For basic questions about limits, derivatives, integrals, and applications, mainly of one-variable functions.

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Integrating trig substitution triangle equivalence

When we integrate certain integrals, such as $$\int \frac{x^2}{\sqrt{16-x^2}}$$ We can make a substitution like $x = 4 \sin \theta$ Then we can simplify the above integral to the following: $$8 ...
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3answers
27 views

Integrate $\frac{x^2}{\sqrt{16-x^2}}$ using trig substitution

During our integration of the following integral, using $x = 4 \sin \theta$ $$\int \frac{x^2}{\sqrt{16-x^2}} dx$$ We eventually come to the following point: $$\int \frac{16 {\sin ^2 \theta} }{4 ...
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1answer
11 views

Finding the directional derivative.

We need to find the directional derivative of the function , $f(x,y) = x^{2}+y^{2}+xy$ at $P(1,-1)$ in the direction towards origin. The direction towards origin form the point $(1,-1)$ is ...
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32 views

Evaluating $~\int_0^1\sqrt{\frac{1+x^n}{1-x^n}}~dx~$ and $~\int_0^1\sqrt[n]{\frac{1+x^2}{1-x^2}}~dx$

How could we prove that $$\int_0^1\sqrt{\frac{1+x^n}{1-x^n}}~dx~=~a\cdot2^{a-1}~\bigg[\frac12~B\bigg(\frac a2,~\frac a2\bigg)~+~B\bigg(\dfrac{a+1}2,~\dfrac{a+1}2\bigg)\bigg],$$ where ...
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1answer
15 views

Calculus III Vectors - Projectile problem

A projectile is fired from ground level with an initial speed of $450 m/sec$ and an angle of elevation of 30 degrees. Use that the acceleration due to gravity is $9.8 m/sec^2$. The range of the ...
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0answers
29 views
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2answers
37 views

$\lim_{x \to \infty} \frac{\sqrt{x^2 -1}}{2x+1}$

So the question is: $$\lim_{x \to \infty} \frac{\sqrt{x^2 -1}}{2x+1}$$ First of all, I know we have to use Lhopital's rule. However, I just don't know how. Second of all, I thought in the end we ...
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39 views

Find the limit: $\lim_{x \to 1}x^2 + 2$ [on hold]

What is the limit as $$\lim_{x\to 1} (x^2+2)?$$ I greatly appreciate your help and I will continue to type to satisfy the minimum length requirement please do not downvote this question as it will ...
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3answers
22 views

Related Rates Shadow Problem

The question is as follows: A man 6 feet tall walks at a rate of 5 feet per second away from a light that is 15 feet above the ground. When he is 10 feet from the base of the light, (a) at what rate ...
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1answer
39 views

Summation of the reciprocals of the product of consecutive integers

It is well known that there is a closed formula for: $$\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \cdots + \frac{1}{(n)(n + 1)}$$ And likewise for: $$\frac{1}{1 \cdot 2 \cdot 3} + \frac{1}{2 \cdot ...
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61 views

How to solve a Definite Integral (calculus 2)

I've begun to learn about these types of questions, however I still have a hard time knowing how to solve them and doing the actual computation. These three integrals are examples of the questions I ...
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2answers
29 views

Domain and range of $f(x,y)=\sqrt{1+x-y^2}$

I need to find the domain and range of $f(x,y)=\sqrt{1+x-y^2}$. Can someone walk me through the proper reasonings in solving this problem? My attempt Domain From looking at the function I get: ...
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0answers
28 views

Need help analytically solving an integral

For one part of a multiple-part problem, I need to analytically find the value of the integral $$I=\int_{x_L}^{x_R}\frac{dx}{\sqrt{-Ax + B - C/x}}$$ My professor gave us notes that: ...
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1answer
17 views

which of the following is an equivalence relation of the set S

which of the following is an equivalence relation of the set S I have solved all except d and need your help please
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1answer
26 views

Parametric curve parametriced by length

Normally you have a parametric curve with a variable t and you increment t to find the point along the curve. Is it possible to have a curve so that given a value it will give you the point on that ...
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2answers
62 views

Why is convergence required for a series to be differentiable?

First of all , I'll let you know that I am really really bad at calculus so please be gentle. Lets have this series: $\sum_{n=0}^\infty \frac{(-1)^{n-1}x^{2n}}{n(2n-1)}$ The thing is I know ...
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1answer
55 views

Conditions on $c$ such that the inequality dont hold.

