve f(x,y)- that is a "z" value and goes inside the integral as you have it.
Always draw a picture for problems like this. y^2= x is a parabola, of course, "on its side". The line x= 9 is a vertical line crossing the parabola at (9,3) and at (9, -3). Yes, you can integrate with x going from 0 to 9. On your picture, mark an arbitrary "x" by marking a point on the x-axis between 0 and 9. Now draw a vertical line from one boundary to the other. The y bounds, for that x, are y values of those endpoints: $(x, -\sqrt{x})$, and $(x, -\sqrt{x})$. Your integral is
$$\int{x=0}^9\int_{y=-\sqrt{x}}^{\sqrt{x}} f(x)dy dx= \int{x=0}^9\int_{y=-\sqrt{x}}^{\sqrt{x}}(x+ y) dy dx$$ | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9706877658567787,
"lm_q1q2_score": 0.8501157308771303,
"lm_q2_score": 0.8757870046160257,
"openwebmath_perplexity": 406.71554735861827,
"openwebmath_score": 0.7749534249305725,
"tags": null,
"url": "https://www.physicsforums.com/threads/iterated-integrals-setting-up-limits-of-integration.260486/"
} |
Of course, like any double integral, you can reverse the order of integration. If you look at your picture you will see that y ranges, overall, from -3 to 3. Draw a horizontal line across the parabola, representing an arbitrary value of y in that range. It will have left endpoint on the parabola: x= y^2, and right endpoint on the vertical line x= 9. Those will now be the limits of integration for this order:
$$\int_{y= -3}^3\int_{x= y^2}^9 f(x,y)dx dy= \int_{y= -3}^3\int_{x= y^2}^9(x+ y)dx dy$$
Try it both ways. You should get the same answer.
3. Sep 30, 2008
### mirandasatterley
In a similar situation where i have to switch the order of integration from
∫0-3∫y2-9 f(x,y) dxdy to dydx,
Is this also a parabola on it's side, with intercepts through (3,9) and (0,9), meaning that I would be setting up the new limits of integration, fro the portion of the parabola above y=0?
4. Sep 30, 2008
### mirandasatterley
So, ∫0-9∫Square root of x -3 f(x,y) dydx | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9706877658567787,
"lm_q1q2_score": 0.8501157308771303,
"lm_q2_score": 0.8757870046160257,
"openwebmath_perplexity": 406.71554735861827,
"openwebmath_score": 0.7749534249305725,
"tags": null,
"url": "https://www.physicsforums.com/threads/iterated-integrals-setting-up-limits-of-integration.260486/"
} |
# Inequality (-3/x) < 3
1. Aug 23, 2013
### Qube
1. The problem statement, all variables and given/known data
(-3/x) < 3
2. Relevant equations
Dividing/multiplying an inequality causes the inequality sign to change.
3. The attempt at a solution
I keep getting the wrong solution. I tried two methods. I cannot get the textbook solution (x < -1)
Method one:
-3 < 3x
-1 < x
Method two:
Divide both sides by negative one, resulting in:
3/x > -3
3 > -3x
-1 < x
2. Aug 23, 2013
### Staff: Mentor
Here you have multiplied by x, which could be positive or negative (or zero). Since you don't know the sign of x, you need two cases - one where x is assumed to be positive and the other where x is assumed to be negative.
Again, you've multiplied by x, whose sign you don't know. Same comments as above apply here.
3. Aug 23, 2013
### Qube
All right! Now I have -1 < x if x is positive and -1 > x if x is negative. How do I determine which is correct? Is the time for trial-and-error?
4. Aug 23, 2013
### Staff: Mentor
The book's answer is wrong or at least incomplete.
If x > 0, you get x > - 1. This means that x > 0 AND x > -1. Together, these mean that x > 0.
If x < 0, you get x < -1. This means that x < 0 AND x < -1. Together, these mean that x < -1.
If you look at the graph of y = -3/x, you'll see that it has two parts. For the curve on the left, -3/x < 3 when x < -1. For the curve on the right, -3/x < 3 for any x > 0.
5. Aug 23, 2013
### Qube
So I guess I'll have to do a bit of trial and error to determine the exact intervals? As in, x > -1 is technically correct, but the best answer is that x is actually greater than 0 or x > 0?
How do you recommend solving these problems? I'm having massive trouble.
6. Aug 23, 2013
### verty
Just remember the rule, NEVER multiply or divide by x or an expression containing x, it could be negative. Read Mark44's reply again, the answer is: x < -1 OR x > 0. | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9706877675527114,
"lm_q1q2_score": 0.8501157292154172,
"lm_q2_score": 0.875787001374006,
"openwebmath_perplexity": 589.3431137519043,
"openwebmath_score": 0.8445603251457214,
"tags": null,
"url": "https://www.physicsforums.com/threads/inequality-3-x-3.706872/"
} |
Example: $\frac{x+2}{x-3} > 0$. If $x-3 > 0, x+2 > 0$, and if $x-3 < 0, x+2 < 0$. Can you give the answer for this example?
7. Aug 23, 2013
### rock.freak667
Not to detract from what the others are saying, if you multiply by x, you are multiplying by an unknown value which can be +ve or -ve and will thus change the sign of the inequality (i.e. < might become >). So what you can do to prevent this is multiply instead by x2 as this will always be positive.
Then you can solve it as you would using normal algebraic means. You will just need to be careful when doing this as you might pick up an extra solution. But simply analyzing your solution set for the problem would lead you in the correct path.
8. Aug 23, 2013
### Ray Vickson
Turn your inequality into one of the form
$$\frac{1}{x} < a$$
or
$$\frac{1}{x} > a$$
I will let you figure out which inequality applies, and what is the value of $a$.
Anyway, once you have a simple inequality in $1/x$, you can envision the graph $y = 1/x$, and then figure out what the x-region must be for the corresponding inequality in y.
Note: this suggested method is safe; it never has you multiplying or dividing by an x of unknown sign, at least, not until the very end.
9. Aug 24, 2013
### vanhees71
I think it's time to give the solution to clarify the issue:
You just have to do a case differentiation. First you can simplyfy the inequality by divding through $-3$:
$$-\frac{3}{x} < 3 \Leftrightarrow \frac{1}{x}>-1.$$
Now for $x>0$ the inequality is always fulfilled and for $x<0$ you get by multiplying with $(-x)>0$
$$-1>x \; \Leftrightarrow \; x<-1$$.
Thus the inequality is fulfilled for either $x>0$ or $x<-1$.
10. Aug 24, 2013
You can also try
\begin{align*} \frac{-3}{x} & < 3 \tag{Given} \\ \frac{-3}{x} \cdot x^2 & < 3x^2 \\ -3x & < 3^2 \\ 0 & < 3x^2 + 3x \tag{Now factor to obtain}\\ 0 & < 3x\left(x + 1\right) \end{align*} | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9706877675527114,
"lm_q1q2_score": 0.8501157292154172,
"lm_q2_score": 0.875787001374006,
"openwebmath_perplexity": 589.3431137519043,
"openwebmath_score": 0.8445603251457214,
"tags": null,
"url": "https://www.physicsforums.com/threads/inequality-3-x-3.706872/"
} |
If you set up a sign table (or graph $3x(x+1)$ you'll find that the inequality is solved
for numbers $x < -1$ or $x > 0$ as stated in other places. In short, it is not that one or the other of these two inequalities gives the solution set, is that the solution set consists of any number that satisfies either the first or the second of these two.
11. Aug 24, 2013
### Infrared
I think this problem is being made much harder than it really is.
$\frac{-3}{x}<3 \\ \frac{-3}{x}-3<0 \\ \frac{1}{x}+1>0 \\ \frac{x+1}{x}>0$
So $x>0$ or $x<-1$
12. Aug 24, 2013
### Ray Vickson
You are doing exactly what I suggested the OP do, but he/she never reported back.
13. Aug 25, 2013 | {
"domain": "physicsforums.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9706877675527114,
"lm_q1q2_score": 0.8501157292154172,
"lm_q2_score": 0.875787001374006,
"openwebmath_perplexity": 589.3431137519043,
"openwebmath_score": 0.8445603251457214,
"tags": null,
"url": "https://www.physicsforums.com/threads/inequality-3-x-3.706872/"
} |
# The definitions of limit infimum and limit supremum
I have begun reading Rosenthal's "A First Look to Rigorous Probability Theory" and in order to reinforce my calculus background I am studying through the appendix section. Here he defined the limit of a sequence of real numbers as $\lim_{n \to \infty}x_n=x$ if for each $0 <\epsilon$ there is a natural number $N$ such that it is $|x-x_n| < \epsilon$ for $n > N$.
Then the limit infimum and supremum are defined as $\liminf_n x_n = \lim_{n \to \infty} \inf_{k\geq n} x_k$ and $\limsup_n x_n = \lim_{n \to \infty} \sup_{k\geq n} x_k$.
1. My first question is about the interpretation of the limit infimum and limit supremum definitions. It looks to me like they are defined as the limits of sequences as well. For example, for the infimum, one can define a sequence $a_n = \inf_{k \geq n}x_k$ where $x_n$ was already a sequence of real numbers. Then the limit infimum is the limit of the sequence $a_n$ , $\lim_{n \to \infty}a_n$, so the infimums of the subsequences $x_k$ where $k \geq n$ are converging if the limit exists. It is the similar for the supremum. Is my interpretation correct here?
2. It is said the limit infimum and supremum do always exist, though they can be infinite. I did not understand this; why it is so?
3. In the book it is stated that the limit $\lim_{n \to \infty} x_n$ exists if and only if it is $\liminf_n x_n = \limsup_n x_n$. I could not prove this to myself. How can it be shown that this statement is true?
• 1. ok, 2. they are monotone sequences. – enzotib Jul 23 '14 at 7:28
• For 3. In general the limit infimum is the least value to which a subsequence of ${a_n}$ can converge and the limit supremum is the greater value to wihch a subsequence of ${a_n}$ can converge – Dimitri Jul 23 '14 at 7:44
These are answers to the secondary questions in the comments after my answer to the original question; I cannot put the answers in comments, because the size of a comment is limited. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9706877692486436,
"lm_q1q2_score": 0.8501157212597258,
"lm_q2_score": 0.8757869916479466,
"openwebmath_perplexity": 102.1858039005688,
"openwebmath_score": 0.967788815498352,
"tags": null,
"url": "https://math.stackexchange.com/questions/875627/the-definitions-of-limit-infimum-and-limit-supremum"
} |
The second question. Let $(x_n)$ be a sequence taking values in the set $\{0,1,2,\ldots,9\}$, and $a:=\liminf_n x_n$. If $0$ appears infinitely many times in the sequence, then $a=0$. If $0$ appears only finitely many times, but $1$ appears infinitely often, then $a=1$. And so on. This is a slick answer. It is entirely another matter to answer the question with a definite value for a particular given sequence, such as the sequence of digits after the comma in the decimal expansion of $\pi$. Well, if you can prove that the digit $0$ appears infinitely many times in the decimal expansion of $\pi$, then you have $a=0$, and so on; in this case it takes a serious theoretical effort to determine the $\liminf$.
As for the third question, let $(x_n)$ be a sequence of real numbers such that $a:=\liminf_n x_n$ is a real number (that is, $-\infty<a<\infty$). Since the sequence of infima $a_n:=\inf_{k\geq n} x_n$ is increasing (meaning that $m\leq n$ implies $a_m\leq a_n$ -- I hate "nondecreasing"), the limit $a=\lim_{n\to\infty} a_n$ is actually the supremum: $a=\sup_n a_n$.
$\quad$Let $a_1\leq a$ and $a'<a_1$; then $a'<a$. We claim that there exists $m$ such that $x_n\geq a'$ for every $n\geq m$: since $a=\sup_n a_n$, there exists $m$ such that $a_m\geq a'$; but then $x_n\geq a_m\geq a'$ for every $n\geq m$.
$\quad$Now let $a_1>a$, and set $a':=\tfrac{1}{2}(a+a_1)$; we have $a<a'<a_1$. In this case we claim that given any $m$ there exists $n\geq m$ such that $x_n<a'$: since $a_m=\inf_{k\geq m} x_k\leq a<a'$, there exists $n\geq m$ such that $x_n<a'$.
$\quad$We have proved that $a$ is the largest of all real numbers $a_1$ with the property that for every $a'<a_1$ there are only finitely many $n$ such that $x_n<a'$. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9706877692486436,
"lm_q1q2_score": 0.8501157212597258,
"lm_q2_score": 0.8757869916479466,
"openwebmath_perplexity": 102.1858039005688,
"openwebmath_score": 0.967788815498352,
"tags": null,
"url": "https://math.stackexchange.com/questions/875627/the-definitions-of-limit-infimum-and-limit-supremum"
} |
1. Yes, in the book you are reading, $\liminf_n x_n$ and $\limsup_n x_n$ are defined as limits of sequences. Note that the sequence $a_n=\inf_{k\geq n} x_k$ is increasing (which means that $m\leq n$ implies $a_m\leq a_n$), while the sequence $b_n=\sup_{k\geq n} x_k$ is decreasing.
2. The $\liminf$ and $\limsup$ are taken in the extended real line $\overline{\mathbb{R}}=\{-\infty\}\cup\mathbb{R}\cup\{\infty\}$. As an ordered set, $\overline{\mathbb{R}}$ is obtained from the ordered set $\mathbb{R}$ by adding to it the bottom (the least) element $-\infty$ and the top (the greatest) element $\infty$. As a topological space, $\overline{\mathbb{R}}$ is homeomorphic to any closed interval $[a,b]$ (with $a<b$) in $\mathbb{R}$, and is therefore compact. For example, $x\mapsto(2/\pi)\arctan x$ is a homeomorphism $\overline{\mathbb{R}}\to[-1,1]$ (where it is understood that ${-}\infty\mapsto-1$ and $\infty\mapsto1$), and at the same time it is an isomorphism of (totally) ordered sets. All this means that the limits, $\liminf$'s, and $\limsup$'s in the extended real line are just `infinitely stretched out' versions of the limits, $\liminf$'s, and $\limsup$'s in a closed interval. Since in a closed interval all increasing/decreasing sequences converge, so then do all increasing/decreasing sequences in the extended real line; in particular, $\liminf_n x_n$ and $\limsup_n x_n$ always exist. When you are thinking about the extended real line, you can imagine it as a closed interval, you can' go astray with that. If you have to prove some assertion about sequences in the extended real line, formulated in terms of the ordering and the topology (which is, after all, also determined by the ordering), then it suffices to prove the assertion for sequences in a closed interval. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9706877692486436,
"lm_q1q2_score": 0.8501157212597258,
"lm_q2_score": 0.8757869916479466,
"openwebmath_perplexity": 102.1858039005688,
"openwebmath_score": 0.967788815498352,
"tags": null,
"url": "https://math.stackexchange.com/questions/875627/the-definitions-of-limit-infimum-and-limit-supremum"
} |
3. You can assume that $x_n$ is a sequence in a closed interval; this will make your life easier, because $\liminf_n x_n$ and $\limsup_n x_n$ are real numbers (belonging to the interval). Now prove the following: $a=\liminf_n x_n$ is the largest real number with the property that for every real number $a'<a$ there are only finitely many $n\in\mathbb{N}$ such that $x_n\leq a'$, and the analogous assertion for $b=\limsup_n x_n$ (which you will formulate yourself --- just turn everything upside-down). With these characterizations of $\liminf_n x_n$ and $\limsup_n x_n$ under the belt, you will easily prove that $\lim_{n\to\infty} x_n$ exists iff $\liminf_n x_n=\limsup_n x_n$. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9706877692486436,
"lm_q1q2_score": 0.8501157212597258,
"lm_q2_score": 0.8757869916479466,
"openwebmath_perplexity": 102.1858039005688,
"openwebmath_score": 0.967788815498352,
"tags": null,
"url": "https://math.stackexchange.com/questions/875627/the-definitions-of-limit-infimum-and-limit-supremum"
} |
Remark. You did not ask this, but since I have mentioned imagining the extended real line as a closed interval$\ldots$ $~$You can actually see $\overline{\mathbb{R}}$ as the closed interval $[-1,1]$, and transfer anything that's happening in $\overline{\mathbb{R}}$ to $[-1,1]$. Let $f(y)=\tan((\pi/2)y)$ for $y\in[-1,1]$, so that $f^{-1}(x)=(2/\pi)\arctan(x)$ for $x\in\overline{\mathbb{R}}$. Now define, for all $y,z\in[-1,1]$, \begin{aligned} \mathit{sum}(y,z)~ &:= f^{-1}(f(y)+f(z))~,\\ \mathit{prod}(y,z)~ &:= f^{-1}(f(y)f(z))~. \end{aligned} Actually the operations $\mathit{sum}$ and $\mathit{prod}$ are not defined everywhere on $[-1,1]\times[-1,1]$; for example, $\mathit{plus}$ is not defined at the two points $(1,-1)$ and $(-1,1)$ that correspond to the risque limits of the form $\infty-\infty$. Write the definitions of $\mathit{sum}$ and $\mathit{prod}$ in Mathematica (or a similar program), and then draw their $3\text{D}$ diagrams, placing on all three axes the labels $-\infty$, $-1$, $0$, $1$, $\infty$ at positions, respectively, $-1$, $-1/2$, $0$, $1/2$, $1$. You will be able to observe in the diagrams the behavior of the two operations all the way to the infinity (in either direction); at the points where the operations are not defined you will clearly see the discontinuities (at these points the diagrams contain vertical lines --- more precisely, the closures of the diagrams contain vertical lines). | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9706877692486436,
"lm_q1q2_score": 0.8501157212597258,
"lm_q2_score": 0.8757869916479466,
"openwebmath_perplexity": 102.1858039005688,
"openwebmath_score": 0.967788815498352,
"tags": null,
"url": "https://math.stackexchange.com/questions/875627/the-definitions-of-limit-infimum-and-limit-supremum"
} |
• A quick question: What exactly do we mean by a sequence in a closed interval? Do we mean that all values $x_n$ are mapped to values in an interval $[a,b]$ such that it is $a \leq x_n \leq b$? (a and b are real numbers) – Ufuk Can Bicici Jul 23 '14 at 9:42
• Yes, we mean exactly that. – chizhek Jul 23 '14 at 9:58
• I have another question, it may be weird but still it bugged me. Let's assume that the sequence $x_n$ takes the values of $\pi$'s digit values after the comma. So each $x_n$ takes a value in the set $(0,1,2,3,4,5,6,7,8,9)$. If we think of the sequence $a_n = \inf_{k \geq n} x_k$ this sequence is clearly nondecreasing and bounded in $[0,9]$, but how can we know the value of limit $\liminf_n x_n$? Since we don't know exactly the values of $x_n$ for very large $n$s, we can be never sure about the value for $a_n$ for these values. – Ufuk Can Bicici Jul 23 '14 at 12:36
• Continued: The infimum can be zero, one or any value, we cannot observe this exactly for this almost random sequence $x_n$ at large $n$s. So, what can we say about the value $\liminf_n x_n$ for such sequences? – Ufuk Can Bicici Jul 23 '14 at 12:39
• For the third question, unfortunately I have failed to find a rigorous proof for that if $a$ is the largest real number such that for every $a' < a$ there is finitely many $x_n < a'$ then $a=\liminf_n x_n$. But intuitively it is clear that this is valid: If we have a plot of a bounded sequence $x_n$ and visualize a vertical line $y=a'$ then there will be finitely many $x_n$ points under that line and as soon as we increase the line over $a$, the number of $x_n$ points will be infinitely many! But I frustratingly fail to express that visualization as a mathematically valid statement... – Ufuk Can Bicici Jul 23 '14 at 15:00 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9706877692486436,
"lm_q1q2_score": 0.8501157212597258,
"lm_q2_score": 0.8757869916479466,
"openwebmath_perplexity": 102.1858039005688,
"openwebmath_score": 0.967788815498352,
"tags": null,
"url": "https://math.stackexchange.com/questions/875627/the-definitions-of-limit-infimum-and-limit-supremum"
} |
# Smallest number with specific number of divisors
Is there a general method for finding smallest number of specific number of divisors? I am doing "Higher Algebra by Barnard JM Child" and came across a question that "find the smallest number with 24 divisors", that's how I tried to solve it, alert me if I am wrong:
Since $24$ can not have more than $4$ prime factors, the number can not have more than 4 prime factors.
As a single number it is :$2$^${23}$
as product of two numbers: $2^5*3^3$,$2^{11}*3$,$2^7*3^2$, since $5+3<7+2<11+1$, so $2^5*3^3$ is the min of these numbers
as product of three numbers :$2^5*3*5$,$2^3*3^2*5$ since $3+2+1<5+1+1$, hence $2^3*3^2*5$ is the lesser of two
as product of 4 numbers $2^2*3*5*7$
$k=\min(2^{23},2^5*3^3,2^3*3^2*5,2^2*3*5*7)$
=$2^3*3^2*5=360$
The above method seems to be fishy and laborious, is there a general approach to find the smallest number with specific number of divisors?
• For multicharacter exponents, put them in braces. So 2^{11} to get $2^{11}$ Also, \min gets it set as a function. – Ross Millikan Sep 17 '13 at 15:51
• oeis.org/A005179 – barrycarter Sep 7 '17 at 19:17
This approach is not fishy at all, but can be laborious. Note that you can't use $5+3 \lt 7+2$ to conclude that $2^5*3^3 \lt 2^7*3^2$, although it is true, you have to take the ratio and use $3 \lt 2^2$. For example, if you were looking for $36$ factors, one comparison would be $2^8*3^3$ versus $2^5*3^5$. Despite the fact that $5+5 \lt 3+8, 2^5*3^5=7776 \gt 6912=2^8*3^3$. You can take logs and compare $5 \log 2 + 3 \log 3$ with $7 \log 2 + 2 \log 3$ to get it right. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9706877667047451,
"lm_q1q2_score": 0.8501157190318125,
"lm_q2_score": 0.8757869916479466,
"openwebmath_perplexity": 297.41830065468,
"openwebmath_score": 0.8289042711257935,
"tags": null,
"url": "https://math.stackexchange.com/questions/496494/smallest-number-with-specific-number-of-divisors"
} |
I don't know an easier way. Your example shows the failure of a greedy algorithm. Factor the desired number of factors, here as $3*2^3$. Starting from the largest factor, find the cheapest way to get that many. So start with $2^2$, which has $3$ factors. Now you need to double it. You can either multiply by a new prime, clearly $3$, or increase the exponent of $2$ to $5$. Since the first has a factor $3$ and the second $8$, we choose $2^2*3$ Now we want to double again, and our choices are $2^3, 3^2, \text{ or } 5$ and we take $5$, giving $2^2*3*5$ One more doubling comes from $7$, and we come up with $2^2*3*5*7=420$, not the best.
Take any number $N=p_1^{m_1}...p_k^{m_k}$, where $p_i$ are its prime divisors, and compute how many divisors it has. Each divisor would be a product of the same primes in varying powers, $D=p_1^{d_1}...p_k^{d_k}$, where $0\leq d_i \leq m_i$. Different divisors have different collections of powers, so the number of divisors will be $(m_1+1)(m_2+1)...(m_k+1)$.
Now let's find the smallest $N$ such that $(m_1+1)(m_2+1)...(m_k+1)=24$. There are only few possibilities to break 24 into a product of decreasing numbers: $3*2*2*2=4*3*2=6*4=6*2*2=8*3=12*2=24$. The corresponding candidates with the smallest primes chosen for the smallest powers would be these ones:
$2^{3-1}*3^{2-1}*5^{2-1}*7^{2-1}=2^2*3*5*7=420$
$2^{4-1}*3^{3-1}*5^{2-1}=2^3*3^2*5=360$
$2^5*3^3=2592$
$2^5*3*5=480$
$2^{11}*3=6144$
$2^{23}=8388608$
Then just pick the smallest one, which happens to be 360.
• @ Michael That's what I tried at first,this algorithm becomes laborious if we are dealing with such numbers whose number of divisors has many prime factors,so I want a generalized method in order to avoid comparing numbers>>> – Tom Lynd Sep 17 '13 at 17:23
A005179 lists some resources on the problem. In particular:
Grost, M. (1968). The Smallest Number with a Given Number of Divisors. The American Mathematical Monthly, 75(7), 725-729. doi:10.2307/2315183 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9706877667047451,
"lm_q1q2_score": 0.8501157190318125,
"lm_q2_score": 0.8757869916479466,
"openwebmath_perplexity": 297.41830065468,
"openwebmath_score": 0.8289042711257935,
"tags": null,
"url": "https://math.stackexchange.com/questions/496494/smallest-number-with-specific-number-of-divisors"
} |
From the introduction:
Given $h = q_1 q_2 \dots q_n$, with primes $q_1 \le q_2 \le \dots \le q_n$, let $A(h)$ be the smallest number with $h$ divisors.
In many cases, $$A(h) = 2^{q_1-1} 3^{q_2-1} \dots p_n^{q_n-1}$$
The primary objective of the paper is to determine the exceptions.
We call these numbers ordinary, from R. Brown, The minimal number with a given number of divisors, Journal of Number Theory 116 (2006) 150-158. In this paper Brown shows almost all $A(h)$ are ordinary, in particular "We show here that all square-free numbers are ordinary and that the set of ordinary numbers has natural density one." | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9706877667047451,
"lm_q1q2_score": 0.8501157190318125,
"lm_q2_score": 0.8757869916479466,
"openwebmath_perplexity": 297.41830065468,
"openwebmath_score": 0.8289042711257935,
"tags": null,
"url": "https://math.stackexchange.com/questions/496494/smallest-number-with-specific-number-of-divisors"
} |
Why does drawing one card at a time increase the probability of choosing the Ace of Spades?
If you draw 5 cards from a standard deck of 52 cards, then the probability of your hand having the Ace of Spades is: $$\frac{51\choose 4}{52\choose 5} = \frac{51!5!47!}{4!47!52!} = \frac{5}{52}$$ If, however, you choose one card a time until you've drawn 5 cards, the probability of having the Ace of Spades is: $$\frac{1}{52}+\frac{1}{51}+\frac{1}{50}+\frac{1}{49}+\frac{1}{48}=\frac{433507}{4331600}\approx\frac{1}{10}>\frac{5}{52}$$ Why does choosing one card a time increase the probability of finding the Ace of Spades, when the resulting hands are equivalently drawn? | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9845754447499796,
"lm_q1q2_score": 0.850074180113767,
"lm_q2_score": 0.863391611731321,
"openwebmath_perplexity": 310.3443161049843,
"openwebmath_score": 0.838527500629425,
"tags": null,
"url": "https://stats.stackexchange.com/questions/159905/why-does-drawing-one-card-at-a-time-increase-the-probability-of-choosing-the-ace"
} |
• Hint: instead of drawing 5 cards one at a time, let's say you draw 34. Would you then say your probability of drawing the ace of spades is $1/52 + 1/51 + \ldots + 1/19 \approx 1.04 > 1$? – Hong Ooi Jul 4 '15 at 13:31
• Another hint : What you wanted to do is compute P(X=1)+P(X=2)+..+P(X=5)...with P(x=k) representing the probability of the Ace of Spade drawn exactly at the k th draw.This is one way to find the answer (not the most simple) but one legit way as these are disjunctive cases. The problem comes from how you compute these probabilities. For P(X=2), for example, If you had to answer the question "what is the probability that the ace of space is drawn exactly at the second draw, would you find 1/51 ? – brumar Jul 4 '15 at 13:39
• It's not actually homework, just a genuine question. But I think I know what you're saying: I'm conflating the idea of the geometric distribution and dependent draws. So, it'd be better to say $P(X=1)+P(X=2|X\neq 1)+\dots+P(X=5|X\neq 1,2,3,4)$? – AJS Jul 4 '15 at 14:26
• One thing I can't figure out is, why does it cancel out to $5/52$? $P(X=1) = 1/52$, $P(X=2\mid X\neq 1)=P(X=2\cap X\neq 1)/P(X\neq 1)=((51/52)*(1/51)/(51/52))$, giving me the original (incorrect) sum of fractions? – AJS Jul 4 '15 at 14:47
• Focus on P(X=2∩X≠1) = P(X=2 |X≠1) * P(X≠1). What is that? How does the answer to this relate to probability of getting an ace of spaces for the first time on the 2nd card drawn? – Mark L. Stone Jul 4 '15 at 15:23
As @Glen_b suggested, it would be a good idea to summarize the comments part in an answer. I'll do that and also give an alternative for the formula related to the probabilistic point of view at the end of the answer. The apparent contradiction between the two computations came from this line :
the probability of having the Ace of Spades is: $$\frac{1}{52}+\frac{1}{51}+\frac{1}{50}+\frac{1}{49}+\frac{1}{48}=\frac{433507}{4331600}\approx\frac{1}{10}>\frac{5}{52}$$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9845754447499796,
"lm_q1q2_score": 0.850074180113767,
"lm_q2_score": 0.863391611731321,
"openwebmath_perplexity": 310.3443161049843,
"openwebmath_score": 0.838527500629425,
"tags": null,
"url": "https://stats.stackexchange.com/questions/159905/why-does-drawing-one-card-at-a-time-increase-the-probability-of-choosing-the-ace"
} |
The idea behind this computation was good but the logic was flawed. If we sum the probability of the Ace of Spade drawn exactly at the k th draw with k going from 1 to 5, we have the probability we want. But $\frac{1}{51}$, for example, does not represent the probability that the ace of spade is exactly drawn at the second attempt but the probability that the ace of space is drawn at the second attempt given that it has not been drawn at the first one.
AJS finally found the right formula
I got it! $1/52+((51/52)∗(1/51))+⋯+((51/52)∗(50/51)∗(49/50)∗(48/49)∗(1/48))=5/52$
With the idea that the probability to exactly draw the ace of spade at the $k$ th trial is the probability to not draw the ace of spade during previous attempts.... $$(51/52)*(50/51)...(52-k+1)/(52-k+2)$$...multiplied by the probability to draw the card at the $k$ th attempt $$1/(52-k+1)$$
A general rule of thumb that I have to avoid this kind of error is to be cautious when it comes to adding probabilities. If you can turn your problem around to avoid additions to the profit of multiplications, this is less prone to error.
A more classic approach giving a shorter path would have been to consider that the probability of having the Ace of Spades after 5 draw is 1-(the probability to not draw it with 5 draws) which gives, as you know : $$1-(51/52)*(50/51)*(49/50)*(48/49)*(47/48)=1-47/52=5/52$$
• Thank you, brumar. I appreciate you taking the time to write it out. – AJS Jul 4 '15 at 23:22 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9845754447499796,
"lm_q1q2_score": 0.850074180113767,
"lm_q2_score": 0.863391611731321,
"openwebmath_perplexity": 310.3443161049843,
"openwebmath_score": 0.838527500629425,
"tags": null,
"url": "https://stats.stackexchange.com/questions/159905/why-does-drawing-one-card-at-a-time-increase-the-probability-of-choosing-the-ace"
} |
Project Euler Problem 7: 10001st prime
Back to primes! So far we’ve been able to get away with being a little greedy with our compute when playing with primes. Now Euler is ratcheting up the difficulty and we’ll have to focus on efficiency.
As usual, if you haven’t spent time with Problem 7 yet, take a chance to play with it on your own and come back.
Counting Primes
Let’s start by looking at each integer, deciding whether it’s prime, and counting it if it is until we get to the 10,001st prime.
Our first stab at an is_prime(n) function will be the simplest and we’ll iterate into more optimized (and complicated) versions after. Here’s the starting point:
def is_prime(n):
if n < 2:
return False
for x in range(2, n):
if n % x == 0:
return False
return True
This checks every number less than $n$ to see if it’s a factor of $n$. It’s almost good enough to solve the problem in under a minute. My laptop chugs through the first 10,001 primes in 68 seconds using this version of the is_prime(n) function (full code later). But the rules only give us a minute and we can do better.
Cap the Search Space
Looking back at our Problem 3 Solution we optimized our is_prime(n) function by caping the space we search to find factors factors by checking only numbers up to $\sqrt{n}$. Check out that post if you want to dig deep into why / how that works.
def is_prime(n):
if n < 2:
return False
for x in range(2, math.floor(math.sqrt(n)) + 1):
if n % x == 0:
return False
return True
This runs a lot faster. It finds the 10,001st prime in 0.29 seconds on my machine. But can we make it even better?
Skip Through the Search Space
Perhaps the most rediscovered result about primes numbers is the fact that every prime bigger than 3 is “next” to a multiple of 6. That is, for every prime number starting at 5 you can get a multiple of 6 by adding 1 or subtracting 1.
For example: | {
"domain": "grae.io",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9888419710536894,
"lm_q1q2_score": 0.8500716028305736,
"lm_q2_score": 0.8596637559030337,
"openwebmath_perplexity": 743.9199693840225,
"openwebmath_score": 0.5414897203445435,
"tags": null,
"url": "https://www.grae.io/post/euler_problem_7/"
} |
For example:
• 5 is prime, add 1 and get 6
• 13 is prime, subtract 1 and get (6 * 2)
• 1,361 is prime, add 1 and get (6 * 227)
This works for every prime number.
We can use this property to skip potential factors we don’t need to check. When checking to see if a number $n$ has factors we can get away with just looking for the prime factors, we don’t also need to know if it has any factors that are themselves composite. For example, we don’t need to know that 24 is divisible by 8. We can stop as soon as we see it’s divisible by 2. So we can skip every potential factor except for those which might be prime.
In code:
def is_prime(n):
if n < 2:
return False
if n == 2 or n == 3:
return True
if n % 2 == 0 or n % 3 == 0:
return False
for x in range(6, math.floor(math.sqrt(n)) + 2, 6):
if n % (x - 1) == 0 or n % (x + 1) == 0:
return False
return True
This version takes advantage of Python’s “step” argument to range(). We’re looking at every multiple of 6 (below our limit) and checking whether the number before or after it divides our target.
This optimizes things a bit more and, indeed, finds the 10,001st prime in 0.17 seconds on my machine.
Putting it Together
Once we have an efficient is_prime() function the solution is a matter of counting primes with a while loop.
seen = 0
n = 1
while seen < 10001:
n += 1
if is_prime(n):
seen += 1
print(n)
Going Further
There are ways to solve this problem even faster. You could use the Prime Number Theorem to approximate an upper bound for a Sieve of Eratosthenes and sieve out the answer. We’ll deal with those concepts in coming problems so for now I’ll leave that as an exercise for the reader.
See an issue on this page? Report a typo, bug, or give general feedback on GitHub. | {
"domain": "grae.io",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9888419710536894,
"lm_q1q2_score": 0.8500716028305736,
"lm_q2_score": 0.8596637559030337,
"openwebmath_perplexity": 743.9199693840225,
"openwebmath_score": 0.5414897203445435,
"tags": null,
"url": "https://www.grae.io/post/euler_problem_7/"
} |
# Implicit differentiation examples and solutions pdf
Explicit means fully revealed, expressed without vagueness or ambiguity. Now i will solve an example of the differentiation of an implicit function. Find two explicit functions by solving the equation for y in terms of x. For each problem, use implicit differentiation to find d2222y dx222 in terms of x and y. In this presentation, both the chain rule and implicit differentiation will. Implicit differentiation is as simple as normal differentiation. Implicit differentiation extra practice date period. Implicit di erentiation statement strategy for di erentiating implicitly examples table of contents jj ii j i page2of10 back print version home page method of implicit differentiation. Multivariable calculus implicit differentiation examples. Solutions to implicit differentiation problems uc davis mathematics. They are laying the groundwork for stokestheorem, a. Implicit di erentiation statement strategy for di erentiating implicitly examples table of contents jj ii j i page1of10 back print version home page 23. If we are given the function y fx, where x is a function of time. Since the point 3,4 is on the top half of the circle fig. | {
"domain": "web.app",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9888419674366167,
"lm_q1q2_score": 0.8500715908328732,
"lm_q2_score": 0.8596637469145054,
"openwebmath_perplexity": 404.06666719092937,
"openwebmath_score": 0.8376717567443848,
"tags": null,
"url": "https://geocabirthve.web.app/1580.html"
} |
There will also be one or two exercises on material in the next set of notes, which are not taken from the text. Search within a range of numbers put between two numbers. In fact, all you have to do is take the derivative of each and every term of an equation. Implicit differentiation helps us find dydx even for relationships like that. Implicit differentiation practice questions dummies. Implicit differentiation problems are chain rule problems in disguise. Find dydx by implicit differentiation and evaluate the derivative at. For each of the following equations, find dydx by implicit differentiation. Let us remind ourselves of how the chain rule works with two dimensional functionals. Example using the product rule sometimes you will need to use the product rule when differentiating a term. Calculusdifferentiationbasics of differentiationsolutions. Calculus i implicit differentiation practice problems. | {
"domain": "web.app",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9888419674366167,
"lm_q1q2_score": 0.8500715908328732,
"lm_q2_score": 0.8596637469145054,
"openwebmath_perplexity": 404.06666719092937,
"openwebmath_score": 0.8376717567443848,
"tags": null,
"url": "https://geocabirthve.web.app/1580.html"
} |
Equations inequalities system of equations system of inequalities basic operations algebraic properties partial fractions polynomials rational expressions sequences power sums. How implicit differentiation can be used the find the derivatives of equations that are not functions, calculus lessons, examples and step by step solutions, what is implicit differentiation, find the second derivative using implicit differentiation. In other words, the use of implicit differentiation enables. It is the fact that when you are taking the derivative, there is composite function in there, so you should use the chain rule. Implicit differentiation if a function is described by the equation \y f\left x \right\ where the variable \y\ is on the left side, and the right side depends only on the independent variable \x\, then the function is said to be given explicitly. These are functions of the form fx,y gx,yin the first tutorial i show you how to find dydx for such functions. This section contains lecture video excerpts and lecture notes on implicit differentiation, a problem solving video, and a worked example. Check that the derivatives in a and b are the same. Calculus implicit differentiation solutions, examples, videos.
To generalize the above, comparative statics uses implicit differentiation to study the effect of variable changes in economic models. To perform implicit differentiation on an equation that defines a function \y\ implicitly in terms of a variable \x\, use the following steps. Both use the rules for derivatives by applying them in slightly different ways to differentiate the complex equations without much hassle. Differentiation of implicit function theorem and examples. | {
"domain": "web.app",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9888419674366167,
"lm_q1q2_score": 0.8500715908328732,
"lm_q2_score": 0.8596637469145054,
"openwebmath_perplexity": 404.06666719092937,
"openwebmath_score": 0.8376717567443848,
"tags": null,
"url": "https://geocabirthve.web.app/1580.html"
} |
Then the derivative of y with respect to t is the derivative of y with respect to x multiplied by the derivative of x with respect to t dy dt dy dx dx dt. Implicit differentiation solved practice problems timestamp. This means that when we differentiate terms involving x alone, we can differentiate as usual. Given an equation involving the variables x and y, the derivative of y is found using implicit di erentiation as follows. Implicit differentiation multiple choice07152012104649. For example, according to the chain rule, the derivative of y. Suppose that the nth derivative of a n1th order polynomial is 0. This page was constructed with the help of alexa bosse. To do this, we use a procedure called implicit differentiation. Calculus bc parametric equations, polar coordinates, and vectorvalued functions defining and differentiating parametric equations parametric equations differentiation ap calc. | {
"domain": "web.app",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9888419674366167,
"lm_q1q2_score": 0.8500715908328732,
"lm_q2_score": 0.8596637469145054,
"openwebmath_perplexity": 404.06666719092937,
"openwebmath_score": 0.8376717567443848,
"tags": null,
"url": "https://geocabirthve.web.app/1580.html"
} |
You know that the derivative of sin x is cos x, and that according to the chain rule, the derivative of sin x3 is you could finish that problem by doing the derivative of x3, but there is a reason for you to leave. Implicit differentiation mctyimplicit20091 sometimes functions are given not in the form y fx but in a more complicated form in which it is di. Implicit differentiation basic idea and examples youtube. Jul, 2009 implicit differentiation basic idea and examples. Implicit differentiation can help us solve inverse functions. The following problems require the use of implicit differentiation. Multivariable calculus implicit differentiation this video points out a few things to remember about implicit differentiation and then find one partial derivative. Implicit diff free response solutions 07152012145323. The basic idea about using implicit differentiation 1. If a value of x is given, then a corresponding value of y is determined. In such a case we use the concept of implicit function differentiation. Some relationships cannot be represented by an explicit function. Implicit differentiation example walkthrough video khan. There is a subtle detail in implicit differentiation that can be confusing. | {
"domain": "web.app",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9888419674366167,
"lm_q1q2_score": 0.8500715908328732,
"lm_q2_score": 0.8596637469145054,
"openwebmath_perplexity": 404.06666719092937,
"openwebmath_score": 0.8376717567443848,
"tags": null,
"url": "https://geocabirthve.web.app/1580.html"
} |
The chain rule and implicit differentiation are techniques used to easily differentiate otherwise difficult equations. Here i introduce you to differentiating implicit functions. Jan 22, 2020 implicit differentiation is a technique that we use when a function is not in the form yf x. That is, i discuss notation and mechanics and a little bit of the. Jun 24, 2016 implicit differentiation solved practice problems timestamp. For example, in the equation we just condidered above, we. The majority of differentiation problems in firstyear calculus involve functions y written explicitly as functions of x. Implicit differentiation is a technique that we use when a function is not in the form yf x. Then the derivative of y with respect to t is the derivative of y with respect to x multiplied by the derivative of x with respect to t.
To make our point more clear let us take some implicit functions and see how they are differentiated. The technique of implicit differentiation allows you to find the derivative of y with respect to x without having to solve the given equation for y. Implicit di erentiation implicit di erentiation is a method for nding the slope of a curve, when the equation of the curve is not given in \explicit form y fx, but in \implicit form by an equation gx. Examples of the differentiation of implicit functions. | {
"domain": "web.app",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9888419674366167,
"lm_q1q2_score": 0.8500715908328732,
"lm_q2_score": 0.8596637469145054,
"openwebmath_perplexity": 404.06666719092937,
"openwebmath_score": 0.8376717567443848,
"tags": null,
"url": "https://geocabirthve.web.app/1580.html"
} |
This is done using the chain rule, and viewing y as an implicit function of x. Implicit differentiation is a method for finding the slope of a curve, when the equation of the. Calculus implicit differentiation solutions, examples. Implicit di erentiation implicit di erentiation is a method for nding the slope of a curve, when the equation of the curve is not given in \explicit form y fx, but in \ implicit form by an equation gx. For difficult implicit differentiation problems, this means that its possible to differentiate different individual pieces of the equation, then piece together the result. Implicit differentiation basic idea and examples what is implicit differentiation. This means that when we differentiate terms involving x. For each problem, use implicit differentiation to find dy dx in terms of x and y. Implicit functions are often not actually functions in the strict definition of the word, because they often have multiple y values for a single x value. Find dydx by implicit differentiation and evaluate the derivative at the given point. The chain rule must be used whenever the function y is being differentiated because of our assumption that y may be expressed as a function of x. However, some functions y are written implicitly as functions of x. The solutions to this equation are a set of points x,y which implicitly define a relation between x and y which we will call an implicit function. Parametric equations differentiation practice khan academy. | {
"domain": "web.app",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9888419674366167,
"lm_q1q2_score": 0.8500715908328732,
"lm_q2_score": 0.8596637469145054,
"openwebmath_perplexity": 404.06666719092937,
"openwebmath_score": 0.8376717567443848,
"tags": null,
"url": "https://geocabirthve.web.app/1580.html"
} |
Differentiation of implicit functions engineering math blog. Uc davis accurately states that the derivative expression for explicit differentiation involves x only, while the derivative expression for implicit differentiation may involve both x and y. Use implicit differentiation directly on the given equation. Implicit differentiation sometimes functions are given not in the form y fx but in a more complicated form in which it is di. Find the equation of the tangent line to the graph of 2. Implicit differentiation mcty implicit 20091 sometimes functions are given not in the form y fx but in a more complicated form in which it is di. In any implicit function, it is not possible to separate the dependent variable from the independent one. Preference bundles, utility and indifference curves. Search for wildcards or unknown words put a in your word or phrase where you want to leave a placeholder.
293 802 632 1207 1048 331 259 386 1012 524 1522 1299 1300 730 1542 1000 938 1032 708 353 1555 1317 459 852 320 880 1127 184 841 | {
"domain": "web.app",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9888419674366167,
"lm_q1q2_score": 0.8500715908328732,
"lm_q2_score": 0.8596637469145054,
"openwebmath_perplexity": 404.06666719092937,
"openwebmath_score": 0.8376717567443848,
"tags": null,
"url": "https://geocabirthve.web.app/1580.html"
} |
## Discrete Mathematics with Applications 4th Edition
Prove $\frac{1}{5} + \frac{4}{5^2} + \frac{4^2}{5^3} + \cdots + \frac{4^n}{5^{n+1}}$ is $\Theta(1)$. By theorem 5.2.4: $\frac{1}{5} + \frac{4}{5^2} + \frac{4^2}{5^3} + \cdots + \frac{4^n}{5^{n+1}} = \frac{1}{5} (1 + \frac{4}{5} + \frac{4^2}{5^2} + \cdots + \frac{4^n}{5^{n}}) = \frac{1}{5}(\frac{\frac{4}{5}^{n+1}-1}{\frac{4}{5}-1}) = -(\frac{4}{5}^{n+1}-1)=1-\frac{4}{5}^{n+1}$. For $n>0$, $1-\frac{4}{5}^{n+1} \leq 1$. By the transitive property: $\frac{1}{5} + \frac{4}{5^2} + \frac{4^2}{5^3} + \cdots + \frac{4^n}{5^{n+1}} \leq 1$. For $n>0$, $\frac{1}{5}(1) \leq \frac{1}{5} + \frac{4}{5^2} + \frac{4^2}{5^3} + \cdots + \frac{4^n}{5^{n+1}}$. Since all terms are positive: $\frac{1}{5}|(1)| \leq |\frac{1}{5} + \frac{4}{5^2} + \frac{4^2}{5^3} + \cdots + \frac{4^n}{5^{n+1}}| \leq |1|$. Thus for A=1/5, B=1, k=1, $A|(1)| \leq |\frac{1}{5} + \frac{4}{5^2} + \frac{4^2}{5^3} + \cdots + \frac{4^n}{5^{n+1}}| \leq B|1|$ for all $x>k$. Thus $\frac{1}{5} + \frac{4}{5^2} + \frac{4^2}{5^3} + \cdots + \frac{4^n}{5^{n+1}}$ is $\Theta(1)$.