I want to find conditions on $c$ such that the inequality don't hold. $$1-ac(a-2)(a-1)^2 < 0 \ \ \ \ \ \ \text{for } a>2, c>0$$ If $\phi(a) = ac(a-2)(a-1)^2 \Rightarrow \phi'(a) = ...
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41 views

Exercise of algebra [on hold]

Given the linear transformation $T:\mathbb{R}^4 \to P_2(\mathbb{R})$ such that: $\ker(T) = \{(x_1,x_1,x_3,x_4) \in \mathbb{R}^4 : 2x_2 - x_3 + x_4 = 0,\, 2x_1 - x_2 = 0\}$ $T(0,1,0,1) = -2x^2 + x$ ...
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1answer
45 views

Exercise of algebra II [on hold]

Can anybody please help me with this exercise?My exam is comming soon :S It says: 1)We define = f: P₂[R] ---> R^2x2 linear transformation whose transformation matrix in basis B and E' is: Mf(over ...
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2answers
16 views

if n is a positive integer let Z be the subset of integer in {1,…,n} which are relatively prime to n

if n is a positive integer let Z be the subset of integer in {1,...,n} which are relatively prime to n my effort to solve this question I''m confused and need help to solve this question please ...
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2answers
21 views

a bijection is an injective (one-to-one) , surjective (onto) map between sets. if S = (0, 1) and T =R, find a map from S to T which is

a bijection is an injective (one-to-one) , surjective (onto) map between sets. if S = (0, 1) and T =R, find a map from S to T which is my effort 1) (a) f(x) = x is a one to one function but it ...
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4answers
83 views

How does $\int (\cos(x))^{-2}dx$ equal to $\tan(x)$?

How does $$\int \frac{1}{\cos^2(x)} dx= \tan(x)+ C$$ ?
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3answers
84 views

If $\{x_i\}_{i=1}^n$ are the roots of $f(x)=a_nx^n + a_{n-1}x^{n-1} + \ldots +a_0$ then $\sum_{i=1}^nx_i^{n-1}$ is independent of $a_0$

I found an interesting conclusion when I did this simple question. Let $$f(x)=(x^2-1)(x+2)=x^3+2x^2-x-2$$ and let $x_i$ for $i=1,2,3$ be the roots of $f(x)$. Find the sum $\sum\limits_{i=1}^3x_i^2$. ...
4
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5answers
94 views

How do I integrate$ \int\frac{1}{e^{2x}+e^x} \,dx $ [on hold]

How do I integrate following function? $$ \int\frac{1}{e^{2x}+e^x} \,dx $$
2
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1answer
62 views

Finding a general integral

$$ \int\limits_{0}^{1}{\frac{\ln(1+{t}^{a})}{1+t} \;\mathrm{d}t} $$ I have tried many tings but I am just not successful in any of them - Feynman, summation inside integral, Beta function ...
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0answers
6 views

To prove that there is no scalar field $f'(a;y)>0$ for a fixed vector $a$ and every nonzero vector $y$.

To prove that there is no scalar field $f'(a;y)>0$ for a fixed vector $a$ and every nonzero vector $y$. I am having difficulty in this problem please help. Here $f'(a;y)$ is the derivative of $f$ ...
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3answers
45 views

Consider $F(x,y)=f(x+3y,2x-y)$…

If $f: \mathbb{R}^2\rightarrow\mathbb{R}$ where $F(x,y)=f(x+3y,2x-y)$ with $f$ is defferentiable and $\nabla f(0,0)=(4,-3)$ compute the derivate at the origin in the direction of unit vector ...
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1answer
20 views

Functional equations with involutions

Having seen that the topic of functional equations is loved by StackExchange, I have constructed this problem hoping that it will please readers. Solve the functional equation $$ ...
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1answer
23 views

Limit in electric field far from a charged disk

Let $\sigma$ be the areal charge density $\frac{Q}{\pi R^2}$ of a disk of radius $R$; then the electric field on the line pependicular to the disk and passing through is centre, if we use its ...
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1answer
16 views

Calculate the rate of change of the slope of a tangent line of a graph, given the equation, value of x, and rate of change of x.

Heree is the full question from Supp. problem 20.26, Beginning Calculus, Schaum's Outlines, 3rd. ed. An object is moving along the graph of y = 3x - x^2, and its x-coordinate is changing at the ...
2
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1answer
39 views

Derivative of $\frac{x}{\|x\|}$ w.r.t. x where $x\in \mathbb{R}$ ($x \neq \theta_n$)

I want to find the Hessian of a function. I have already computed the gradient of the function. So, I have to again differentiate it w.r.t. $x \in \mathbb{R}^n$ to get the hessian, but I am facing a ...
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0answers
21 views

The highest direction of the trace operator

Let $W$ be a real and symmetric matrix ${m \times m}$ from the set $\widetilde{W_m}$, and $T:\widetilde{W_m} \rightarrow \mathbb{R}$ a function defined by $T(W) = trace(W^3)$. We are interested to ...
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1answer
38 views

what's the greatest volume of a cylinder using calculus?