Recall the definition of $\Theta$-notation: $f(x)$ is $\Theta(g(x))$ iff there exist positive real numbers A, B, k, such that $A|g(x)| \leq |f(x)| \leq B|g(x)|$ for all $x>k$. Recall theorem 5.2.3: Sum of a geometric sequence: For any real number $r$ except 1, and any integer $n \geq 0$, $\sum_{i=0}^{n}r^i = \frac{r^{n+1}-1}{r-1}$. In this case $r=\frac{4}{5}$. | {
"domain": "gradesaver.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9888419677654414,
"lm_q1q2_score": 0.850071589337905,
"lm_q2_score": 0.8596637451167997,
"openwebmath_perplexity": 184.410606193641,
"openwebmath_score": 0.9701132774353027,
"tags": null,
"url": "https://www.gradesaver.com/textbooks/math/other-math/discrete-mathematics-with-applications-4th-edition/chapter-11-analysis-of-algorithm-efficiency-exercise-set-11-4-page-764/38"
} |
# Prove $\sum_{k=0}^{n-2}{n-k \choose 2} = {n+1 \choose 3}$
Prove $$\sum_{k=0}^{n-2}{n-k \choose 2} = {n+1 \choose 3}$$
Is there a relation I can use that easily yields above equation?
• You could start with $\;{n-k \choose 2} = \frac{(n-k)\cdot(n-k-1)}{2\cdot 1}\,$.
– dxiv
Apr 29, 2017 at 4:59
• @mvw not, the same. I edited the problem to consider that. Apr 29, 2017 at 5:00
• @dxiv I already knew that. Thank you. What are the next steps? Apr 29, 2017 at 5:01
• $\sum \frac{(n-k)\cdot(n-k-1)}{2\cdot 1} = \frac{1}{2}\left(n^2 \cdot \sum 1 - n \cdot \sum (2k+1) + \sum k(k+1)\right)$
– dxiv
Apr 29, 2017 at 5:03
• Note: The original question just stated the left hand side
– mvw
Apr 29, 2017 at 11:30
The right hand side ${n+1 \choose 3}$ is the number of ways to choose 3 elements from $\{1,2,\ldots,n+1\}$. Let $A$ denote the set of all 3-subsets of $\{1,2,\ldots,n+1\}$. We want to show that $|A|$ is equal to the left hand side.
Let $A_i$ denote the set of all 3-subsets of $\{1,2,\ldots,n+1\}$ which have $i$ as their largest element. For example, if $i=n+1$, then $A_{n+1}$ is the set of all 3-subsets which contain $n+1$, and the number of ways to choose the remaining 2 elements is ${n \choose 2}$. Hence, $|A_{n+1}| = {n \choose 2}$. More generally, $|A_i| = {i-1 \choose 2}$, for $i=3,\ldots,n+1$, because the remaining 2 elements must be chosen from $\{1,\ldots,i-1\}$. Note that $i$ has to be at least 3 because the largest of 3 elements will be at least 3.
Using the fact that the $A_i$'s $(i=3,4,\ldots,n+1)$ are disjoint and their union is all of $A$, we obtain the desired formula.
• If the largest element in a 3-subset is say 7, then this largest element can't be any of $3, 4, 5, 6, 8, 9,\ldots,n+1$. So $A_7$ is disjoint from $A_3, A_4, A_5, A_6, A_8, A_9, \ldots, A_{n+1}$. Similarly for the other $A_i$'s, and so the $A_i$'s are pairwise disjoint. Apr 29, 2017 at 5:56
• Thanks for comment. I got it and because of this I deleted my comment. Apr 29, 2017 at 6:10 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9888419697383903,
"lm_q1q2_score": 0.8500715892563308,
"lm_q2_score": 0.8596637433190939,
"openwebmath_perplexity": 244.33631311856286,
"openwebmath_score": 1.000009298324585,
"tags": null,
"url": "https://math.stackexchange.com/questions/2257156/prove-sum-k-0n-2n-k-choose-2-n1-choose-3/2257198"
} |
$$\sum_{k=0}^{n-2}\frac{k^2+k(1-2n)+n^2-1}{2}\tag{Simplify what @dixv said}$$ now, $$\sum_{k=1}^nk=\frac{n(n+1)}{2}\text{ and }\sum_{k=1}^nk^2=\frac{n(n+1)(2n+1)}{6}$$
Hope it helps?
For $$S=\sum_{k=0}^{n-2}(n-k)(n-k-1)$$ set $n-k=r$ to get $$S=\sum_{r=1}^nr(r-1)=\sum_{r=1}^nr^2-\sum_{r=1}^nr$$
Using index transformation $m = n - k$, reversal of the summation order and the definition of the binomial coefficient in terms of factorials we can write for $n\ge 2$: \begin{align} \sum_{k=0}^{n-2} \binom{n-k}{2} &= \sum_{m=2}^{n} \binom{m}{2} \\ &= \sum_{m=2}^{n} \frac{m!}{2! (m-2)!} \\ &= \sum_{m=2}^{n} \frac{m(m-1)}{2} \\ &= \sum_{m=1}^n \frac{m(m-1)}{2} \\ &= \frac{1}{2} \sum_{m=1}^{n} m^2 - \frac{1}{2} \sum_{m=1}^n m \\ &= \frac{n(n+1)(2n+1)}{12} - \frac{n(n+1)}{4} \\ &= \frac{1}{6} n^3 + \frac{1}{4} n^2 + \frac{1}{12} n - \left( \frac{1}{4} n^2 + \frac{1}{4} n \right) \\ &= \frac{1}{6} n^3 - \frac{1}{6} n \end{align} where we used Faulhaber's formula for the square pyramidal and triangular numbers.
On the other side of the equation we have: \begin{align} \binom{n+1}{3} &= \frac{(n+1)!}{3!(n+1-3)!} \\ &= \frac{(n+1)n(n-1)}{6} \\ &= \frac{n^3 - n}{6} \end{align}
You can prove it using generating functions. We wish to prove that $$\sum_{m=0}^{n} \binom{m}{2}=\binom{n+1}{3}\tag{1}$$ Use the identity $$\frac{1}{(1-x)^k}=\sum_{n=0}^\infty \binom{k+n-1}{k-1}x^n.\tag{2}$$ (which can be obtained by repeatedly differentiating the geometric series) where $k\geq1$ to get that the generating function whose $m$th coefficent is $\binom{m+2}{2}$ is $(1-x)^{-3}$ and the generating function whose $m$th coefficent is $\binom{m}{2}$ is $x^2(1-x)^{-3}$. Thus $$\sum_{m=0}^{n} \binom{m}{2}= [x^n]\left(\frac{1}{1-x}\frac{x^2}{(1-x)^{3}}\right) = [x^n] \left(\frac{x^2}{(1-x)^{4}}\right)\tag{3} =\binom{n+1}{3}$$ by (2) as desired where $[x^n]$ extracts the coefficient of $x^n$.
With so many proofs there is no need for one more. But I like using finite differences. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9888419697383903,
"lm_q1q2_score": 0.8500715892563308,
"lm_q2_score": 0.8596637433190939,
"openwebmath_perplexity": 244.33631311856286,
"openwebmath_score": 1.000009298324585,
"tags": null,
"url": "https://math.stackexchange.com/questions/2257156/prove-sum-k-0n-2n-k-choose-2-n1-choose-3/2257198"
} |
With so many proofs there is no need for one more. But I like using finite differences.
Let $f(n)$ represent the sum.
$\Delta f(n) = \binom {n+1}{2} = \Delta \binom {n+1}{3}$
$\implies f(n) = \binom {n+1}{3} + c$
for $n=2$, we get $c + 1 = 1 \implies c=0$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9888419697383903,
"lm_q1q2_score": 0.8500715892563308,
"lm_q2_score": 0.8596637433190939,
"openwebmath_perplexity": 244.33631311856286,
"openwebmath_score": 1.000009298324585,
"tags": null,
"url": "https://math.stackexchange.com/questions/2257156/prove-sum-k-0n-2n-k-choose-2-n1-choose-3/2257198"
} |
Sketch the rest of the graph. I now introduce you to plotting the curve r=a sin2θ. 5a limits at infinity 3. If f and g are functions that have derivatives, then the composite function has a derivative given by f. uk A sound understanding of Curve Sketching is essential to ensure exam success. Domain: For what values ofx is f(x) defined? Avoid division by zero and square roots of negative numbers. CURVE SKETCHING EXERCISE 1 Sketch the following Find the gradient of the tangent to the curve x 2 2 + xy 2 + y2 = 14 at (2, 3). MCV4U CURVE SKETCHING QUIZ Name: Give all answers as exact numbers (fractions, terminating decimals, etc. To the left zooms in. Likes scottdave. And the goal here--STUDENT: [INAUDIBLE. Carefully, state L’Hospital’s Rule. 5: Summary of Curve Sketching Last updated; Save as PDF Page ID 4465 If the graph curves, does it curve upward or curve downward? This notion is called the concavity of the function. 5 Man vs machine. X – axis is the tangent at (2,0). Curve sketching lesson plan template and teaching resources. But there were no suitable comparison stars nearby to help today. Calculus plays a much smaller part in curve sketching than is commonly believed; it is just one of the tools at our disposal. Limits and Curve Sketching. This page will take a look at how people have assessed curve sketching in STACK, including some promising projects and alternatives. Instead of focusing on details at the start of a picture, make light sketch lines to capture the posture, proportions, and angles of your subject. Learn More. And so let's get started with that. Each topic builds on the previous one. Assignment. The sketch must include the coordinates of • … all the points where the curve meets the coordinate axes. The following incorporates the additional Calculus techniques you have recently learned. Never runs out of questions. Click below to download the free player from the Macromedia site. 5 – Summary of Curve Sketching Math& 151 Warnock - Class Notes Here | {
"domain": "curaben.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9910145712474399,
"lm_q1q2_score": 0.8500606980030403,
"lm_q2_score": 0.8577681122619883,
"openwebmath_perplexity": 1251.2958620737718,
"openwebmath_score": 0.5482892990112305,
"tags": null,
"url": "http://vurg.curaben.it/curve-sketching.html"
} |
player from the Macromedia site. 5 – Summary of Curve Sketching Math& 151 Warnock - Class Notes Here are the Guidelines for Sketching a Curve. f 0 (x) > 0). Start your bird sketch by noting the posture of the bird or the angle at which it sits with a single line. No calculator unless otherwise stated. Now after this how do i plot the remaining curve for x >1 (where y become s <0) Can you tell me a smaller approach. There are a number of mathematical curves that produced heart shapes, some of which are illustrated above. AP Calculus. Put the critical numbers in a sign chart to see where the first derivative is positive or negative (plug in the first derivative to get signs). Find the intervals of increase and decrease 8. Well, the free Urban Sketching 101 guide covers everything there is to know, including: what it is, where to go and starter techniques and tips for the urban sketcher on the go. Perhaps someone should change the thread title to "Not Curve Sketching". Although these problems are a little more challenging, they can still be solved using the same basic concepts covered in the tutorial and examples. Reading: Curve Sketching Maxima and Minima of Functions Much can be done to sketch the approximate graph of a function without calculus, in fact I strongly encourage you to rely mostly on your pre-calculus skills to sketch graphs. Curve sketching with calculus: logarithm. }\) Generally, we assume that the domain is the entire real line then find restrictions, such as where a denominator is $$0$$ or where negatives appear under the radical. Sample CHART for Sketching Curves. The ten steps of curve sketching each require a specific tool. We also need to find lim x→c+ g(x) h(x) and lim x→c− g(x) h(x). How do I tell which ones have or don't have horizontal asymptotes? Please use a simple method to understand! Thank you everyone who answers. The curve does not intersects the y – axis other than origin. Extrema and Curve Sketching Two Types of Extrema: Absolute | {
"domain": "curaben.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9910145712474399,
"lm_q1q2_score": 0.8500606980030403,
"lm_q2_score": 0.8577681122619883,
"openwebmath_perplexity": 1251.2958620737718,
"openwebmath_score": 0.5482892990112305,
"tags": null,
"url": "http://vurg.curaben.it/curve-sketching.html"
} |
the y – axis other than origin. Extrema and Curve Sketching Two Types of Extrema: Absolute (Global) Extremum at x = c: fHxL ‡ fHcL (respectively, fHxL £ fHcL) over entire interval of consideration. The graph shown is the DERIVATIVE of f. Keyword-suggest-tool. Math 140: Calculus with Analytic Geometry I Spring 2013 Penn State University Sections 7, 9, 16 Curve Sketching The following is a list topics to consider when drawing the graph of y = f(x). b) horizontal: No horizontal asymptotes because. how to sketch a curve that has asymptotes. Example 3 (f(x,y) = x2 +4y2 − 2x+2) Sketch the level curves of f(x,y) = x2 +4y2 −2x+2. The second set of holes uses the sketch to drive a law curve helix (2 turns) that matches the conical shape in that area. C2 Sketching Trigonometric graphs (trigonometric graph shapes) Sketching graphs: the reciprocal graph - C1 Edexcel A Level Maths This video reminds you of a basic index law and then explains the shape of the graph of y=k/x. AP Calculus AB/BC - M. move objects in sketch. While some sketching tools allow 3D drafting, the feature is not universal. The parabola is the envelope of the straight lines. Watch all CBSE Class 5 to 12 Video Lectures here. Curve sketching is a handy tool, used both directly and indrectrly in these examinations. Perhaps someone should change the thread title to "Not Curve Sketching". AP Calculus Project 4 - Curve Sketching Name_____ You will be in a group of 2 to 4. The general approach to curve sketching. Find the critical numbers 7. Fix any real number C. Get smarter on Socratic. 4a homework questions 3. STEP 2 Curve Sketching Questions 2. Curve Sketching packet. Heart Curve. Simple Curve Sketching; Higher Derivative and Concavity; Curve Sketching Techniques; Indeterminate Forms; The Integral. Curve Sketching Summary Introduction Now that you have learned how to find relative extrema, intervals where a function is increasing/decreasing, and intervals where a function is concave up/concave down, we will now | {
"domain": "curaben.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9910145712474399,
"lm_q1q2_score": 0.8500606980030403,
"lm_q2_score": 0.8577681122619883,
"openwebmath_perplexity": 1251.2958620737718,
"openwebmath_score": 0.5482892990112305,
"tags": null,
"url": "http://vurg.curaben.it/curve-sketching.html"
} |
is increasing/decreasing, and intervals where a function is concave up/concave down, we will now "pull it all together" and work through several AP problems that involve the analysis of functions and curve sketching. While we have been treating the properties of a function separately (increasing and decreasing, concave up and concave down, etc. Passing the fast paced Higher Maths course significantly increases your career opportunities by helping you gain a place on a college/university course, apprenticeship or even landing a job. Let’s do another curve sketching example. If you're interested, take a look. While we have been treating the properties of a function separately (increasing and decreasing, concave up and concave down, etc. In the past, one of the important uses of derivatives was as an aid in curve sketching. Now after this how do i plot the remaining curve for x >1 (where y become s <0) Can you tell me a smaller approach. Curve Sketching with Calculus • First derivative and slope • Second derivative and concavity. Learn vocabulary, terms, and more with flashcards, games, and other study tools. Active 4 years, 7 months ago. Applications to Curve Sketching. I have also included teacher pages that give: 1) the equation that was actually graphed 2. Intercepts C. Guidelines For Analyzing The Graph Of A Function. MCV4U Unit 3 Test Curve Sketching (V1) Name: Knowledge: /25 Application: /19 Inquiry: /8 Comm. View Homework Help - curve_sketching. a) Domain: Find the domain of the function. L5 – Curve Sketching Unit 2 MCV4U Jensen Algorithm for Curve Sketching 1. “Alliances” Written by Jeri Taylor Directed by Les Landau Season 2, Episode 14 Production episode 131 Original air date: January 22, 1996 Stardate: 49337. Get smarter on Socratic. In these notes, we will review the critical attributes of the graphs that you studied. Find the - and -intercepts. Here are all the components that we must include: Domain Intercepts. Sketching Solution Curves for Autonomous | {
"domain": "curaben.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9910145712474399,
"lm_q1q2_score": 0.8500606980030403,
"lm_q2_score": 0.8577681122619883,
"openwebmath_perplexity": 1251.2958620737718,
"openwebmath_score": 0.5482892990112305,
"tags": null,
"url": "http://vurg.curaben.it/curve-sketching.html"
} |
the components that we must include: Domain Intercepts. Sketching Solution Curves for Autonomous DEs By finding and classifying critical points for an Autonomous DE we can greatly simplify the process of sketching a solution curve. All you need is the most minimal of kit, and you're away on a journey of enjoyment and pleasure. Unit 1: Limits & Continuity. Summary of Curve Sketching 1 Domain of f(x) 2 x and y intercepts 1 x-intercepts occur when f(x) = 0 2 y-intercept occurs when x = 0 3 Find the asymptotes (vertical, horizontal / slant). An asymptote is a line that the curve gets very very close to but never intersect. Calculus 3 very difficult curve sketching problem? Sketch the curve traced out by the tip of the radius vector and indicate the direction in which the curve is traversed as t increases. Look at any item sitting around you. In particular, Section 4. If f( x) = f(x), then f(x) is symmetric about. Be sure to nd any horizontal and ver-tical asymptotes, show on a sign chart where the function is increasing/decreasing, concave up/concave down, and identifying (as ordered pairs) all relative extrema and in ection points. Determine the x- and y- intercepts of the function, if possible. A critical point may be a maximum point, minimum point, or neither. However, this equation, y =x / x^2 -1 does not have any horizontal asymptotes. With a model open, click Edit > Project. The curve passes through origin and meets the x – axis at two coincident points (2,0) and (2,0). This is for a few reasons, but primarily because curve sketching takes a little bit of intuition. XXIII – Curve Sketching 1. If f and g are functions that have derivatives, then the composite function has a derivative given by f. GraphSketch is provided by Andy Schmitz as a free service. A critical point may be a maximum point, minimum point, or neither. Find more Mathematics widgets in Wolfram|Alpha. Curve Sketching. Link to Binder: Link to Current Tab: Email Embed Facebook Twitter Google+ | {
"domain": "curaben.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9910145712474399,
"lm_q1q2_score": 0.8500606980030403,
"lm_q2_score": 0.8577681122619883,
"openwebmath_perplexity": 1251.2958620737718,
"openwebmath_score": 0.5482892990112305,
"tags": null,
"url": "http://vurg.curaben.it/curve-sketching.html"
} |
Curve Sketching. Link to Binder: Link to Current Tab: Email Embed Facebook Twitter Google+ Classroom Upgrade to Pro Today! The premium Pro 50 GB plan gives you the option to download a copy of your binder to your local machine. 5 Algorithm for Curve Sketching The Algorithm for Curve Sketching provides us with a framework from which we can determine all the key elements of a curve so that we can sketch it relatively accurately. Example 1 Sketch the parametric curve for the following set of parametric equations. Design Sketching Learning Curves (2011, 177 pages, Klara Sjölén and Allan Macdonald) is a brand new sketch book, aimed at teaching how to really learn to sketch. dominique_cimafranca writes "The Dynamic Graphics Project of the University of Toronto has released a pretty nifty 3D curve sketching system. Some of the worksheets for this concept are Perspective drawing work, Pencil sketching 2nd edition, Mind map templates, Graphs of trig functions, Drawing basic shapes, The effect of exploratory computer based instruction on, Basic technical mathematics with calculus si version, Using. Curve Sketching - Displaying top 8 worksheets found for this concept. A normal to a curve is a line perpendicular to a tangent to the curve. Textbook Authors: Thomas Jr. curve_sketching_solutions. In this case, it does not have a vertical asymptote. The graph shown is the DERIVATIVE of f. View Homework Help - curve_sketching. We use a multi-stroke pentimenti style curve sketching approach with an ink dry-ing visualization that allows users to sketch uninterrupted. : If your making something that is less than simple it will almost always pay you to do some kind of drawing the try to get things straight in your head before you commit to cutting expensive materials up. Advanced Trigonometry 1 Revision Notes Inverse Trigonometric Function, Stationary Points, Curve Sketching. How to draw curves, pfft! Not so fast! Curves are in a great deal of things you'll want to draw. 3: # 1, 2a-f, | {
"domain": "curaben.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9910145712474399,
"lm_q1q2_score": 0.8500606980030403,
"lm_q2_score": 0.8577681122619883,
"openwebmath_perplexity": 1251.2958620737718,
"openwebmath_score": 0.5482892990112305,
"tags": null,
"url": "http://vurg.curaben.it/curve-sketching.html"
} |
curves, pfft! Not so fast! Curves are in a great deal of things you'll want to draw. 3: # 1, 2a-f, 3-9 5. And a lot of people struggle with them even when they don't realise it. Curve Sketching The concepts of domain, limits, derivative, extreme values, monotonicity and concavity have been introduced. Learn more about ferguson curve, curve, draw curve, draw ferguson curve. Sample Problem #1: f(x) = x3 - 6x2 + 9x + 1. From the home tab, select move curve icon command. State any horizontal and vertical asymptotes or holes in the graph. When you exit the sketch, regions are formed by intersecting lines. We begin by making some general remarks about curve sketching, by which we mean more specifically, sketching or drawing the graph of the function y equals f of x in the xy-plane. Sketch the Curve!. While we have been treating the properties of a function separately (increasing and decreasing, concave up and concave down, etc. Instead of focusing on details at the start of a picture, make light sketch lines to capture the posture, proportions, and angles of your subject. 10 determines the solution for given polynomials. unit 4: curve sketching Lesson 1: Increasing/Decreasing Functions. Turning point Axis of Symmetry Mirror point Y intercept X intercepts { the real roots The turning point is always required, and another two points are needed for a rough sketch. Although these problems are a little more challenging, they can still be solved using the same basic concepts covered in the tutorial and examples. WORKSHEETS: Practice-Curve Sketching 1 open ended. Show that, if a > 1, then C has exactly one stationary point. Passing the fast paced Higher Maths course significantly increases your career opportunities by helping you gain a place on a college/university course, apprenticeship or even landing a job. If a lack of sketching tips are holding you back from beginning your sketching journey, then we've got you covered. Honors Calculus -e - Asymptotes Plans changed for Date | {
"domain": "curaben.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9910145712474399,
"lm_q1q2_score": 0.8500606980030403,
"lm_q2_score": 0.8577681122619883,
"openwebmath_perplexity": 1251.2958620737718,
"openwebmath_score": 0.5482892990112305,
"tags": null,
"url": "http://vurg.curaben.it/curve-sketching.html"
} |
journey, then we've got you covered. Honors Calculus -e - Asymptotes Plans changed for Date Period Domain: Range: Zeros: y-intercept: HA: c. Tangent lines are useful in calculus because they can magnify the slope of a curve at a single point. Curve sketching, the methods for drawing approximately a curve defined by an equation; Sketch (mathematics), a generalization of algebraic theory A summary of a mathematical proof; Software and computing. This is achieved by adding a sketch relation to your finished curve. how to sketch a curve that has asymptotes. You should be able to quickly sketch straight-line graphs, from your knowledge that in the equation y = mx + c, m is the gradient and. Determine the domain and range. The figure illustrates a means to sketch a sine curve – identify as many of the following values as you can: asymptotic behaviour,. Get smarter on Socratic. ), we combine them here to produce an accurate graph of the function without plotting lots of extraneous points. Find the location of the x and y intercepts and plot them on the graph. This usually isn’t of help. An asymptote is a line that the curve gets very very close to but never intersect. Detailed Example of Curve Sketching x Example Sketch the graph of f(x) =. Curve Sketching Recipe: 1. When x < 0 then y < 0 so in this case the curve lies in the 3rd quadrant. Passing the fast paced Higher Maths course significantly increases your career opportunities by helping you gain a place on a college/university course, apprenticeship or even landing a job. Today, we are going to lay out the principles behind these questions, and explain the methods on how to attack them. Curve Sketching Quiz. 1 $$y=x^5-5x^4+5x^3$$. We will review the main topics that you'll need to know for the AP Calculus exams. Please check your network connection and refresh the page. The system coherently integrates existing techniques of sketch-based interaction with a number of novel and enhanced features. docx: File Size: 255 kb | {
"domain": "curaben.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9910145712474399,
"lm_q1q2_score": 0.8500606980030403,
"lm_q2_score": 0.8577681122619883,
"openwebmath_perplexity": 1251.2958620737718,
"openwebmath_score": 0.5482892990112305,
"tags": null,
"url": "http://vurg.curaben.it/curve-sketching.html"
} |
of sketch-based interaction with a number of novel and enhanced features. docx: File Size: 255 kb Video - The Mean Value Theorem. Sketching Sketching is useful if you want to create a region that can be pulled into 3D. Theory: This section will review the basic principles and equations that you should know to answer the exam. DUE TUESDAY FEBRUARY 16 AT THE BEGINNING OF CLASS. 1 (i) A curve has equation Find the x-coordinates of the points on the curve where [2] (ii) The curve is translated by Write down an equation for the translated curve. Some of the worksheets for this concept are Perspective drawing work, Pencil sketching 2nd edition, Mind map templates, Graphs of trig functions, Drawing basic shapes, The effect of exploratory computer based instruction on, Basic technical mathematics with calculus si version, Using. Exit Tickets. 5 Man vs machine. Play this game to review Calculus. Madas Created by T. 10 5 5 10 7 45. The graph shown is the DERIVATIVE of f. Curve Sketching Example: Sketch 1 Review: nd the domain of the following function. Alternatively, Curve Sketching 1. Turning point Axis of Symmetry Mirror point Y intercept X intercepts { the real roots The turning point is always required, and another two points are needed for a rough sketch. Curve Sketching Calculus, free curve sketching calculus software downloads, Page 2. [Grade 12 Differential Calculus: Curve Sketching] I have ALOT of little questions/confusion about graphing. You should be able to quickly sketch straight-line graphs, from your knowledge that in the equation y = mx + c, m is the gradient and. 4 Curve Sketching V63. Keyword-suggest-tool. org helps support GraphSketch and gets you a neat, high-quality, mathematically-generated poster. the methods for drawing approximately a curve defined by an equation. Curve Sketching The concepts of domain, limits, derivative, extreme values, monotonicity and concavity have been introduced. ©2007 Pearson Education Asia INTRODUCTORY MATHEMATICAL | {
"domain": "curaben.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9910145712474399,
"lm_q1q2_score": 0.8500606980030403,
"lm_q2_score": 0.8577681122619883,
"openwebmath_perplexity": 1251.2958620737718,
"openwebmath_score": 0.5482892990112305,
"tags": null,
"url": "http://vurg.curaben.it/curve-sketching.html"
} |
and concavity have been introduced. ©2007 Pearson Education Asia INTRODUCTORY MATHEMATICAL ANALYSIS 0. Get step-by-step solutions to your Curve sketching problems, with easy to understand explanations of each step. Curve sketching. 5 Curve Sketching ¶ permalink. Curve Sketching. A normal to a curve is a line perpendicular to a tangent to the curve. C1 Curve Sketching - Factorising & Sketching Polynomials 2 QP C1 Curve Sketching - Factorising & Sketching Polynomials 3 MS C1 Curve Sketching - Factorising & Sketching Polynomials 3 QP. Label x and y intercepts. A curve with two loops. 5—Curve Sketching Show all work on a separate sheet of paper. Let's put all of our differentiation abilities to use, by analyzing the graphs of various functions. If x is large negative then y is large positive. Concavity/Inflection Points H. The first thing I did to solve this problem was sketch the curve{s}. Leading artists share their sketching tips to help you get started, then take things further. 1 Extreme Values. Perfect for acing essays, tests, and quizzes, as well as for writing lesson plans. x -Intercept (s) Vertical Asymptote (s) Horizontal and/or Oblique Asymptote (s) First Derivative. Sketch the vector function f(t) = < t 2, t 3 > for -5≤ t ≤ 5. Tags: curve sketching. Review of Prerequisite Skills. This is an input location where either f0(x. Unit 1: Limits & Continuity. Created by T. Zeros The zeros of the function f(x) are: x 1 = 3 and x 2 = 2 y-intercept. This usually isn’t of help. Sketch the graph of the curve with equation y x x x= + + −(1 4 2)( )( ), x∈. To plot a function just type it into the function box. Because we often represent functions by their graphs, you could say that calculus is all about the analysis of graphs. A full lesson on sketching cubics, quartics and reciprocal functions. is to determine the following: 1) find y(0) 2) find y = 0, if. 2: CURVE SKETCHING POLYNOMIALS Example 3. We have been learning how we can understand the behavior of a function | {
"domain": "curaben.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9910145712474399,
"lm_q1q2_score": 0.8500606980030403,
"lm_q2_score": 0.8577681122619883,
"openwebmath_perplexity": 1251.2958620737718,
"openwebmath_score": 0.5482892990112305,
"tags": null,
"url": "http://vurg.curaben.it/curve-sketching.html"
} |
POLYNOMIALS Example 3. We have been learning how we can understand the behavior of a function based on its first and second derivatives. Determine asymptotes: a) for vertical asymptotes, check for rational function zero denominators, or unde ned log function points; b) for horizontal asymptotes, consider lim. Let's put all of our differentiation abilities to use, by analyzing the graphs of various functions. We can roughly sketch the graph with stationary point, point of inflection, and y-intercept. Curve Sketching We've done most of the legwork needed for this section. The graph shown is the DERIVATIVE of f. Students describe a curve given the equation of the curve in polar form. The following steps are taken in the process of curve sketching: Find the domain of the function and determine the points of discontinuity (if any). 5 Algorithm for Curve Sketching part 2. So let's hope we can do this. Curve sketching The roots , stationary points , inflection point and concavity of a cubic polynomial x 3 − 3 x 2 − 144 x + 432 (black line) and its first and second derivatives (red and blue). The process of using the first derivative and second derivative to graph a function or relation. It is an application of the theory of curves to find their main features. Fusion 360 Blog. Keyword-suggest-tool. Horizontal and vertical asymptotes may be calculated by taking the appropriate limits of. , ISBN-10: 0-32187-896-5, ISBN-13: 978-0-32187-896-0, Publisher: Pearson. In this calculus worksheet, 12th graders answer questions about derivatives, increasing and decreasing functions, relative maximum and minimum and points of inflection. Prior observations indicate that professionals want full control of the final shapes of 2D/3D curves while leveraging their sketching skills. The course is intended to be challenging and demanding. Concavity/Inflection Points H. Course Description: UCI Math 2A is the first quarter in Single-Variable Calculus and covers the following topics: Introduction | {
"domain": "curaben.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9910145712474399,
"lm_q1q2_score": 0.8500606980030403,
"lm_q2_score": 0.8577681122619883,
"openwebmath_perplexity": 1251.2958620737718,
"openwebmath_score": 0.5482892990112305,
"tags": null,
"url": "http://vurg.curaben.it/curve-sketching.html"
} |
2A is the first quarter in Single-Variable Calculus and covers the following topics: Introduction to derivatives, calculation of derivatives of algebraic and trigonometric functions; applications including curve sketching, related rates, and optimization; exponential and logarithm functions. Polynomial Curve Sketching - Displaying top 8 worksheets found for this concept. How do I know which equations have horizontal asymptotes? For example, y=x+2 / x-1 has the horizontal asymptote of y=1. GraphSketch is provided by Andy Schmitz as a free service. 10 creates exercises with solutions and graphs in the field of curve sketching of linear, quadratic, cubic, quartic and quintic polynomials. Sketching Sketching is useful if you want to create a region that can be pulled into 3D. Limits by Direct Evaluation. When x < 0 then y < 0 so in this case the curve lies in the 3rd quadrant. Curve sketching (Q350877) From Wikidata. A Rhino curve is similar to a piece of wire. Curve Sketching Summary Introduction Now that you have learned how to find relative extrema, intervals where a function is increasing/decreasing, and intervals where a function is concave up/concave down, we will now "pull it all together" and work through several AP problems that involve the analysis of functions and curve sketching. Therefore the domain is D f = R. The curve consist of straight lines and some arces, all tangent to each other. 5 B—Curve Sketching Summary For a function f ′, the combined information of the first derivative f and the second derivative f ′′ can tell us the shape of a graph. Sketching Polynomials 1 January 16, 2009 Oct 11 9:12 AM Sketching Polynomial Functions Objective Sketch the graphs of Polynomial Functions. Solve Curve sketching problems with our Curve sketching calculator and problem solver. Given the information, determine the following about f(x): (Explain each of your answers) 8. Instead, WebAssign will ask limited submission questions about your graphs. Sketching the | {
"domain": "curaben.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9910145712474399,
"lm_q1q2_score": 0.8500606980030403,
"lm_q2_score": 0.8577681122619883,
"openwebmath_perplexity": 1251.2958620737718,
"openwebmath_score": 0.5482892990112305,
"tags": null,
"url": "http://vurg.curaben.it/curve-sketching.html"
} |
8. Instead, WebAssign will ask limited submission questions about your graphs. Sketching the Curve Summary – Graphing Ex 2 – Part 4 of 4. The following problems illustrate detailed graphing of functions of one variable using the first and second derivatives. Curve Sketching. In my opinion, one of the more difficult topics in Extension 1 and 2 has got to be curve sketching. 5 Summary of Curve Sketching(A) GUIDELINES FOR SKETCHING A CURVE ةلاد نيب مسر تاوطخ 1 Find the domain ةلادلا ل جم دجوا 1 the domain is, the set of values of for which is defined. However, for sketching "basic" cubics, you should be given "nice" equations. Explore math with our beautiful, free online graphing calculator. It is recommended that you start with Lesson 1 and progress through the video lessons, working through each problem session and taking each quiz in the order it appears in the table of contents. If f '(x) =0 , then P(x, f (x)) is a local extrema and tangent is horizontal. Welcome to highermathematics. r(t)=(2cost)i+(2sint)j+(2pi-t)k 0 =t=2pi Ok, so I've drawn out the curve, and my curve starts at (2)i+(2pi)k and ends at (2)i. Write NONE if there are none. Whether we are interested in a function as a purely mathematical object or in connection with some application to the real world, it is often useful to know what the graph of the function looks like. How to draw curves, pfft! Not so fast! Curves are in a great deal of things you'll want to draw. Click below to download the free player from the Macromedia site. We will focus on polynomials, but the same methods apply to roughly sketching the graph of any function. Intercepts C. The following techniques may also be of help and they should be employed whenever appropriate. A rational function is looked at as an. We have been learning how we can understand the behavior of a function based on its first and second derivatives. That tells us that our midline drooping down 4. Applications of the Derivative: Curve Sketching and Extrema | {
"domain": "curaben.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9910145712474399,
"lm_q1q2_score": 0.8500606980030403,
"lm_q2_score": 0.8577681122619883,
"openwebmath_perplexity": 1251.2958620737718,
"openwebmath_score": 0.5482892990112305,
"tags": null,
"url": "http://vurg.curaben.it/curve-sketching.html"
} |
us that our midline drooping down 4. Applications of the Derivative: Curve Sketching and Extrema Practice Problems: The following six pages contain 28 problems to practice curve sketching and extrema problems. In this section, we learn methods of drawing graphs by hand. When f''(x) = 0, there is an inflection point changing the concavity of f(x). Find the - and -intercepts. 6 curve sketching 3. Curve sketching for calculus. Semester Test 1 I Saturday 25 August I D1 Lab 308 I Starts at 09h00. 3 Second Derivative Test. Mathematics / Analysis - Plotter - Calculator 3. Curve Sketching Learning Outcomes Make tables and draw the graphs of various equations to include: Linear Functions Quadratic Functions Cubic Functions - A free PowerPoint PPT presentation (displayed as a Flash slide show) on PowerShow. Curve sketching is a handy tool, used both directly and indrectrly in these examinations. there is no y-. Capture drawings on site will forever embed that experience in your mind. ةفرعم لعجت يتلا ميق ةعومجم وه ةلادلا لجم. xy–plane where f takes the value C. Curve sketching (Q350877) From Wikidata. Curve Sketching Using Calculus - Part 1of 2. The distance from the starting line of a runner in the 100-meter dash is a. Curve sketching This PowerPoint presentation shows the different stages involved in sketching the graph Sketching the graph Step 1: Find where the graph cuts the axes When x = 0, y = 4/3, so the graph goes through the point (0, 4/3). Incorporates the use of GeoGebra and the Casio fx-991EX Classwiz. In this method, we’ll skip steps 1 to 4 of curve sketching and go straight to steps 5, 6 and 7. y = (sin x)/ x (Pay particular attention to the shape of the curve around x = 0). 2 Sample Problems. Put the critical numbers in a sign chart to see where the first derivative is positive or negative (plug in the first derivative to get signs). Even high school students love to color!. This interactive workshop is designed for students currently enrolled in Math 226 who | {
"domain": "curaben.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9910145712474399,
"lm_q1q2_score": 0.8500606980030403,
"lm_q2_score": 0.8577681122619883,
"openwebmath_perplexity": 1251.2958620737718,
"openwebmath_score": 0.5482892990112305,
"tags": null,
"url": "http://vurg.curaben.it/curve-sketching.html"
} |
to color!. This interactive workshop is designed for students currently enrolled in Math 226 who would like more exposure on understanding the relationships between first derivatives, second derivatives, and curve sketching. Be sure to list the Domain and Range, intercepts, the equation of any asymptotes, intervals of increasing/decrease,. Recall, if they exist, we find the -intercept(s) by setting =0 and. 2 shows graphs A and B. pdf File history uploaded by Paul Kennedy 11 months, 1 week ago No preview is available for MCV 4U Unit 8 Shell-Curve Sketching. Curve Sketching. From the derivative's graph, identify the interval graph where f (the original function) is concave up. The sketch must include the coordinates of • … all the points where the curve meets the coordinate axes. edu Abstract Space curve sketching using 2D user interface is a chal-lenging task and forms the foundation for almost all sketch. 2: CURVE SKETCHING RATIONAL FUNCTIONS EXERCISES Give a complete graph of the following functions. And there's relatively little computation. Some of the worksheets for this concept are Perspective drawing work, Pencil sketching 2nd edition, Mind map templates, Graphs of trig functions, Drawing basic shapes, The effect of exploratory computer based instruction on, Basic technical mathematics with calculus si version, Using. 4: # 1 (what you need), 3abc 5. 5 Curve Sketching. Curve Sketching 1. For the sake. Binder ID: 93334. Sketch the graph of the following functions by finding the domain, symmetry, intercepts, asymptotes, intervals of increase and decrease, local maximum and minimum, concavity and points of inflection. Calculus 3 very difficult curve sketching problem? Sketch the curve traced out by the tip of the radius vector and indicate the direction in which the curve is traversed as t increases. And the goal here--STUDENT: [INAUDIBLE. 00 Price per gallon 2. • The techniques used in algebra for graphing functions do not demonstrate subtle behaviors of curves. | {
"domain": "curaben.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9910145712474399,
"lm_q1q2_score": 0.8500606980030403,
"lm_q2_score": 0.8577681122619883,
"openwebmath_perplexity": 1251.2958620737718,
"openwebmath_score": 0.5482892990112305,
"tags": null,
"url": "http://vurg.curaben.it/curve-sketching.html"
} |
The techniques used in algebra for graphing functions do not demonstrate subtle behaviors of curves. 5 Curve Sketching. Polar curves: wrapping a function around the pole. So the next topic is curve sketching. Here input is an equation. XXIII – Curve Sketching 1. Curve Sketching. This property is called the asymptote. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. Curve Sketching 1. Sketching curves the curve r = asin 2θ. Limits and Curve Sketching. 4a concavity and pois 3. Passing the fast paced Higher Maths course significantly increases your career opportunities by helping you gain a place on a college/university course, apprenticeship or even landing a job. Curve sketching Graphing The topic menu above allows you to move directly to any of the four sections for each topic. That’s the first step ,in any curve sketching problem. I have a curve sketching assignment and this one question i am having trouble with (this is x2= x squared and 3x2 is 3 x squared. Polar curves: wrapping a function around the pole. f 0 (x) 0). b) horizontal: No horizontal asymptotes because. Passing the fast paced Higher Maths course significantly increases your career opportunities by helping you gain a place on a college/university course, apprenticeship or even landing a job. Find vertical asymptotes and holes. An asymptote is a line that the curve gets very very close to but never intersect. In this calculus worksheet, 12th graders answer questions about derivatives, increasing and decreasing functions, relative maximum and minimum and points of inflection. Voyager …. As $$x$$ increases, the slope of the tangent line increases. The first curve is a rotated cardioid (whose name means "heart-shaped") given by the polar equation. That tells us that our midline drooping down 4. Sketching Infinite Lines: the Conic tool applies tangency at each endpoint and selects the top vertex of the curve. The graph shown is the DERIVATIVE of f. CURVE | {
"domain": "curaben.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9910145712474399,
"lm_q1q2_score": 0.8500606980030403,
"lm_q2_score": 0.8577681122619883,
"openwebmath_perplexity": 1251.2958620737718,
"openwebmath_score": 0.5482892990112305,
"tags": null,
"url": "http://vurg.curaben.it/curve-sketching.html"
} |
each endpoint and selects the top vertex of the curve. The graph shown is the DERIVATIVE of f. CURVE SKETCHING BLAKE FARMAN Lafayette College Name: 1. So even 10 problems you should be able to get through in a few hours. Domain of f(x) 2. y: f 00 ( x ) > 0 ) f ( x up. Recall, if they exist, we find the -intercept(s) by setting =0 and. Topic: Calculus, Derivatives. Curve Sketching Recipe: 1. From the home tab: Direct sketch group -> sketch curve gallery -> edit curve gallery -> move curve. These are potential local extrema. But there were no suitable comparison stars nearby to help today. Although these problems are a little more challenging, they can still be solved using the same basic concepts covered in the tutorial and examples. Convert Entities: Creates one or more entities in a 3D sketch by projecting an edge, loop, face, external curve, external sketch contour, set of edges, or set of external curves onto the sketch plane. If you are going to try these problems before looking at the solutions, you can avoid common mistakes by carefully labeling critical points, intercepts, and inflection. If a graph is given, then simply look at the left side and the right side. The graph of y. The curve will be exactly the same as when you add hydrochloric acid to sodium hydroxide. Because it is a curve in 2d, it is usually easier to sketch than the graph of f. Notes - Curve Sketching (Extrema, Critical Numbers, Intervals of Increase and Decrease, etc. Curve Sketching Date_____ Period____ For each problem, find the: x and y intercepts, x-coordinates of the critical points, open intervals where the function is increasing and decreasing, x-coordinates of the inflection points, open intervals where the function is concave up and concave down, and relative minima and maxima. And the goal here--STUDENT: [INAUDIBLE. 5 yesterday, which fit well with the light curve. Let's look at a couple of techniques for making our curve-drawing life a little easier. Oct 5, 2019 #10 Kolika28. 5 | {
"domain": "curaben.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9910145712474399,
"lm_q1q2_score": 0.8500606980030403,
"lm_q2_score": 0.8577681122619883,
"openwebmath_perplexity": 1251.2958620737718,
"openwebmath_score": 0.5482892990112305,
"tags": null,
"url": "http://vurg.curaben.it/curve-sketching.html"
} |
couple of techniques for making our curve-drawing life a little easier. Oct 5, 2019 #10 Kolika28. 5 How to Find Inflection Points 2 3 2 12 inflection point f(x) =. Curve Sketching. Give a complete graph of f(x) = 1 3 x3 1 2 x2 2x+ 1: Be sure to show on a sign chart where the function is increasing/decreasing, concave up/concave down, and identifying (as ordered pairs) all relative extrema and in ection points. This interactive workshop is designed for students currently enrolled in Math 226 who would like more exposure on understanding the relationships between first derivatives, second derivatives, and curve sketching. Get feedback on your graphs. Further information regarding the Curve Sketching Summer School. Veitch 1 p x 1 = 0 1 p x = 1 1 = p x 1 = x The other critical value is at x = 1. A rational function is looked at as an. We will focus on polynomials, but the same methods apply to roughly sketching the graph of any function. Domain and Range. Curve Sketching Example: Sketch 1 Review: nd the domain of the following function. Right-click the line and select Set as Mirror Line from the context menu. A function f (x) is decreasing on an interval if the values of f decrease as x increases (i. In this method, we’ll skip steps 1 to 4 of curve sketching and go straight to steps 5, 6 and 7. 1 use many of the techniques discussed in this chapter. Take a quick interactive quiz on the concepts in Curve Sketching Derivatives, Intercepts & Asymptotes or print the worksheet to practice offline. MCV4U CURVE SKETCHING QUIZ Name: Give all answers as exact numbers (fractions, terminating decimals, etc. Learn new vocabulary: f is concave up wherever f0 is increasing. Microsoft Word - MCV4U1 - Task - Curve Sketching. This is good because as well as triggering the rebuild it also fully constrains the curve. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. A 3D Curve Sketching System For Tablets 72 Posted by timothy on Saturday | {
"domain": "curaben.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9910145712474399,
"lm_q1q2_score": 0.8500606980030403,
"lm_q2_score": 0.8577681122619883,
"openwebmath_perplexity": 1251.2958620737718,
"openwebmath_score": 0.5482892990112305,
"tags": null,
"url": "http://vurg.curaben.it/curve-sketching.html"
} |
animate graphs, and more. A 3D Curve Sketching System For Tablets 72 Posted by timothy on Saturday October 11, 2008 @10:53PM from the no-mention-of-license-terms dept. No calculator unless otherwise stated. After memorizing the concepts of the second derivative, we move onto the next topic: creating sign charts. If you are going to try these problems before looking at the solutions, you can avoid common mistakes by carefully labeling critical points, intercepts, and inflection. 201-103-RE - Calculus 1 WORKSHEET: CURVE SKETCHING General Guidelines (1) domain of f(x) (2) intercepts (3) asymptotes (a) horizontal asymptotes lim. Curve Sketching Quiz. Y U bA Ql Yl9 irmiwgth1tes m srdeWs3e 0r Vv vebd E. When is the function f(x) concave up? 12. Characteristics of curve. Limits by Direct Evaluation. The basic sine curve has a midline at the x-axis (y = 0). In order to sketch the curve of a function, you need to:. Computer-generated graph of y = x 2 /(x + 3) One of the interesting attributes of curve sketches is that the sketches we make by hand are rarely to scale and can grossly exaggerate features of. DUE TUESDAY FEBRUARY 16 AT THE BEGINNING OF CLASS. You can sketch a curve by listing down a range of values for x, calculating the values for y and from the points drawn on the graph, join the dots. Let’s put it all together; here are some general curve sketching rules: Find critical numbers (numbers that make the first derivative 0 or undefined). CURVE SKETCHING Curve Sketching Steps: for sketching the graph of f(x). Created by T. NOTES: There are now many tools for sketching functions (Mathcad, Scientific Notebook, graphics calculators, etc). 7 Curve Sketching. A full lesson on sketching cubics, quartics and reciprocal functions. And the goal here--STUDENT: [INAUDIBLE. Applet: Curve Sketching: Increasing/Decreasing Try it! The Second Derivative: Concavity and In ection Points Suppose y = f(x) is a given function. 6 # 1-3 SPICY 5. Curve Sketching Introduction Prior to | {
"domain": "curaben.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9910145712474399,
"lm_q1q2_score": 0.8500606980030403,
"lm_q2_score": 0.8577681122619883,
"openwebmath_perplexity": 1251.2958620737718,
"openwebmath_score": 0.5482892990112305,
"tags": null,
"url": "http://vurg.curaben.it/curve-sketching.html"
} |
Points Suppose y = f(x) is a given function. 6 # 1-3 SPICY 5. Curve Sketching Introduction Prior to learning calculus, you studied functions of various types, and you learned how to sketch their graphs with and without the support of a calculator. In my opinion, one of the more difficult topics in Extension 1 and 2 has got to be curve sketching. From the derivative's graph, identify the interval graph where f (the original function) is concave up. Polar curves: wrapping a function around the pole. Likes scottdave. Graphs reveal the behaviour of functions and are used for many purposes in mathematics, science and engineering. To zoom, use the zoom slider. This is a graph of the derivative of function h(x). Similarly, we set x = 0 to find the y- intercept. But some of the steps are closely related. (If f ˜˜x˚ ˚ f ˜x˚, the graph is symmetric with respect to the y-axis; if f ˜˜x˚ ˚˜f ˜x˚, the graph is symmetric with respect to the origin). C2 Sketching Trigonometric graphs (trigonometric graph shapes) Sketching graphs: the reciprocal graph - C1 Edexcel A Level Maths This video reminds you of a basic index law and then explains the shape of the graph of y=k/x. Displaying all worksheets related to - Polynomial Curve Sketching. Sample CHART for Sketching Curves. 2 First Derivative Test. The parabola is the envelope of the straight lines. Concavity/Inflection Points H. Sketch the curve using the information for the previous items: Sketch the asymptotes as dashed lines. These are general guidelines for all curves, so each step may not always apply to all functions. org helps support GraphSketch and gets you a neat, high-quality, mathematically-generated poster. These cannot be graded by WebAssign. 201-103-RE - Calculus 1 WORKSHEET: CURVE SKETCHING General Guidelines (1) domain of f(x) (2) intercepts (3) asymptotes (a) horizontal asymptotes lim. Polar curves: wrapping a function around the pole. Created by T. Solution: 1. You can access the circle tools from the Sketch tab of | {
"domain": "curaben.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9910145712474399,
"lm_q1q2_score": 0.8500606980030403,
"lm_q2_score": 0.8577681122619883,
"openwebmath_perplexity": 1251.2958620737718,
"openwebmath_score": 0.5482892990112305,
"tags": null,
"url": "http://vurg.curaben.it/curve-sketching.html"
} |
around the pole. Created by T. Solution: 1. You can access the circle tools from the Sketch tab of command manager. dvi Created Date. 707\) and then switch back to concave down at. Let's look at a couple of techniques for making our curve-drawing life a little easier. ) - Domain - Symmetry (Plug in "-x" for every "x" in the equation, and if the equation doesn't change, then it's an even. Never runs out of questions. (b) Critical Numbers — numbers a in the domain of f where f′(a) is 0 or undefined. We'll cover two types of curves. how to sketch a curve that has asymptotes. 5 Curve Sketching. 2 First Derivative Test. Andrew Cuomo Saturday said he was signing an executive order allowing the state's roughly 5,000 independent pharmacies to. Add HW points so we can figure out your Unit 3 HW grade. So now, happily in this subject, there are more pictures and it's a little bit more geometric. Let's take a look at an example to see one way of sketching a parametric curve. a) 2f(x) = x3 + x - x + 4 5 b) g(x) = 5x - 3x3 + 3 2. In this video I discuss the following topics to help produce the graph of a function: domain, x-y intercepts, symmetry of the function, intervals of. Title: math142weekinreview6. Powered by Create your own unique website with customizable templates. Some things that might keep your lines from being more fluid loose: using a guide to draw the lines (I would not recommend using a French curve for this reason), drawing the line slowly to maintain precision, pressing too hard (usually goes along with previous). 3 Higher Degree Polynomials and Curve Sketching Name_____ Period____ ©E q2O0e1V7_ jKruStYaB wSuoxfZtGwma^rFe] CLvLeCW. 4 Curve Sketching V63. Curve Sketching. Give x- and y-intercepts. Check your answers with 1t calculator. November 18, 2013. Before we move onto using concavity as a part of curve sketching, we note that using a function’s concavity can be a helpful tool for classifying its extrema. First Derivative Test Find where dy/dx (the | {
"domain": "curaben.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9910145712474399,
"lm_q1q2_score": 0.8500606980030403,
"lm_q2_score": 0.8577681122619883,
"openwebmath_perplexity": 1251.2958620737718,
"openwebmath_score": 0.5482892990112305,
"tags": null,
"url": "http://vurg.curaben.it/curve-sketching.html"
} |
can be a helpful tool for classifying its extrema. First Derivative Test Find where dy/dx (the deriviative, which is the slope) is zero or undefined; find the critcal numbers for the function. We use a multi-stroke pentimenti style curve sketching approach with an ink dry-ing visualization that allows users to sketch uninterrupted. Let's suppose the function you need to sketch is x^3-8/(x^2-4). 6 A Summary of Curve Sketching 209 Section 3. 147 seventh pages Chapter 3 Curve Sketching How much metal would be required to make a 400-mL soup can? What is the least amount of cardboard needed to build a box that holds 3000. Buying a poster from posters. There are a number of mathematical curves that produced heart shapes, some of which are illustrated above. If you're interested, take a look. Analysis of graphs (or curve sketching) includes finding: Domain and range. pdf: File Size: 12 kb: File Type: pdf: Download File. Plot the intercepts, maximum and minimum points, and in ection points. Okay, so in the last blog, we went over tips for curve-sketching. Brief Notes for STEP Section 06 – Curve Sketching Curve-sketching is a challenging exercise, one with much variety, and one which represents a good opportunity to display analytical skill. Maximum-minimum by LearnOnline Through OCW. At this point the graph starts to decrease and will continue to decrease until we hit $$x = 1$$. Well, the free Urban Sketching 101 guide covers everything there is to know, including: what it is, where to go and starter techniques and tips for the urban sketcher on the go. We will review the main topics that you'll need to know for the AP Calculus exams. Because it is a curve in 2d, it is usually easier to sketch than the graph of f. That is, this is when f intercepts the x-axis. With a model open, click Edit > Project. y = x sin(1/x) (In particular, what does x sin(1/x) tend to as x tends to 0? The answer is not 1, as a cursory application of might lead you to believe). Multiple-version | {
"domain": "curaben.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9910145712474399,
"lm_q1q2_score": 0.8500606980030403,
"lm_q2_score": 0.8577681122619883,
"openwebmath_perplexity": 1251.2958620737718,
"openwebmath_score": 0.5482892990112305,
"tags": null,
"url": "http://vurg.curaben.it/curve-sketching.html"
} |
to 0? The answer is not 1, as a cursory application of might lead you to believe). Multiple-version printing. These are potential local extrema. Give the domain. 1 Increasing and Decreasing Functions. quadratic: 6: PDF: Practice-Curve Sketching 2 open ended. Algebra and pre-calculus. Connecting a function, its first derivative, and its second derivative. Curve Sketching. These are are the sampe problems that we did in class. Determine y-intercept and x-intercepts, if possible. Design Sketching Learning Curves (2011, 177 pages, Klara Sjölén and Allan Macdonald) is a brand new sketch book, aimed at teaching how to really learn to sketch. Curve sketching, the methods for drawing approximately a curve defined by an equation; Sketch (mathematics), a generalization of algebraic theory A summary of a mathematical proof; Software and computing. How to draw curves, pfft! Not so fast! Curves are in a great deal of things you'll want to draw. Titration curves for weak acid v strong base. On-screen applet instructions: For the function shown, the applet identifies the relationship between the derivative (positive, negative, or zero) and the function (increasing, decreasing, max or min) that can aid in sketching a graph of the function. We now look at an example of sketching curves with asymptotes, i. Curve Sketching: Level 4 Challenges on Brilliant, the largest community of math and science problem solvers. So the next topic is curve sketching. MAXIMUM, MINIMUM, AND INFLECTION POINTS: CURVE SKETCHING - Applications of Differential Calculus - Calculus AB and Calculus BC - is intended for students who are preparing to take either of the two Advanced Placement Examinations in Mathematics offered by the College Entrance Examination Board, and for their teachers - covers the topics listed there for both Calculus AB and Calculus BC. The Reference panel opens. This is the graph of the second derivative of a function. Never runs out of questions. Sketching Solution Curves for | {
"domain": "curaben.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9910145712474399,
"lm_q1q2_score": 0.8500606980030403,
"lm_q2_score": 0.8577681122619883,
"openwebmath_perplexity": 1251.2958620737718,
"openwebmath_score": 0.5482892990112305,
"tags": null,
"url": "http://vurg.curaben.it/curve-sketching.html"
} |
of the second derivative of a function. Never runs out of questions. Sketching Solution Curves for Autonomous DEs By finding and classifying critical points for an Autonomous DE we can greatly simplify the process of sketching a solution curve. This handout contains three curve sketching problems worked out completely. edu Abstract Space curve sketching using 2D user interface is a chal-lenging task and forms the foundation for almost all sketch. Sketch the Curve!. A function can have two, one, or no asymptotes. If it appears that the curve levels off, then just locate the y-coordinate to which the curve seems to be. Likes scottdave. edu is a platform for academics to share research papers. 2: CURVE SKETCHING POLYNOMIALS Example 3. Determine the x- and y- intercepts of the function, if possible. Concavity and inflection points Critical points (maxima, minima, inflection) Video transcript. Look for any. We can obtain a good picture of the graph using certain crucial information provided by derivatives of the function and certain. Video Lesson Part 2. It cannot have "no solution" since a cubic curve has to cross the x-axis at least once. Heart Curve. x 3 - 3 x 2 - 9 x + 5; The sign of the derivative can be used to determine where a function is monotonic, i. Put the critical numbers in a sign chart to see where the first derivative is positive or negative (plug in the first derivative to get signs). Plot the intercepts, maximum and minimum points, and in ection points. These are general guidelines for all curves, so each step may not always apply to all functions. My claim is that we can write $\cos x + \sin x$ as $\sqrt{2}\cos\left(x-\frac{\pi}{4}\right)$. Technical Sketching and Drawing. asymptotes: Polynomial functions do not have asymptotes: a) vertical: No vertical asymptotes because f(x) continuous for all x. Math 170 Curve Sketching I Notes All homework problems will require that you create both a sign chart and a graph. Now if g(c) 6= 0, then x= cis a Vertical | {
"domain": "curaben.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9910145712474399,
"lm_q1q2_score": 0.8500606980030403,
"lm_q2_score": 0.8577681122619883,
"openwebmath_perplexity": 1251.2958620737718,
"openwebmath_score": 0.5482892990112305,
"tags": null,
"url": "http://vurg.curaben.it/curve-sketching.html"
} |
will require that you create both a sign chart and a graph. Now if g(c) 6= 0, then x= cis a Vertical Asymptote to the curve y= f(x). 1 Crit #s and Abs Extrema 3. com ©3 r2I0 E1K3 A YKTurt fa V 9S eo Rfbt NwraWrie A PLyL 5C Q. These are are the sampe problems that we did in class. This is the definitive app for calculus!Simply insert your function into The Calculus. Int: V I ) Max: Min:. geometry, curve sketching (or curve tracing) includes techniques that can be used to produce a rough idea of overall shape of a plane curve given its equation without computing the large numbers of points required for a detailed plot. 2) Curve Sketching Color by Number - In this activity, students practice finding the characteristics of curves. When f''(x) = 0, there is an inflection point changing the concavity of f(x). And there's relatively little computation. Math 170 Curve Sketching II Notes This homework is, once again, mostly about sign charts and graphing. 5 Man vs machine. 5 - Summary of Curve Sketching Math& 151 Warnock - Class Notes Here are the Guidelines for Sketching a Curve. We will focus on polynomials, but the same methods apply to roughly sketching the graph of any function. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. This section deals with recognizing when a curve is symmetric by performing a simple. [Grade 12 Differential Calculus: Curve Sketching] I have ALOT of little questions/confusion about graphing. Mark any asymptotes (if the limit of f(x) (as x approaches positive or negative infinity equals a y-value, then the y-value is a horizontal asymptote). 4 Curve Sketching V63. 5 How to Find Inflection Points 2 3 2 12 inflection point f(x) =. Comet Swan continues to brighten - posted in Sketching: This mornings observation was hampered by windy weather. c O + + +. 8 – A free PowerPoint PPT presentation (displayed as a Flash slide show) on PowerShow. ; Standard Deviants (Performing group); Goldhil Video | {
"domain": "curaben.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9910145712474399,
"lm_q1q2_score": 0.8500606980030403,
"lm_q2_score": 0.8577681122619883,
"openwebmath_perplexity": 1251.2958620737718,
"openwebmath_score": 0.5482892990112305,
"tags": null,
"url": "http://vurg.curaben.it/curve-sketching.html"
} |
as a Flash slide show) on PowerShow. ; Standard Deviants (Performing group); Goldhil Video (Firm);] -- How does art figure into calculus? This program illustrates applications of the derivative through graphing. My claim is that we can write $\cos x + \sin x$ as $\sqrt{2}\cos\left(x-\frac{\pi}{4}\right)$. In Curve Sketching 2, we have learned the different properties of quadratic functions that can help in sketching its graphs. 1, Relative Maxima and Minima: Curve Sketching 1 Increasing and Decreasing Functions We say that a function f (x) is increasing on an interval if the values of f increase as x increases (i. If you're interested, take a look. MCV4U CURVE SKETCHING QUIZ Name: Give all answers as exact numbers (fractions, terminating decimals, etc. y ≠ 0 for any values of x, so the graph does not cut the x axis. Convert Entities: Creates one or more entities in a 3D sketch by projecting an edge, loop, face, external curve, external sketch contour, set of edges, or set of external curves onto the sketch plane. Curve Sketching. | {
"domain": "curaben.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9910145712474399,
"lm_q1q2_score": 0.8500606980030403,
"lm_q2_score": 0.8577681122619883,
"openwebmath_perplexity": 1251.2958620737718,
"openwebmath_score": 0.5482892990112305,
"tags": null,
"url": "http://vurg.curaben.it/curve-sketching.html"
} |
# Generate points in circle uniformly
## Introduction
This chapter’s problem concerns uniformly generating a (potentially large) number of random points in a circle of a certain radius. Despite its simplicity the problem poses some unexpected challenges. We will discuss the best approach to this problem as well as one solution that many candidates provide which, whilst initially appearing correct actually fails in one crucial aspect (spoiler: it does not distribute points uniformly).
## Problem statement
Write a function that, given a circle of radius $$r$$ and centered at $$(x,y)$$ where $$r,x,y \in \mathcal{R}$$ returns a uniformly distributed point in the circle.
## Clarification Questions
What exactly does it mean for the point to be uniformly distributed?
It means that every point of the circle has the same probability of being picked/generated by the function.
## Discussion
Before discussing solutions it is worth mentioning that the fact that the circle is centered at $$(x,y)$$ makes very little difference and we can continue our discussion as if it were centered at $$(0,0)$$. This is the case because all the points we generate can then be translated to $$(x,y)$$ by simply adding $$x$$ and $$y$$ to the $$x$$-coordinate and $$y$$-coordinate of the generated point.
### Polar Coordinates - The wrong approach
Let’s start by discussing an intuituve, but ultimately incorrect, approach. One might think that in order to pick a point in the circle it is sufficient to
1. Pick a random angle $$\theta \in [0, 2\pi[$$
2. Pick a random radius $$\overline{r} \in [0,r]$$
3. Generate the Cartesian coordinates of the point given the radius and the angle (polar coordinates [@cit:wiki:polarcoordinates]) as (see Figure 22.1): $\begin{gathered} x=\overline{r}\sin(\theta) \\ y=\overline{r}\cos(\theta) \end{gathered}$
[fig:random_points_in_cirle:polar_coordinates] | {
"domain": "davidespataro.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846678676151,
"lm_q1q2_score": 0.850046803550406,
"lm_q2_score": 0.8688267830311355,
"openwebmath_perplexity": 434.14399494217605,
"openwebmath_score": 0.750119686126709,
"tags": null,
"url": "https://davidespataro.it/codinginterviewessentials/18.Generate_points_in_circle_uniformly.html"
} |
[fig:random_points_in_cirle:polar_coordinates]
Unfortunately, despite its appealing simplicity, this approach is wrong as it fails to produce points that are distributed uniformly in the circle. Before examining the mathematical proof it is instructive to have a look at Figure 22.2 which is drawing a large number of points on the circle generated using this incorrect solution. As you can see, the points are not generated uniformly as their density is higher towards the center. The bottom line is, do not use this solution in an interview. A possible matlab implementation of this buggy approach is shown at [list:random_points_in_circle:buggy]
Listing 1: Non-uniform random point in a circle generation using Matlab
function [px, py] = buggy_random_point(radius, x,y)
theta = rand()*2*pi;
px = r * sin(theta);
py = r *cos(theta);
endfunction
[fig:random_points_in_cirle:buggy]
### Loop approach
A good way to ensure that the point density is uniform across on the surface of the circle is to pick a point randomly in an enclosing square and make sure that we discard all the points that lie outside the circle. In other words, we keep asking for a random point $$(p_x=\text{rand()}, p_y=\text{rand()})$$ in the enclosing square until the following is true: $$p_x^2 + p_y^2 \leq r$$. In this way we are guaranteed to generate uniformly distributed points because we pick those points from a set of points that are already uniformly distributed in a square, and we exclude those which are not inside the circle. This method is also known as the exclusion method. Figure 22.3 depicts a large number of points generated with this method. | {
"domain": "davidespataro.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846678676151,
"lm_q1q2_score": 0.850046803550406,
"lm_q2_score": 0.8688267830311355,
"openwebmath_perplexity": 434.14399494217605,
"openwebmath_score": 0.750119686126709,
"tags": null,
"url": "https://davidespataro.it/codinginterviewessentials/18.Generate_points_in_circle_uniformly.html"
} |
The downside here is that we might need to generate a number of points in the square before getting lucky and picking one lying in the circle. We need to make on average $$\approx 1.2732$$ tries before getting a point in the circle. This number is the ratio between the are of enclosing square and the area of the enclosed circle i.e. $$\frac{(2r)^2}{\pi r^2} = \frac{4}{\pi}$$.
[fig:random_points_in_cirle:loop]
A Matlab implementation of this approach is shown in Listing [list:random_points_in_circle:loop].
Listing 2: Random point in a circle generation using the exclusion method.
function [px, py, t] = random_point_loop(radius, x,y)
px = 100;
py = 100;
t=0;
while (px*px + py*py > 1)
signx = 2*randi([0,1])-1;
signy = 2*randi([0,1])-1;
t=t+1;
endwhile
endfunction
### Polar Coordinates - The right approach
In order for the points to be distributed uniformly it is necessary that the average distance between the points be the same regardless of how far they lie from the center of the circle. This means that, looking at the points generated on a circumference of radius $$2$$, there have to be twice as many points as the the number of points on a circumference of radius $$1$$. A circumference that is twice as long translates to needing twice as many points to maintain the same density. Another intuitive way to understand why simply picking a random angle and a random radius is not enough would be to think about having to distribute $$10$$ points at random on a circle of radius $$1$$ and $$2$$. It is clear that the circumference of radius $$2$$ would look emptier than the one with radius $$1$$ simply because there is more circumference to be filled but a constant amount of points. | {
"domain": "davidespataro.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846678676151,
"lm_q1q2_score": 0.850046803550406,
"lm_q2_score": 0.8688267830311355,
"openwebmath_perplexity": 434.14399494217605,
"openwebmath_score": 0.750119686126709,
"tags": null,
"url": "https://davidespataro.it/codinginterviewessentials/18.Generate_points_in_circle_uniformly.html"
} |
The fundamental problem with the appraoch described in Section 22.3.1 is that the area of the circle is not uniformly covered. The random radius cuts through the area of the circle and this is the only parameter that affects how the points are going to be distributed across the full area of the circle. Therefore we should focus our attention how we can pick a better radius by making sure that larger radii are picked more often to accommodate for the larger area they define. In other words, we need to ensure that our random function for picking the radius takes the area of our circle into account.
Consider the area $$A$$ of a circle of radius $$r$$ i.e. $$A = \pi r^2$$. We can rearrange the formula so that $$r = \sqrt{\frac{A}{\pi}}$$. What this formula is really telling us is that the radius is proportional to $$\sqrt{A}$$. Now we have a way of choosing the radius that depends on the area of the circle. We can simply pick an area at random and then calculate the radius accordingly. This will make sure that the radius is picked taking into consideration the area of the circle. Figure 22.4 shows many points generated using this method. As you can see the points are generated uniformly across the area of the circle and the picture looks similar to Figure 22.3.
A C++ implementation of this method is shown in Listing [list:random_points_in_circle:sqrtcpp]. Details on the random number generation in Modern C++ can be found in [@cit::std::random].
Listing 3: C++ implementation of the function for generating a random point in a circle described in Section \ref{random_points_in_circle:sec:polar_sqrt}
auto generate_random_point_in_circle()
{
static std::uniform_real_distribution<double> dist_angle(0, 2 * M_PI);
const auto theta = dist_angle(rnd);
const auto x = r * cos(theta);
const auto y = r * sin(theta);
return std::make_pair{x + x_center, y + y_center};
}
[fig:random_points_in_cirle:polar_sqrt] | {
"domain": "davidespataro.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846678676151,
"lm_q1q2_score": 0.850046803550406,
"lm_q2_score": 0.8688267830311355,
"openwebmath_perplexity": 434.14399494217605,
"openwebmath_score": 0.750119686126709,
"tags": null,
"url": "https://davidespataro.it/codinginterviewessentials/18.Generate_points_in_circle_uniformly.html"
} |
[fig:random_points_in_cirle:polar_sqrt]
A Matlab implementation of this approach is shown in Listing [list:random_points_in_circle:sqrt].
Listing 4: Random point in a circle generation using polar coordinates and the $\approx \sqrt{A}$ dependency of the radius on the area of the circle.
function [px, py] = random_sqrt_area(radius, x,y)
r = sqrt(area/pi);
theta = rand()*2*pi;
px = r * sin(theta);
py = r *cos(theta);
endfunction
### Conclusion
For both the viable methods for generating random points withing a circle which we have discussed the time and space complexity is constant although the one presented in Section 22.3.3 will probably have better performance when included in a hot path i.e. in a loop for the generation of many random points.
All the code used to generate the Figures in this chapter is shown in Listing [list:random_points_in_circle:drivercode].
Listing 5: Matlab driver code for the generation of all figures in Chapter \ref{ch:random_points_in_circle}
function draw_points(n)
clf(1);
% n is the total number of points
px = zeros(1,n);
py = zeros(1,n);
tries = 0;
for i =0:n
%[x,y] = buggy_random_point(1,0,0);
% [x,y,t] = random_point_loop(1,0,0);
[x,y] = random_sqrt_area(1,0,0);
% tries = tries + t;
px(i+1) = x;
py(i+1) = y;
endfor
average = tries/n
% Plot a circle.
angles = linspace(0, 2*pi, n);
xCenter = 0;
yCenter = 0;
cx = radius * cos(angles) + xCenter;
cy = radius * sin(angles) + yCenter;
% Plot circle.
plot(cx, cy, 'b-', 'LineWidth', 2);
% Plot center.
hold on;
plot(xCenter, yCenter, 'k+', 'LineWidth', 2, 'MarkerSize', 16);
grid on;
axis equal;
xlabel('X', 'FontSize', 16);
ylabel('Y', 'FontSize', 16);
% Plot random points.
plot(px, py, 'r.', 'MarkerSize', 1);
rectangle('Position',[-1 -1 2 2], 'LineWidth',3, 'EdgeColor' , [0 .5 .5])
endfunction | {
"domain": "davidespataro.it",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846678676151,
"lm_q1q2_score": 0.850046803550406,
"lm_q2_score": 0.8688267830311355,
"openwebmath_perplexity": 434.14399494217605,
"openwebmath_score": 0.750119686126709,
"tags": null,
"url": "https://davidespataro.it/codinginterviewessentials/18.Generate_points_in_circle_uniformly.html"
} |
# Rewrite so that the denominator does not have any root expressions: $\frac{\sqrt[3]{49} +\sqrt[3]{7x} + \sqrt[3]{x^2}}{\sqrt[3]{x} -\sqrt[3]{7}}$
I am struggling with rewriting the following so that the denominator does not have any root expressions:
$$\frac{\sqrt[3]{49} +\sqrt[3]{7x} + \sqrt[3]{x^2}}{\sqrt[3]{x} -\sqrt[3]{7}}$$
I guess I should start with the denominator and try to get rid of the cube root expressions. But I cannot really get how one would do that easily. Is there another way to solve this problem?
Thank you kindly for your help!
-
Remember $x^3-y^3=(x-y)(x^2+xy+y^2)$. – Maesumi Feb 24 '13 at 15:47
@Maesumi Post it as an answer? – Git Gud Feb 24 '13 at 15:50
Hint:$a^3-b^3=(a-b)(a^2+ab+b^2)$
Let $a=x^{1/3},b=7^{1/3}$
$x-7=a^3-b^3=(a-b)(a^2+ab+b^2)=(x^{1/3}-7^{1/3})(7^{2/3}+(7x)^{1/3}+x^{2/3})$
$\displaystyle \frac{(7^{2/3}+(7x)^{1/3}+x^{2/3})}{(x^{1/3}-7^{1/3})}=\frac{(7^{2/3}+(7x)^{1/3}+x^{2/3})}{(x^{1/3}-7^{1/3})}\frac{(7^{2/3}+(7x)^{1/3}+x^{2/3})}{(7^{2/3}+(7x)^{1/3}+x^{2/3})}=\frac{(7^{2/3}+(7x)^{1/3}+x^{2/3})^2}{(x-7)}$
-
Thank you for your answer! I would very much appreciate if you would write the full answer. I want to study in what order you would do the arithmetic. – Lukas Arvidsson Feb 24 '13 at 15:58
Now is it okay?????? @LukasArvidsson – Abhra Abir Kundu Feb 24 '13 at 16:04
Thank you very much! – Lukas Arvidsson Feb 24 '13 at 16:05
You are welcome @LukasArvidsson – Abhra Abir Kundu Feb 24 '13 at 16:06
Since $u^3-v^3=(u-v)(u^2+uv+v^2)$, then $$\frac{v^2+uv+u^2}{u-v}=\frac{u^2+uv+v^2}{u-v}=\frac{\left(u^2+uv+v^2\right)^2}{u^3-v^3}.$$ What are $u$ and $v$ in your particular case?
- | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846628255123,
"lm_q1q2_score": 0.8500467908620429,
"lm_q2_score": 0.8688267745399465,
"openwebmath_perplexity": 546.4246400125473,
"openwebmath_score": 0.895392656326294,
"tags": null,
"url": "http://math.stackexchange.com/questions/312981/rewrite-so-that-the-denominator-does-not-have-any-root-expressions-frac-sqrt"
} |
# Is a prime number still a prime when in a different base?
Is a prime number in the decimal system still a prime when converted to a different base?
-
What do you mean exactly? 4 is divisible by 2 independently of base. – Il-Bhima Sep 4 '10 at 9:11
Think of representing a number as a pile of rocks. If a number n can be factored as n = a*b, then we can arrange the pile of rocks into an a by b rectangle. If a number n is prime then it can only take the form of a trivial a 1 by n or n by 1 rectangle. Notice we haven't made any mention of a base. – yjj Sep 4 '10 at 9:11
Developing on yjj's answer: n being a prime number is a property of the number in terms of arithmetic operations (e.g. multiplications). A decimal representation is just a way of representing the number: the representation doesn't affect any of its "arithmetical" properties. As the Bard said, "a rose by any other name would smell as sweet." – Niel de Beaudrap Sep 4 '10 at 10:17
On a semi-related note, Mersenne primes have the trivial but fun property that if $2^p -1$ is a Mersenne prime, then it can be represented as $p$ $1$'s in base $2$. – Joshua Shane Liberman Sep 4 '10 at 13:53
Here is the same question which has been asked at Mathforum: Here is the link
http://mathforum.org/library/drmath/view/55880.html
-
It would have been helpful if you quoted the answer here in addition to the link. – Sniper Clown May 14 '12 at 4:45 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846681827465,
"lm_q1q2_score": 0.8500467888704321,
"lm_q2_score": 0.8688267677469952,
"openwebmath_perplexity": 527.6393870965749,
"openwebmath_score": 0.8890319466590881,
"tags": null,
"url": "http://math.stackexchange.com/questions/3999/is-a-prime-number-still-a-prime-when-in-a-different-base/198752"
} |
Alas, the accepted answer is misleading (and arguably incorrect). The reason that primality (or any other purely arithmetic property) is preserved in radix representation is simply that such representation faithfully preserves all of the arithmetic operations on integers. More precisely, $\:$ if $\rm\;n\to r(n)\;$ is the map from $\rm\:n\:$ to its radix $\rm\:d\;$ representation, then it preserves addition $\rm\;r(m+n) = r(m) + r(n),\;$ and multiplication $\rm\;r(mn) = r(m)\ r(n),\;$ and $\rm\;r\;$ has an inverse $\rm\;s\;$ that similarly also preserves addition and multiplication (technically: $\rm\:r\:$ is a ring isomorphism). This readily implies that the relation of divisibility is faithfully preserved in radix representation, because the relation of divisibility can be expressed as an equation involving only arithmetic (ring) operations (namely multiplication), and such equations are necessarily preserved by the maps $\rm\;r\;$ and $\rm\;s\;$ - indeed these maps are defined precisely so to preserve these fundamental operations.
It's instructive to examine more closely the preservation of divisibility. First, we recall the standard notation $\rm\;a|b\; := \: a\;$ divides $\rm\:b\:,\;\;$ i.e. $\rm\:\exists \:n\in\mathbb Z : \ an = b\:,\;$ i.e. there exists an integer $\rm\:n\:$ such that $\rm\;an = b\;$.
LEMMA $\rm\;\quad a|b \iff r(a)|r(b)\quad\quad$ (Divisibility Preservation by Isomorphisms)
Proof: $\rm\;(\;\Rightarrow\;)\quad a|b \;\Rightarrow\; \exists\:n\in\mathbb Z: \: an = b \;\Rightarrow\; r(a)\ r(n) = r(an) = r(b) \;\Rightarrow\; r(a)|r(b)$
$\rm\;(\Leftarrow)\quad r(a)|r(b) \;\Rightarrow\;\exists\:c\in r(\mathbb Z): \: r(a)\: c = r(b)\;\Rightarrow\; r(a)\ r(n) = r(b) \;\Rightarrow\; a n = b\;$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846681827465,
"lm_q1q2_score": 0.8500467888704321,
"lm_q2_score": 0.8688267677469952,
"openwebmath_perplexity": 527.6393870965749,
"openwebmath_score": 0.8890319466590881,
"tags": null,
"url": "http://math.stackexchange.com/questions/3999/is-a-prime-number-still-a-prime-when-in-a-different-base/198752"
} |
The final $\;\Rightarrow\;$ above is by applying $\rm\;r\:$'s inverse $\rm\:s\:$ so to cancel the $\rm\;r\:$'s using $\rm\;\: sr = 1 =\;$ identity map.
Note: this employs $\rm\:s\:$'s preservation of multiplication, $\:$ viz. $\rm\:s(r(a)\: r(n)) \:=\: sr(a)\: sr(n) \:=\: a\: n\;$
As a corollary, we conclude that primes (i.e. irreducibles) are also preserved, since they are definable purely in terms of divisibilty, viz. $\rm\;p\;$ is prime $\;\: := \:\rm\;p = ab \;\Rightarrow\; p|a\;$ or $\rm\;p|b\;$ and $\rm\;p\;$ is not a unit, i.e. not $\rm\;p|1\;$.
So your question reduces to the more fundamental why is radix representation a ring isomorphism, i.e. why really does radix representation preserve the addition and multiplication operations? This is a very good question that deserves a thoughtful answer. It's a serious pedagogical oversight that this topic is rarely discussed in algebra textbooks. Although most students understand this fact subconsciously, many have difficulty providing a rigorous proof (or, worse, they overlook the fact that it does require a rigorous proof). Your question would attract much more interest and receive much more interesting replies if you rephrase it in this manner. Thus I propose the following:
NOTE to much more experienced readers: this problem is not as trivial as you might think at first glance (and certainly less trivial for novices). For example, the analogous problem for real numbers (or p-adics) is the subject of a famous paper [1]. For an introduction see e.g. here. Its closing line is quite apropos:
This is very much in keeping with Rota’s thinking that mathematics is not just a quest to solve problems, it is also a quest to understand the mathematical universe as clearly and as deeply as possible
[1] F. Faltin, N. Metropolis, B. Ross, G.-C. Rota,
The real numbers as a wreath product.
Advances in Math. 16 (1975), 278-304 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846681827465,
"lm_q1q2_score": 0.8500467888704321,
"lm_q2_score": 0.8688267677469952,
"openwebmath_perplexity": 527.6393870965749,
"openwebmath_score": 0.8890319466590881,
"tags": null,
"url": "http://math.stackexchange.com/questions/3999/is-a-prime-number-still-a-prime-when-in-a-different-base/198752"
} |
-
Bill, the accepted answer is just a link. It links to an answer stating in part that: "The fact of being prime or composite is just a property of the number itself, regardless of the way you write it." This answers the OP's question succinctly and accurately. – Robin Chapman Sep 4 '10 at 18:23
Fine, Bill, let us agree to disagree. I will continue to interpret the OP's question as he/she wrote it, while you can interpret it as a question about your calculator ("calculator" being a word appearing neither in the original question, nor in your reply). – Robin Chapman Sep 4 '10 at 18:43
@Robin: I interpret the question less trivially, e.g. "If I perform a primality test using radix R algorithms and it tells me that N is prime, how do I know that the result doesn't depend upon the radix?" It's not merely an issue of syntax ("how you write it" in the linked answer) but also of semantics, i.e. the meaning of the notations and the correctness of the radix algorithms. I think that the linked answer completely misses the essence of the matter. – Bill Dubuque Sep 4 '10 at 18:51
@Robin: Perhaps our different interpretations reflect our different backgrounds. I come from a strong constructive background, having done much work in computational algebra and number theory. So I have encountered many similar such student questions that do have the non-trivial interpretation that I gave above. I think you may have a less constructive background, so perhaps to you that might not be the most natural initial interpretation of the question. – Bill Dubuque Sep 4 '10 at 19:33
+1 for not just being a link. – grieve Jan 30 '11 at 0:21
Is a prime number in the decimal system still a prime when converted to a different base?
The base is a numbers symbology (display representation).
A prime number is a prime by defination, irrespective of base. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846681827465,
"lm_q1q2_score": 0.8500467888704321,
"lm_q2_score": 0.8688267677469952,
"openwebmath_perplexity": 527.6393870965749,
"openwebmath_score": 0.8890319466590881,
"tags": null,
"url": "http://math.stackexchange.com/questions/3999/is-a-prime-number-still-a-prime-when-in-a-different-base/198752"
} |
A prime number is a prime by defination, irrespective of base.
-
Sorry for such an abrupt answer. A logical answer; Next to the Ones place, is the Base place, each place after this is just a power of the Base. So the only number in any Base that can be Prime, is by definition, the Base itself. Of Interest would be a Non-linear Base system; Next to the Ones place, would be the first prime (2), all powers of 2 removed, would leave the next prime (3), and so on. Like a sieve program eliminating all powers of each Base. Each place becomes waited, by its own Base. All successive numbers are either powers of a place, or belong in a place of its own. – Optionparty Sep 29 '12 at 7:45
We should distiguish between numbers, on the one hand, and numerals , on the other, which are used to represent numbers. So, e.g., 13 = 15(octal) = D hexadecimal = XIII = treize, in French word(s) = τρισκαίδεκα. It is prime, no matter how you represent it. Some notations may be more convenient than others; and which, might depend on the user!
- | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9783846681827465,
"lm_q1q2_score": 0.8500467888704321,
"lm_q2_score": 0.8688267677469952,
"openwebmath_perplexity": 527.6393870965749,
"openwebmath_score": 0.8890319466590881,
"tags": null,
"url": "http://math.stackexchange.com/questions/3999/is-a-prime-number-still-a-prime-when-in-a-different-base/198752"
} |
# How do I write these in summation/product notation?
I have difficulties in writing some equations in summation/product notation.
I want to write this in summation notation. $$p_1p_2+p_1p_3+...+p_1p_n+p_2p_3+p_2p_4+...+p_2p_n+...+p_{n-1}p_n$$
is it $$\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}p_ip_j \quad ?$$
or
$$\sum_{i=1}^{n-1}\sum_{j=1}^{n}p_ip_{i+j} \quad ?$$
or both wrong?
I want to write this in product notation. $$(1-p_1p_2)(1-p_1p_3)...(1-p_1p_n)(1-p_2p_3)(1-p_2p_4)...(1-p_2p_n)...(1-p_{n-1}p_n)$$
is it
$$\prod_{i=1}^{n-1}\prod_{j=i+1}^{n}(1-p_ip_j) \quad ?$$
or
$$\prod_{i=1}^{n-1}\prod_{j=1}^{n}(1-p_ip_{i+j}) \quad ?$$
or both wrong?
• try with $\sum_{1\le i<j\le n}$ – Exodd Apr 2 '17 at 9:39
The first double sum $$\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}p_ip_j$$ is quite ok.
It is useful to recall the validity of \begin{align*} \sum_{i=1}^{n-1}\sum_{j=i+1}^{n}p_ip_j=\sum_{1\leq i<j\leq n}p_ip_j=\sum_{j=2}^n\sum_{i=1}^{j-1}p_ip_j \end{align*}
The second double sum $$\sum_{i=1}^{n-1}\sum_{j=1}^{n}p_ip_{i+j}$$ is not correct. When looking at the term with index $i=n-1$ and $j=n$ we obtain $$p_{n-1}p_{2n-1}$$ which is not part of $$p_1p_2+p_1p_3+...+p_1p_n+p_2p_3+p_2p_4+...+p_2p_n+...+p_{n-1}p_n$$
We have quite the same situation when looking at the products. Again, it is useful to recall the validity of \begin{align*} \prod_{i=1}^{n-1}\prod_{j=i+1}^{n}(1-p_ip_j)=\prod_{1\leq i<j\leq n}(1-p_ip_j)=\prod_{j=2}^n\prod_{i=1}^{j-1}(1-p_ip_j) \end{align*}
For the sums: your first expression is correct, while your second expression should take the limit of the $j$-sum to be $n-i$.
Likewise the products: the limit of your second expression's $j$-product should be $n-i$.
As Exodd says, it's clearer just to use $$\sum_{1 \leq i < j \leq n}$$ | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9744347875615795,
"lm_q1q2_score": 0.8500456368091288,
"lm_q2_score": 0.8723473829749844,
"openwebmath_perplexity": 716.8366889244468,
"openwebmath_score": 0.9517501592636108,
"tags": null,
"url": "https://math.stackexchange.com/questions/2214221/how-do-i-write-these-in-summation-product-notation"
} |
When is the image of a proper map closed?
A map is called proper if the pre-image of a compact set is again compact.
In the Differential Forms in Algebraic Topology by Bott and Tu, they remark that the image of a proper map $$f: \mathbb{R}^n \to \mathbb R^m$$ is closed, adding the comment "(why?)".
I can think of a simple proof in this case for continuous $$f$$:
If the image is not closed, there is a point $$p$$ that does not belong to it and a sequence $$p_n \in f(\mathbb R^n)$$ with $$p_n \to p$$. Since $$f$$ is proper $$f^{-1}(\overline {B_\delta(p)})$$ is compact for any $$\delta$$. Let $$x_n$$ be any point in $$f^{-1}(p_n)$$ and wlog $$x_n \in f^{-1}(\overline{B_\delta(p)})$$. Since in $$\mathbb{R}^n$$ compact and sequentially compact are equivalent, there exists a convergent subsequence $$x_{n_k}$$ of $$x_n$$. From continuity of $$f$$: $$f(x_{n_k}) \to f(x)$$ for some $$x$$. But $$f(x_{n_k})=p_{n_k} \to p$$ which is not supposed to be in the image and this gives a contradiction.
My problem is that this proof is too specific to $$\mathbb{R}^n$$ and uses arguments from basic analysis rather than general topology.
So the question is for what spaces does it hold that the image of a proper map is closed, how does the proof work, and is it necessary to pre-suppose continuity?
• Often map already implies continuity. I'd check the text for this. Jan 8, 2016 at 12:14
First of all the definition of a proper map assumes continuity by convention (I have not come across texts that say otherwise)
Secondly, here is a more general result -
Lemma : Let $$f:X\rightarrow Y$$ be a proper map between topological spaces $$X$$ and $$Y$$ and let $$Y$$ be locally compact and Hausdorff. Then $$f$$ is a closed map.
Proof : Let $$C$$ be a closed subset of $$X$$. We need to show that $$f(C)$$ is closed in $$Y$$ , or equivalently that $$Y\setminus f(C)$$ is open. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9744347875615794,
"lm_q1q2_score": 0.8500456351921954,
"lm_q2_score": 0.8723473813156294,
"openwebmath_perplexity": 84.26977895098601,
"openwebmath_score": 0.9617077708244324,
"tags": null,
"url": "https://math.stackexchange.com/questions/1604210/when-is-the-image-of-a-proper-map-closed"
} |
Let $$y\in Y\setminus f(C)$$. Then $$y$$ has an open neighbourhood $$V$$ with compact closure. Then $$f^{-1}(\bar{V})$$ is compact.
Let $$E=C\cap f^{-1}(\bar{V})$$ . Then clearly $$E$$ is compact and hence so is $$f(E)$$. Since $$Y$$ is Hausdorff $$f(E)$$ is closed.
Let $$U=V\setminus f(E)$$. Then $$U$$ is an open neighbourhood of $$y$$ and is disjoint from $$f(C)$$.
Thus $$Y\setminus f(C)$$ is open. $$\square$$
I hope this helps.
EDIT: To clarify the statement $$U$$ is disjoint from $$f(C)$$ -
Suppose $$z\in U\cap f(C)$$ Then there exists a $$c\in C$$ such that $$z=f(c)$$. This means $$c\in f^{-1}(U)\subseteq f^{-1}(V)\subseteq f^{-1}(\bar V)$$. So $$c\in C\cap f^{-1}(\bar V)=E$$. So $$z=f(c)\in f(E)$$ which is a contradiction as $$z\in U$$.
• Why is $U$ disjoint from $f(C)$? From your definition it is clear that $E \subseteq C$. So $f(E) \subseteq f(C)$. Hence $V \setminus f(C) \subseteq V \setminus f(E) = U$. If the containment is proper then $U$ may contain some element of $f(C)$. Who knows that? Isn't it so @R_D? Jan 25, 2019 at 10:26
• Is it fine now? @Dbchatto67
– R_D
Jan 26, 2019 at 3:34
• Yeah @R_D it's now absolutely fine. Thanks so much. Jan 26, 2019 at 6:04
• Why is $f(E)$ compact? Mar 18, 2020 at 3:26
• @XiuyiYang continuous image of a compact set is compact. $E$ is compact (being the closed subset of the compact set $f^{-1}(\bar V)$) and $f$ is continuous so $f(E)$ is compact.
– R_D
Mar 18, 2020 at 14:58
One may generalize the result in R_D's answer even further: | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9744347875615794,
"lm_q1q2_score": 0.8500456351921954,
"lm_q2_score": 0.8723473813156294,
"openwebmath_perplexity": 84.26977895098601,
"openwebmath_score": 0.9617077708244324,
"tags": null,
"url": "https://math.stackexchange.com/questions/1604210/when-is-the-image-of-a-proper-map-closed"
} |
One may generalize the result in R_D's answer even further:
A proper map $f:X\to Y$ to a compactly generated Hausdorff space is a closed map (A space $Y$ is called compactly generated if any subset $A$ of $Y$ is closed when $A\cap K$ is closed in $K$ for each compact $K\subseteq Y$).
Proof: Let $C\subseteq X$ be closed, and let $K$ be a compact subspace of $Y$. Then $f^{-1}(K)$ is compact, and so is $f^{-1}(K)\cap C =: B$. Then $f(B)=K\cap f(C)$ is compact, and as $Y$ is Hausdorff, $f(B)$ is closed. Since $Y$ is compactly generated, $f(C)$ is closed in $Y$.
A locally compact space $Y$ is compactly generated: If $A\subset Y$ intersects each compact set in a closed set, and if $y\notin A$, then $A$ intersects the compact neighborhood $K$ of $y$ in a closed set $C$. Now $K\setminus C$ is a neighborhood of $y$ disjoint from $A$, hence $A$ is closed. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9744347875615794,
"lm_q1q2_score": 0.8500456351921954,
"lm_q2_score": 0.8723473813156294,
"openwebmath_perplexity": 84.26977895098601,
"openwebmath_score": 0.9617077708244324,
"tags": null,
"url": "https://math.stackexchange.com/questions/1604210/when-is-the-image-of-a-proper-map-closed"
} |
A quadric quadrilateral is a convex quadrilateral whose four vertices all lie on the perimeter of a square. Example: the perimeter of this rectangle is 7+3+7+3 = 20. Perimeter of a rhombus formula. Area of quadrilateral formula can be divided into three categories based on given values. where all angels are the right angels. Question 5: Find out the height of a cylinder with a circular base of radius 70 cm and volume 154000 cubic cm. Quadrilateral circumscribing a circle (also called tangential quadrilateral) is a quadrangle whose sides are tangent to a circle inside it. Knowing how to find the perimeter or area of a shape can be useful in everyday ... A trapezium is a quadrilateral with one pair of parallel sides. Using the distance formula, it is possible to find the length of each side of the polygon, which then makes it possible to determine its perimeter. Square is a quadrilateral with four equal sides and angles. 3. Example: the perimeter of this regular pentagon is:. perimeter of a circle, we call it by the special name of circumference. A Watt quadrilateral is a quadrilateral with a pair of opposite sides of equal length. Rectangle formula – Area and perimeter of a rectangle. Perimeter of Quadrilateral Calculator. Perimeter Formula. 3+3+3+3+3 = 5×3 = 15 Because, the area of the quadrilateral is never negative. Mar 20, 2012 - Discuss Perimeter of Rectangle along with the formula and solved examples. Note : If you get the area of a quadrilateral as a negative value, take it as positive. Quadrilateral formula. There are two types of quadrilaterals — regular and irregular quadrilaterals. Heron's Formula depends on knowing the semiperimeter, or half the perimeter, of a triangle. Yes No. In Concave, the interior angles are greater than 180 degrees whereas, in Convex the interior angles are less than 180 degrees. An equilic quadrilateral has two opposite equal sides that when extended, meet at 60°. A Quadrilateral is a polygon with 4 side. Free Quadrilateral | {
"domain": "referencement-offshore.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9744347831070321,
"lm_q1q2_score": 0.8500456313062827,
"lm_q2_score": 0.8723473813156294,
"openwebmath_perplexity": 454.91438127777985,
"openwebmath_score": 0.690194845199585,
"tags": null,
"url": "http://referencement-offshore.com/russian-marriage-ikbes/8566de-quadrilateral-formula-perimeter"
} |
sides that when extended, meet at 60°. A Quadrilateral is a polygon with 4 side. Free Quadrilateral Perimeter Calculator - calculate the perimeter of a quadrilateral step by step This website uses cookies to ensure you get the best experience. By … Properties of a Rhombus. Perimeter is the distance around a two-dimensional shape. Thanks! Sometimes, it is also referred to as equilateral quadrilateral because of its characteristic of equivalency of length. Another solution to finding the rhombus perimeter requires the diagonal lengths: Rhombus Perimeter = 2 * √(e² + f²) Try deriving the formula yourself. Square. The formula for the perimeter, area and diagonal of the square Perimeter formula for … Properties. Find the value of 'x', substitute it in the linear expression and determine each side length of the quadrilateral. On the Ellipse page we looked at the definition and some of the simple properties of the ellipse, but here we look at how to more accurately calculate its perimeter.. Perimeter. (c) Length of side c.(d) Length of side d.Semi-perimeter (s): The calculator returns the semi-perimeter in meters. Perimeter of an Ellipse. Some examples of the quadrilaterals are square, rectangle, rhombus, trapezium, and parallelogram. Hence, the perimeter of a given square-shaped chocolate piece is $$\ 1+1+1+1=4\ in$$ The perimeter formula is, It is classified into two types : concave and convex. Algebra in Perimeter of Quadrilaterals | Level 2. Rhombus. Below, you can find three different formulas to calculate area of a quadrilateral. Example 1: Find the perimeter of the figure below 8 11 14 4 Solution: It is tempting to just start adding of the numbers given together, Opposite angles are in … The area of a quadrilateral inscribed in a circle is given by the Bret Schneider’s formula as: Area = √[s(s-a) (s-b) (s – c) (s – c)] where a, b, c and d are the side lengths of the quadrilateral. Join now. Join now. The Perimeter of an Irregular Quadrilateral The formula is a + b + | {
"domain": "referencement-offshore.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9744347831070321,
"lm_q1q2_score": 0.8500456313062827,
"lm_q2_score": 0.8723473813156294,
"openwebmath_perplexity": 454.91438127777985,
"openwebmath_score": 0.690194845199585,
"tags": null,
"url": "http://referencement-offshore.com/russian-marriage-ikbes/8566de-quadrilateral-formula-perimeter"
} |
Join now. Join now. The Perimeter of an Irregular Quadrilateral The formula is a + b + c+ d = perimeter. We now have the approximate length of side AH as 13.747 cm, so we can use Heron's Formula to calculate the area of the other section of our quadrilateral. Log in. A quadrilateral whose four sides are all congruent in length is a rhombus. What is Area of a Quadrilateral? Question. Therefore, the perimeter of a parallelogram formula is as follows: We know that the opposite sides of a parallelogram are parallel and equal to each other. s = Semi perimeter of the quadrilateral = 0.5(a + b + c + d) Let’s get an insight of the theorem by solving a … INSTRUCTIONS: Choose units and enter the following: (a) Length of side a. Since we don’t have straight sides to add up for the circumference (perimeter) of a circle, we have a formula for calculating this. Given a general quadrilateral, that has the lengths of its.It is interesting to 1. Not Helpful 31 Helpful 31. Form an equation using the perimeter and the side lengths of the quadrilaterals featured in this set of printable worksheets for 6th grade, 7th grade, and 8th grade students. ‹ Derivation / Proof of Ptolemy's Theorem for Cyclic Quadrilateral up Derivation of Formula for Radius of Circumcircle › Log in or register to post comments 12260 reads Use Heron's Formula. Thus, the formula for finding the perimeter of a parallelogram is given by: So, the perimeter of Parallelogram, P = a + b + a + b units. You use this formula for all trapezoids, including isosceles trapezoids. If given a polygon with different side lengths, then the general perimeter formula for it will look like the sum of the lengths of all sides: P = a+b+d+c OwlCalculator.com Geometry - Calculate Quadrilateral perimeter A Rectangle is a four sided-quadrilateral having all the internal angles to be right-angled Be sure the.The formula for the area of a triangle can be developed by making an exact. Calculates the area and perimeter of a quadrilateral given | {
"domain": "referencement-offshore.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9744347831070321,
"lm_q1q2_score": 0.8500456313062827,
"lm_q2_score": 0.8723473813156294,
"openwebmath_perplexity": 454.91438127777985,
"openwebmath_score": 0.690194845199585,
"tags": null,
"url": "http://referencement-offshore.com/russian-marriage-ikbes/8566de-quadrilateral-formula-perimeter"
} |
can be developed by making an exact. Calculates the area and perimeter of a quadrilateral given four sides and two opposite angles. All sides are of the same length. The perimeter of a square = 4L. Example: Find the perimeter of triangle EFG given the coordinates of its vertices E (-2, -2), F (1, 2), and G (4, -2). A demonstration of the formula $A = \frac{1}{2} d_1 d_2$ The formula for the area of a quadrilateral with perpendicular diagonals is . A quadrilateral is a polygon we obtain by joining four vertices, and it has four sides and four angles. The semiperimeter is used most often for triangles; the formula for the semiperimeter of a triangle with side lengths a, b, and c is = + +. When diagonals and angle between them are given. For our MAH, the three sides measure: MA = 7 cm; AH = 13.747 cm; HM = 14 cm The quadrilateral area formulas are as follows: Note: The median of a trapezoid is the segment that connects the midpoints of the legs.Its length equals the average of the lengths of the bases. Brahmaguptas Formula says that the semi-perimeter of a cyclic quadrilateral is s.DWITE Online Computer Programming Contest. The diagonals of quadrilateral are perpendicular to each other, and the lengths are 15 cm and 20 cm. That is, we always take the area of quadrilateral as positive. So, area of the given quadrilateral is 28 square units. Substitute the value of “a” in the formula, we get Area of a square = 10 2 A = 10 x 10 = 100 Therefore, the area of a square = 100 cm 2 The perimeter of a square = 4a units P = 4 x 10 =40 Therefore, the perimeter of a square = 40 cm. Perimeter. The Semi-perimeter of a Quadrilateral calculator computes the semi-perimeter of a quadrilateral based on the length of the four sides.. Log in. Without knowing any angles, you cannot find sides, perimeter or area. Without knowing any angles, you can not find sides, perimeter area. Equal sides and two opposite angles - 8552681 1 vertices, and parallelogram below with measuring as. Name of | {
"domain": "referencement-offshore.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9744347831070321,
"lm_q1q2_score": 0.8500456313062827,
"lm_q2_score": 0.8723473813156294,
"openwebmath_perplexity": 454.91438127777985,
"openwebmath_score": 0.690194845199585,
"tags": null,
"url": "http://referencement-offshore.com/russian-marriage-ikbes/8566de-quadrilateral-formula-perimeter"
} |
and two opposite angles - 8552681 1 vertices, and parallelogram below with measuring as. Name of circumference making an exact the linear expression and determine each side length of the quadrilateral of cylinder. Vertices, and the lengths are 15 cm and 20 cm of quadrilateral formula can be by... The given quadrilateral is 28 square units obtain by joining four vertices, parallelogram...: find out the height of a trapezoid - How to find the area of formula! Of quadrilateral - 8552681 1 given four sides and four angles referred to as equilateral quadrilateral because its. Of equal length categories based on the perimeter of an ellipse is very difficult to calculate! concave! Not find sides, perimeter or area mar 20, 2012 - Discuss perimeter of rectangle along the! All trapezoids, including isosceles trapezoids quadrilateral calculator computes the semi-perimeter of a quadrilateral four. Four sides of opposite sides of equal length + c+ d = perimeter cylinder with a of. Strangely, the area of a cyclic quadrilateral is s.DWITE Online Computer Programming Contest is a b. Categories based on given values each side length of the quadrilateral is 28 square units 2-dimensional figures is as. Lengths are 15 cm and 20 cm we obtain by joining four vertices, and the lengths 15... It in the linear expression and determine each side length of side a you get the area of the is! 1 in area of a cyclic quadrilateral is never negative opposite sides equal. Examples of the four sides are all congruent in length is a quadrangle whose sides are tangent to circle. The lengths are 15 cm and 20 cm different formulas to calculate area of the given quadrilateral is 28 units... The length of side a 20 cm is, we call it by the special name of circumference is... 2-Dimensional figures is measured as the sum of its characteristic of equivalency of length a cyclic is! Lengths are 15 cm and 20 cm square, rectangle, rhombus, trapezium, and parallelogram semi-perimeter a! Regular and Irregular | {
"domain": "referencement-offshore.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9744347831070321,
"lm_q1q2_score": 0.8500456313062827,
"lm_q2_score": 0.8723473813156294,
"openwebmath_perplexity": 454.91438127777985,
"openwebmath_score": 0.690194845199585,
"tags": null,
"url": "http://referencement-offshore.com/russian-marriage-ikbes/8566de-quadrilateral-formula-perimeter"
} |
cm square, rectangle, rhombus, trapezium, and parallelogram semi-perimeter a! Regular and Irregular quadrilaterals greater than 180 degrees whereas, in convex the interior angles are less than 180.. The value of ' x ', substitute it in the linear expression and determine each length! And volume 154000 cubic cm quadrilateral given four sides: concave and convex the! 7+3+7+3 = 20 s.DWITE Online Computer Programming Contest the diagonals of quadrilateral - 8552681 1 quadrilateral because of characteristic. With the formula and solved examples equivalency of length are all congruent in is. Knowing any angles, you agree to our Cookie Policy are tangent to a circle inside.! Height of a quadrilateral calculator computes the semi-perimeter of a quadrilateral whose vertices! Are greater than 180 degrees using this website, you can find three different formulas calculate. Given values and angles says that the semi-perimeter of a quadrilateral with four equal sides and two angles., area of the four sides a pair of opposite sides of equal length whose... Types: concave and convex If you get the area of a trapezoid - How to find area! Circle inside it, trapezium, and the lengths are 15 cm and 20.. Substitute it in the linear expression and determine each side length of the given quadrilateral a. And convex with the formula is a rhombus a quadrangle whose sides are tangent to circle. | {
"domain": "referencement-offshore.com",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9744347831070321,
"lm_q1q2_score": 0.8500456313062827,
"lm_q2_score": 0.8723473813156294,
"openwebmath_perplexity": 454.91438127777985,
"openwebmath_score": 0.690194845199585,
"tags": null,
"url": "http://referencement-offshore.com/russian-marriage-ikbes/8566de-quadrilateral-formula-perimeter"
} |
# Does $\sum_{n=1}^{\infty}(-1)^n\frac{n}{\sqrt{n^3 + 6}}$ converge?
Test the series for convergence or divergence. $$\sum_{n=1}^{\infty}(-1)^n\frac{n}{\sqrt{n^3 + 6}}$$
Because this is an alternating series, I decided to use the alternating series test. This theorem states that if $\lvert a_n \rvert = b_n$ satisfies the conditions:
$b_{n+1} ≤ b_n$ for all $n$
$\lim \limits_{n \to \infty} b_n = 0$
This expression clearly approaches $0$ because $n$ grows asymptotically slower than the denominator of the expression, $\sqrt{n^3 + 6}$, as $n$ approaches $\infty$.
But I think the expression fails to meet the first condition.
While $b_{n+1} ≤ b_n$ is true for all $n>1$, I don't think it is true for $n$. Consider the following values of $b_n$ with values substituted in:
When $n=1$: $$\frac{1}{\sqrt{1^3 + 6}} = \frac{2}{\sqrt{7}} \approx 0.377964473$$ When $n=2$: $$\frac{2}{\sqrt{2^3 + 6}} = \frac{2}{\sqrt{14}} \approx 0.534522483$$ When $n=3$: $$\frac{3}{\sqrt{3^3 + 6}} = \frac{2}{\sqrt{33}} \approx 0.5222329679$$
So, since $0.377964473 < 0.534522483 > 0.5222329679$ I don't think this series passes the alternating series test and is therefore divergent. But I got this question wrong. Apparently, this series converges.
What am I doing wrong? Thanks for your help! | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9744347860767305,
"lm_q1q2_score": 0.8500456241952917,
"lm_q2_score": 0.8723473713594991,
"openwebmath_perplexity": 136.38483873027855,
"openwebmath_score": 0.8766361474990845,
"tags": null,
"url": "https://math.stackexchange.com/questions/1235115/does-sum-n-1-infty-1n-fracn-sqrtn3-6-converge"
} |
What am I doing wrong? Thanks for your help!
• What happens with the first $N = 3$ or $300$ or $3,000,000$ terms doesn't matter. The sum of the first $N$ terms is always finite. The question is: does there exist an $N$ such that the tests hold for all $n > N$? (Yes!) – Simon S Apr 14 '15 at 23:48
• Not only does it converge, it's absolutely convergent (meaning that the absolute value series converges too). Your only problem is that your test values are too early in the sequence. – Brian Tung Apr 14 '15 at 23:48
• This series is not absolutely convergent. – Simon S Apr 14 '15 at 23:49
• Okay that makes sense. According to the theorem, I thought it had to hold true for all values of $n$. Really, the theorem should be: $b_{n+1}≤b_n$ for all $n>N$. – James Taylor Apr 14 '15 at 23:50
Remember exactly what the alternating series test says:
If [some conditions hold] then the series converges.
So, we can never use the alternating series test to conclude that a series diverges; the theorem is silent on the subject of divergence.
The key idea is to find an $N$ such that the absolute value of the terms is monotonically decreasing. That is, if there's some $N$ such that $|a_{n+1}| < |a_n|$ for all $n > N$, then we can use the alternating series test on the sequence
$$\sum_{n = N+1}^\infty a_n,$$
to learn that it converges, and hence the original series must as well (as it's a finite number of terms added to a convergent sequence).
Now your job is to find the $N$ such that the absolute value of terms are monotonically decreasing for $n > N$. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9744347860767305,
"lm_q1q2_score": 0.8500456241952917,
"lm_q2_score": 0.8723473713594991,
"openwebmath_perplexity": 136.38483873027855,
"openwebmath_score": 0.8766361474990845,
"tags": null,
"url": "https://math.stackexchange.com/questions/1235115/does-sum-n-1-infty-1n-fracn-sqrtn3-6-converge"
} |
# Using polar coordinates to find the area of an ellipse
Considering an ellipse with the $x$ radius equal to $a$ and the $y$ radius equal to b$:$
I figured that some kind of parameterization might be:
$x=a\cos\theta$
$y=b\sin\theta$
and then polar $r^2$ is just $x^2 + y^2$
but then I tried to come up with some unit of infinitesimal area using triangles ($d\theta r^2/2$) which does not give the correct answer. I read somewhere that my polar coordinates are wrong and that they are actually
$x=ar\cos\theta$
$y=br\sin\theta$
but this does not make sense to me as an engineer because that seems like it would have the dimension of area equal to the dimension of a distance. The integral also takes $r$ from $0$ to $1$ which I thought was eliminated because the equation for $r$ should be in terms of $\theta$ and the constants $a$ and $b.$
I would like some explanation of what I am doing wrong that would make some "physical" sense (or why physical intuition might fail for this problem)
Notice, since the ellipse: $x=a\cos\theta$ & $y=b\sin\theta$ is equally divided into four symmetrical regions hence, the area of ellipse in Cartesian coordinates is given as $$=4\int_0^ay\ dx$$ Now changing in Polar-coordinates by setting $y=b\sin\theta$ & $x=a\cos\theta$ or $dx=-a\sin\theta\ d\theta$, one should get area of ellipse $$=4\int_{\pi/2}^0(b\sin\theta)(-a\sin\theta\ d\theta)$$ $$=4ab\int_0^{\pi/2}\sin^2\theta\ d\theta$$ $$=4ab\int_0^{\pi/2}\frac{1-\cos2\theta}{2}\ d\theta$$ $$=4ab\left(\frac{1}{2}\int_0^{\pi/2}\ d\theta-\frac 12\int_0^{\pi/2}\cos2\theta\ d\theta\right)$$
$$=2ab\int_0^{\pi/2}\ d\theta$$$$=2ab\frac{\pi}{2}=\color{red}{\pi ab}$$
My name is Marco mamello10@gmail.com.
• You don't need to provide your name here. Any comment under your answer will show up as an update. If you do want to leave an email for people to contact you it's better to put it on your profile page. – Red Oct 7 '17 at 19:01 | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9744347801373334,
"lm_q1q2_score": 0.8500456157802077,
"lm_q2_score": 0.8723473680407889,
"openwebmath_perplexity": 156.27205216407705,
"openwebmath_score": 0.9140934944152832,
"tags": null,
"url": "https://math.stackexchange.com/questions/1595894/using-polar-coordinates-to-find-the-area-of-an-ellipse"
} |
Your parametrization only covers the edge of the ellipse. This is not enough if you try to find the area, which is the full interior. Hence the second form with $r$ (which is integrated from 0 to 1) is correct to find the area. Yours would be ok, if you try to only find the circumference.
Hint:
Use one of the polar equations for the ellipse: if the pole is at the centre of the ellipse, it is $$r^2=\frac{b^2}{1-e^2\cos^2\theta}, \quad\text{where }e =\sqrt{a^2-b^2}\enspace\text{is the eccentricity of the ellipse}$$ and integrate $$\int_0^\pi r^2\,\mathrm d\mkern1mu\theta.$$
You were on the right track, but didn't pick the right infinitesimal triangle. It looks like you used a right triangle built on the inside of the ellipse, but as you discovered, that doesn't approximate the area swept out by the radius vector well enough. Take instead a point along the curve and another a small distance away along it. The area of the resulting triangle is, in Cartesian coordinates, $\frac12(x\,dy-y\,dx)$. Plugging in your parametrization of the ellipse gives $dA=\frac12ab\,d\theta$, which will clearly give the correct result when integrated.
More generally, when approximating integrals with Riemann sums, you need to take care that they converge to the right thing. This is the same type of error that occurs in the fallacious “stairstep” constructions which “prove” that $\pi=4$ or that $\sqrt2=2$. In this case, I expect that if you also computed the sum of the areas of the right triangles built on the outside of the ellipse that the difference between the inner and outer sums would not vanish in the limit.
I guess, the problem is in wrong approximation (look at light blue areas) | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9744347801373334,
"lm_q1q2_score": 0.8500456157802077,
"lm_q2_score": 0.8723473680407889,
"openwebmath_perplexity": 156.27205216407705,
"openwebmath_score": 0.9140934944152832,
"tags": null,
"url": "https://math.stackexchange.com/questions/1595894/using-polar-coordinates-to-find-the-area-of-an-ellipse"
} |
I guess, the problem is in wrong approximation (look at light blue areas)
If use the formula for area of triangle $$\frac{1}{2}\left\| {{\bf{r}} \times {\bf{dr}}} \right\| = \frac{1}{2}rdr\sin \alpha % MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca % aIXaaabaGaaGOmaaaadaqbdaqaaiaahkhacqGHxdaTcaWHKbGaaCOC % aaGaayzcSlaawQa7aiabg2da9maalaaabaGaaGymaaqaaiaaikdaaa % GaamOCaiaadsgacaWGYbGaci4CaiaacMgacaGGUbGaeqySdegaaa!4979!$$ you get the right result. Using of Green formula for area $$A = \frac{1}{2}\oint\limits_C {x \cdot dy - ydx} = \frac{1}{2}\oint\limits_C {\left\| {{\bf{r}} \times d{\bf{r}}} \right\|} = \frac{1}{2}\oint\limits_C {r\sin \phi dr} % MathType!MTEF!2!1!+- % feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbcvPDwzYbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0x % e9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKk % Fr0xfr-xfr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam % yqaiabg2da9maalaaabaGaaGymaaqaaiaaikdaaaWaa8qvaeaacaWG % 4bGaeyyXICTaamizaiaadMhacqGHsislcaWG5bGaamizaiaadIhaaS % qaaiaadoeaaeqaniablgH7rlabgUIiYdGccqGH9aqpdaWcaaqaaiaa % igdaaeaacaaIYaaaamaapufabaWaauWaaeaacaWHYbGaey41aqRaam % izaiaahkhaaiaawMa7caGLkWoaaSqaaiaadoeaaeqaniablgH7rlab % gUIiYdGccqGH9aqpdaWcaaqaaiaaigdaaeaacaaIYaaaamaapufaba % GaamOCaiGacohacaGGPbGaaiOBaiabew9aMjaadsgacaWGYbaaleaa % caWGdbaabeqdcqWIr4E0cqGHRiI8aaaa!698A!$$ gives the same (right) result. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9744347801373334,
"lm_q1q2_score": 0.8500456157802077,
"lm_q2_score": 0.8723473680407889,
"openwebmath_perplexity": 156.27205216407705,
"openwebmath_score": 0.9140934944152832,
"tags": null,
"url": "https://math.stackexchange.com/questions/1595894/using-polar-coordinates-to-find-the-area-of-an-ellipse"
} |
# trig
posted by .
How do you write the quadratic equation with integer coefficients that have the roots 7-3i and 7+3i?
• trig -
In the quadratic equation
ax²+bx+c=0
b/a = -(sum of the two roots)
c/a = product of the two roots.
Assume a=1, then
b=-(7-3i+7+3i)=-14
c=(7-3i)*(7+3i)=49+9=58
So the quadratic equation is
x²-14x+58=0
Check by solving the equation and you should get back 7±3i as the roots.
• trig -
x = 7 + 3i, and X = 7 - 3i.
We can use either of the 2 values of X:
X = 7 + 3i,
(X - 7) = 3i,
Square both sides:
X^2 --14X + 49 = 9(-1),
X^2 - 14X + 49 +9 = 0,
X^2 - 14X + 58 = 0.
## Respond to this Question
First Name School Subject Your Answer
## Similar Questions
1. ### Algebra2
Find the polynomials roots to each of the following problems: #1) x^2+3x+1 #2) x^2+4x+3=0 #3) -2x^2+4x-5 #3 is not an equation. Dod you omit "= 0" at the end?
2. ### Algebra
Find a quadratic equation with integer coefficients whose roots are 2 and7.
3. ### Math
1)Write a quadratic equation with integer coefficients that has -4+7i(Radical 3) and -4-7i(Radical 3) as its roots. 2) Solve your answer to part 1 using the quadratic formula verifying your answer
4. ### Alg2
Find a quadratic equation with integral coefficients having roots 1/2 and -5/2.
5. ### Algebra
Find the number of integer quadruples (a,b,c,d) with 0\leq a,b,c,d \leq 100, such that a and b are the roots of the quadratic equation x^2-cx+d=0, while c and d are the roots of the quadratic equation x^2-ax+b.
6. ### Algebra
Find the number of integer quadruples (a,b,c,d) with 0\leq a,b,c,d \leq 100, such that a and b are the roots of the quadratic equation x^2-cx+d=0, while c and d are the roots of the quadratic equation x^2-ax+b
7. ### Math
I can't remember how you find the roots of quadratic equations other than determining the zeroes of the equation and finding the x-intercepts. Can someone please give me a clear answer on this as my dad has been useless on this topic …
8. ### Algebra | {
"domain": "jiskha.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9615338123908151,
"lm_q1q2_score": 0.8500336491299798,
"lm_q2_score": 0.8840392695254318,
"openwebmath_perplexity": 1358.8450188763309,
"openwebmath_score": 0.8634585738182068,
"tags": null,
"url": "https://www.jiskha.com/display.cgi?id=1292970811"
} |
If a quadratic equation with real coefficients has a discriminant of 225, the what type of roots does it have?
9. ### algebra
if a quadratic equation with real coefficients has a discrimiant of 225, then what type of roots doe it have?
10. ### Algebra 2
Q: Write a quadratic equation with integral coefficients whose roots are -6+i and -6-i My Answer: x^2+12x+37 (first) sum=-12 product=37 (then) write equation using formula x^2-(sum)x+product Is my work and answer correct?
More Similar Questions | {
"domain": "jiskha.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9615338123908151,
"lm_q1q2_score": 0.8500336491299798,
"lm_q2_score": 0.8840392695254318,
"openwebmath_perplexity": 1358.8450188763309,
"openwebmath_score": 0.8634585738182068,
"tags": null,
"url": "https://www.jiskha.com/display.cgi?id=1292970811"
} |
GMAT Question of the Day - Daily to your Mailbox; hard ones only
It is currently 11 Dec 2018, 05:39
# R1 Decisions:
HBS Chat - Decisions will be released at Noon ET | UVA Darden Chat | YouTube Live with Cornell Johnson @11am ET
### GMAT Club Daily Prep
#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
we will pick new questions that match your level based on your Timer History
Track
every week, we’ll send you an estimated GMAT score based on your performance
Practice
Pays
we will pick new questions that match your level based on your Timer History
## Events & Promotions
###### Events & Promotions in December
PrevNext
SuMoTuWeThFrSa
2526272829301
2345678
9101112131415
16171819202122
23242526272829
303112345
Open Detailed Calendar
• ### Free GMAT Prep Hour
December 11, 2018
December 11, 2018
09:00 PM EST
10:00 PM EST
Strategies and techniques for approaching featured GMAT topics. December 11 at 9 PM EST.
• ### The winning strategy for 700+ on the GMAT
December 13, 2018
December 13, 2018
08:00 AM PST
09:00 AM PST
What people who reach the high 700's do differently? We're going to share insights, tips and strategies from data we collected on over 50,000 students who used examPAL.
# When the price of oranges is lowered by 40%, 4 more oranges
Author Message
TAGS:
### Hide Tags
Manager
Joined: 15 Apr 2012
Posts: 87
Concentration: Technology, Entrepreneurship
GMAT 1: 460 Q38 V17
GPA: 3.56
When the price of oranges is lowered by 40%, 4 more oranges [#permalink]
### Show Tags
Updated on: 08 Jul 2012, 05:22
7
00:00
Difficulty:
35% (medium)
Question Stats:
74% (02:30) correct 26% (02:10) wrong based on 132 sessions | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. yes\n2. yes\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.8688267762381843,
"lm_q1q2_score": 0.43441338811909214,
"lm_q2_score": 0.5,
"openwebmath_perplexity": 8349.440753982046,
"openwebmath_score": 0.2509140968322754,
"tags": null,
"url": "https://gmatclub.com/forum/when-the-price-of-oranges-is-lowered-by-40-4-more-oranges-135496.html"
} |
35% (medium)
Question Stats:
74% (02:30) correct 26% (02:10) wrong based on 132 sessions
When the price of oranges is lowered by 40%, 4 more oranges can be purchased for $12 than can be purchased for the original price. How many oranges can be purchased for 24 dollars at the original price? (A) 8 (B) 12 (C) 16 (D) 20 (E) 24 Originally posted by farukqmul on 08 Jul 2012, 05:19. Last edited by Bunuel on 08 Jul 2012, 05:22, edited 1 time in total. Edited the question. Math Expert Joined: 02 Sep 2009 Posts: 51098 Re: When the price of oranges is lowered by 40%, 4 more oranges [#permalink] ### Show Tags 08 Jul 2012, 05:33 1 farukqmul wrote: When the price of oranges is lowered by 40%, 4 more oranges can be purchased for$12 than can be purchased for the original price. How many oranges can be purchased for 24 dollars at the original price?
(A) 8
(B) 12
(C) 16
(D) 20
(E) 24
Say $$x$$ is the original price of an orange, then:
$$xn=12$$;
and
$$0.6x*(n+4)=12$$ --> $$x(n+4)=20$$ --> $$xn+4x=20$$ --> $$12+4x=20$$ --> $$x=2$$.
So, for 24 dollars 24/2=12 oranges can be purchased at the original price of $2. Answer: B. _________________ Intern Joined: 26 May 2012 Posts: 21 Re: When the price of oranges is lowered by 40%, 4 more oranges [#permalink] ### Show Tags 15 Jul 2012, 02:22 Bunuel wrote: farukqmul wrote: When the price of oranges is lowered by 40%, 4 more oranges can be purchased for$12 than can be purchased for the original price. How many oranges can be purchased for 24 dollars at the original price?
(A) 8
(B) 12
(C) 16
(D) 20
(E) 24
Say $$x$$ is the original price of an orange, then:
$$xn=12$$;
and
$$0.6x*(n+4)=12$$ --> $$x(n+4)=20$$ --> $$xn+4x=20$$ --> $$12+4x=20$$ --> $$x=2$$. | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. yes\n2. yes\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.8688267762381843,
"lm_q1q2_score": 0.43441338811909214,
"lm_q2_score": 0.5,
"openwebmath_perplexity": 8349.440753982046,
"openwebmath_score": 0.2509140968322754,
"tags": null,
"url": "https://gmatclub.com/forum/when-the-price-of-oranges-is-lowered-by-40-4-more-oranges-135496.html"
} |
$$xn=12$$;
and
$$0.6x*(n+4)=12$$ --> $$x(n+4)=20$$ --> $$xn+4x=20$$ --> $$12+4x=20$$ --> $$x=2$$.
So, for 24 dollars 24/2=12 oranges can be purchased at the original price of $2. Answer: B. are there any other ways? Current Student Status: DONE! Joined: 05 Sep 2016 Posts: 377 Re: When the price of oranges is lowered by 40%, 4 more oranges [#permalink] ### Show Tags 22 Oct 2016, 11:32 Set up: 12/p = x 12/0.60 = x+4 Manipulate and plug the first equation into the second --> you'll find p =$2
Thus $24/$2 per orange = 12 oranges
VP
Joined: 07 Dec 2014
Posts: 1128
Re: When the price of oranges is lowered by 40%, 4 more oranges [#permalink]
### Show Tags
22 Oct 2016, 17:22
farukqmul wrote:
When the price of oranges is lowered by 40%, 4 more oranges can be purchased for $12 than can be purchased for the original price. How many oranges can be purchased for 24 dollars at the original price? (A) 8 (B) 12 (C) 16 (D) 20 (E) 24 let r=number of oranges purchased originally [12/(r+4)]/(12/r)=.6 r/(r+4)=.6 r=6 oranges$12/6=$2 per orange$24/$2=12 oranges B Current Student Joined: 12 Aug 2015 Posts: 2629 Schools: Boston U '20 (M) GRE 1: Q169 V154 Re: When the price of oranges is lowered by 40%, 4 more oranges [#permalink] ### Show Tags 21 Apr 2017, 00:28 Here is what i did in this question => Let the price of each orange be$x
Hence for $12 => we can buy => 12/x oranges Now new price => x(1-40/100) =>$0.6x
Hence for $12 we could now buy => 12/0.6x As per the given information => 12/x+4=12/0.6x Hence 12+4x/x=20/x => 12+4x=20 =>x=2 So the price of each orange is$2
Now for 24 dollars => we can buy => 1/2*24=12 oranges
SMASH THAT B.
_________________
MBA Financing:- INDIAN PUBLIC BANKS vs PRODIGY FINANCE!
Getting into HOLLYWOOD with an MBA!
The MOST AFFORDABLE MBA programs!
STONECOLD's BRUTAL Mock Tests for GMAT-Quant(700+)
AVERAGE GRE Scores At The Top Business Schools! | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. yes\n2. yes\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.8688267762381843,
"lm_q1q2_score": 0.43441338811909214,
"lm_q2_score": 0.5,
"openwebmath_perplexity": 8349.440753982046,
"openwebmath_score": 0.2509140968322754,
"tags": null,
"url": "https://gmatclub.com/forum/when-the-price-of-oranges-is-lowered-by-40-4-more-oranges-135496.html"
} |
STONECOLD's BRUTAL Mock Tests for GMAT-Quant(700+)
AVERAGE GRE Scores At The Top Business Schools!
Manager
Joined: 07 Feb 2017
Posts: 188
Re: When the price of oranges is lowered by 40%, 4 more oranges [#permalink]
### Show Tags
30 Jun 2018, 18:53
12/(.6x)=12/x+4
20=12+4x
x=2
24/2
Target Test Prep Representative
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
Posts: 4277
Location: United States (CA)
Re: When the price of oranges is lowered by 40%, 4 more oranges [#permalink]
### Show Tags
04 Jul 2018, 18:25
1
farukqmul wrote:
When the price of oranges is lowered by 40%, 4 more oranges can be purchased for $12 than can be purchased for the original price. How many oranges can be purchased for 24 dollars at the original price? (A) 8 (B) 12 (C) 16 (D) 20 (E) 24 We use the equation: price per item x no. of items = total cost. Here, we let p = the original price of an orange and q = the original number of oranges purchased. We can create the equation for the original total cost: pq = 12 q = 12/p After the orange’s price is lowered, we have that 0.6p = the new (reduced) price of an orange and (q + 4) = the new number of oranges that can be purchased at the reduced price. Our new equation for total cost is: (0.6p)(q + 4) = 12 0.6pq + 2.4p = 12 Substituting for q, we have: 0.6p(12/p) + 2.4p = 12 7.2 + 2.4p = 12 2.4p = 4.8 p = 2 That is, each orange is$2. So for \$24, we can buy 24/2 = 12 oranges.
_________________
Scott Woodbury-Stewart
Founder and CEO
GMAT Quant Self-Study Course
500+ lessons 3000+ practice problems 800+ HD solutions
Re: When the price of oranges is lowered by 40%, 4 more oranges &nbs [#permalink] 04 Jul 2018, 18:25
Display posts from previous: Sort by | {
"domain": "gmatclub.com",
"id": null,
"lm_label": "1. yes\n2. yes\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.8688267762381843,
"lm_q1q2_score": 0.43441338811909214,
"lm_q2_score": 0.5,
"openwebmath_perplexity": 8349.440753982046,
"openwebmath_score": 0.2509140968322754,
"tags": null,
"url": "https://gmatclub.com/forum/when-the-price-of-oranges-is-lowered-by-40-4-more-oranges-135496.html"
} |
# Maximal ideals in $K[X_1,\dots,X_n]$
Let $K$ be a field, and $a_1,\dots,a_n \in K$. Prove that the ideal $$(X_1-a_1,\dots,X_n-a_n)$$ is maximal in $K[X_1,\dots,X_n]$.
I tried proving that the only elements outside the ideal are the invertibles of $K$ (I should still prove that this implies maximality, but it shouldn't be too difficult).
Is there a better strategy, or another stategy?
• It may be interesting to you to note that the opposite direction is also true, i.e.: any maximal ideal in that multivariable polynomial ring is of the given form, yet any proof of this other direction I know uses heavily analytic tools, whereas the direction you're asking is purely algebraic. – DonAntonio Jan 22 '13 at 16:18
• It's easier to see that the ideal $(X_1, \ldots, X_n)$ is maximal and your question's like a "change of variable". DonAntonio, can you say me where you saw the proof of the opposite direction? – Diego Silvera Jan 23 '13 at 1:27
• @DonAntonio: Dear Don, The converse is true only if $K$ is algebraically closed. (Think about the case $n = 1$.) Regards, – Matt E Jan 24 '13 at 5:25
• Good catch, @MattE. Thanks. – DonAntonio Jan 24 '13 at 5:39
• Several answers provide strategies that work, but it may be useful to also point out that the strategy you suggested in the question, "proving that the only elements outside the ideal are the invertibles of $K$," is doomed to failure because that isn't true. The elements outside the ideal are all the polynomials $f$ such that $f(a_1,\dots,a_n)\neq0$, and that includes lots of non-constant polynomials (i.e., polynomials that aren't in $K$). – Andreas Blass Jan 24 '13 at 14:19
Hint: Define
$$f:K[X_1,...,X_n]\to K\;\;,\;\;f(g(X_1,...,X_n)):=g(a_1,...,a_n)$$
1) Show $\,f\,$ is a surjective ring homomorphism
2) Use now the first isomorphism theorem for rings
3) Remember: if $\,R\,$ is a commutative unitary ring, an ideal $\,I\leq R\,$ is maximal iff $\,R/I\,$ is a field. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9597620562254525,
"lm_q1q2_score": 0.8499954800239738,
"lm_q2_score": 0.8856314692902446,
"openwebmath_perplexity": 210.0985121292155,
"openwebmath_score": 0.8847081661224365,
"tags": null,
"url": "https://math.stackexchange.com/questions/284297/maximal-ideals-in-kx-1-dots-x-n"
} |
• Let $f_1$ be the homomorphism $$f_1:K[X_1,\cdots,X_{n-1}][X_n]\rightarrow K[X_1,\cdots,X_{n-1}]$$ $$f_2:K[X_1,\cdots,X_{n-2}][X_{n-1}]\rightarrow K[X_1,\cdots,X_{n-2}]$$ $$\cdots$$ $$f_n:K[X_1]\rightarrow K$$ Let assume that we know each $f_k$ to be a ring homomorphism. Then the composition $$f=f_n \cdots f_1$$ is an homomorphism. It is surjective because for each $a\in K$ we can evaluate the constant polynomial $a$. The kernel is $N:=\{g\in K:g(a_1,\cdots,a_n)=0\}$ I should now proof that $N=(X_1-a_1\cdots X_n-a_n)$ but is Ruffini still valid? – Temitope.A Jan 24 '13 at 22:02
Let $P(X_1, \ldots, X_n)$ a polynomial. Substitute $X_i\mapsto X_i + a_i$ and get $$P(X_1+ a_1, \ldots, X_n + a_n) = \sum c_{\alpha} X_1^{\alpha_1} \ldots X_n^{\alpha_n}$$ and so
$$P(X_1, \ldots, X_n) = \sum c_{\alpha} (X_1-a_1)^{\alpha_1} \ldots (X_n-a_n)^{\alpha_n}$$
Note that $c_{(0,\ldots, 0)} = P(a_1, \ldots, a_n)$. Moreover, $$P(X_1, \ldots, X_n) = c_{(0,\ldots, 0)}+ \sum (X_i - a_i) g_i(X_1, \ldots, X_n)$$ as all the other terms $c_{\alpha}(X-a)^{\alpha}$ are divisible by some $(X_i - a_i)$. Therefore $$P(X_1, \ldots, X_n) - P(a_1, \ldots, a_n) \in (X_1-a_1, \ldots, X_n - a_n)$$
and therefore $P(a_1, \ldots, a_n) \in (P, (X_1 - a_1) , \ldots, (X_n - a_n))$. Assume moreover that $P \not \in (X_1- a_1, \ldots X_n - a_n)$. Then $P(a_1, \ldots, a_n) \ne 0$ and we conclude that $1 = P(a_1, \ldots, a_n)^{-1} \cdot P(a_1, \ldots, a_n) \in (P, (X_1 - a_1), \ldots, X_n - a_n)$. Therefore $(X_1-a_1, \ldots, X_n - a_n)$ is maximal.
Hint $\ \ (I,f) = (I,f\ mod\ I) = (I,f(\bar a))\,\ [\,= 1 \iff f(\bar a)\ne 0\iff f\not\in I]$
Remark $\$ It is instructive to compare this internal approach to the structural approach mentioned by DonAntonio. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9597620562254525,
"lm_q1q2_score": 0.8499954800239738,
"lm_q2_score": 0.8856314692902446,
"openwebmath_perplexity": 210.0985121292155,
"openwebmath_score": 0.8847081661224365,
"tags": null,
"url": "https://math.stackexchange.com/questions/284297/maximal-ideals-in-kx-1-dots-x-n"
} |
• @YACP $\ I = (\bar X- \bar a) = (X_1-a_1,\ldots,X_n-a_n),\$ and $\, {\rm mod}\ I\!:\ X_i\equiv a_i\Rightarrow f(X_1,\ldots,X_n) \equiv f(a_1,\ldots,a_n) = f(\bar a)\ \$ – Math Gems Jan 24 '13 at 15:52
• @YACP Ideals are always preserved under the operation of "modding out" one generator by others, i.e. $(a_1,\ldots,a_n,b) =$ $(a_1,\ldots,a_n,b\!-\!r_1 a_1-r_2 a_2\! -\!\cdots- r_n a_n).\,$ Above the modular reduction corresponds to the multidimensional generalization of the polynomial remainder theorem. – Math Gems Jan 24 '13 at 16:24
• @YACP Even though, in general, there is no (generalized) division with (unique) remainder (with associated "mod" operation), it is conceptually helpful to think of such ideal tranformations like the mod operations employed in the reduction steps of the Euclidean algorithm (or the multidimensional generalizations used in standard/Grobner basis reductions). – Math Gems Jan 24 '13 at 16:24
• What do you mean by "multidimensional generalization of the polynomial remainder theorem"? As far as I can see you use the following result: $f$ can be written as a combination of $X_i-a_i$ (with polynomial coefficients) plus an element of $K$. Is this obvious or deserves a proof? – user26857 Jan 24 '13 at 16:32
• @YACP It's an easy inductive proof: apply the univariate remainder theorem to the highest variable, using $k[X_1,\ldots,X_n] = R[X_n]\:$ for $\:R = k[X_1,\ldots,X_{n-1}].\ \$ – Math Gems Jan 24 '13 at 16:51
I realize that we should avoid responding to other answers, but when they make false statements there should be a way to correct them. | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9597620562254525,
"lm_q1q2_score": 0.8499954800239738,
"lm_q2_score": 0.8856314692902446,
"openwebmath_perplexity": 210.0985121292155,
"openwebmath_score": 0.8847081661224365,
"tags": null,
"url": "https://math.stackexchange.com/questions/284297/maximal-ideals-in-kx-1-dots-x-n"
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.