I have a rectangle that has the perimeter of 38cm. I need to make this rectangle into a baseless cylinder and find the greatest volume of it, by deriving. so far i came with this: for the rectangle ...
2
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1answer
28 views

How to determine the function from the following?

The graph of a certain function contains the point $ (0,2)$ and has the property that for each number 'p' the line tangent to $y = f(x)$ at $(p, f(p))$ intersect the x-axis at p + 2. Find $f(x)$ The ...
2
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1answer
38 views

$\lim_{x\to 0} \frac{f(x)}{x}=-1 \implies \lim_{x\to 2}\frac{f(x^2-4)}{x-2}=-4$.

I'm trying to prove if $\lim_{x\to 0} \frac{f(x)}{x}=-1$, then $\lim_{x\to 2}\frac{f(x^2-4)}{x-2}=-4$. I've tried everything, substitution, limit composition, etc. Anyone could help me to solve this ...
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0answers
20 views

For the classical diffusion equation ut = r (5ru) (in 3 space dimensions)

fi nd TWO changes of variables which changes the di ffusion constant from 5 to D = 1 for the new coordinate system?
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3answers
92 views

Why does WolframAlpha's expression for $\int\frac{dx}{x\sqrt{x^4-4}}$ disagree with my own?

$$\int\frac{1}{x\sqrt{x^4-4}}$$ My teacher gave us these notes and I'm unsure if they're correct. Wolfram gives a different answer, and when I derive I might have messed up. Thanks.
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1answer
18 views

If $\nabla f(x,y,z) $ is always parallel to $xi+yj+zk$, them $f$ must be equal values at the points $(0,0,a)$ and $(0,0,-a)$.

If $\nabla f(x,y,z) $ is always parallel to $xi+yj+zk$, them $f$ must be equal values at the points $(0,0,a)$ and $(0,0,-a)$. I am having difficulty in the problem. Please help.
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0answers
9 views

Inquiry about Second order condition for Lagrange Equations

Consider the programming problem: $ min\hspace 3mm x_1^2 + x_2^2$ subject to $ q = x_1+x_2$. The choice variables are $x_1$ and $x_2$. Establish that the second order necessary condition holds ...
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2answers
15 views

Average Value of a Surface

For a project I am working on I'm attempting to get the average height of a surface. That is, for a function $z = f(r, \theta)$ I would like to obtain the average z value within a specified radius and ...
0
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1answer
44 views

Maximum of $xy^3z^7$ in the plane $x+y+z=1$

A friend gave to me this problem and on having seen that I could not solve it in the first instance helped me with the hint of using the AM-GM inequality. PROBLEM.- To maximize the product $xy^3z^7$ ...
4
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1answer
41 views

Splitting up a double integral

I need to compute the following integral: $$ 2\pi\nu^2\int^a_be^{x^2}\int_{-\infty}^xerfcx(-y)dydx, $$ where $erfcx(x)=e^{x^2}erfc(x)$, $erfc(x)=1 - erf(x)$, and $erf(x)$ is the error function. The ...
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3answers
28 views

A question about the formulation of the definition of a limit for sequences

So I know the definition of a limit of a the sequence is: $a$ is a limit of a sequence $\{x_n\}$ if given $\epsilon>0$ there exists a positive integer $N$ such that $|x_n-a|<\epsilon$ for all ...
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0answers
72 views

Closed form for $\sum^\infty_{n=1}\frac{H_n}{2^n\,(2n+1)^2}$

(This is a slight variation of another question, already answered) Can we find a closed form of the following series? $$S=\sum^\infty_{n=1}\frac{H_n}{2^n\,(2n+1)^2}\tag1$$ Using some non-rigorous ...
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1answer
35 views

how to show that $f_n \uparrow f$

How to show that $f_n \uparrow f$ where $$f_n(x)=\min\left(\frac{\lfloor 2^nf(x)\rfloor}{2^n},n\right)$$ It is clear to me that $f_n(x) \leq f(x)$ But how do I show that the limit is indeed $f$ ? ...
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1answer
36 views

Vector calculus course: Find parametric equations? [on hold]

-Find parametric equations "x=t"-style -Also opposite orientation parametric form.(C and -C) A.) Part of line that goes through points $(2,5)$ and $(3,2)$ and $y∈[1,2]$. ...
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0answers
31 views

Find the maximum value of $\log_{10}(\frac {c_2}{x})\log_{10}(x+1-c_1)$, where $c_1 ,c_2$ are real constants and x is a real number,$x\in [c_1,c_2]$

What is the maximum value of: $$\log_{10}\left(\frac {c_2}{x}\right)\log_{10}(x+1-c_1)$$ where $c_1$ & $c_2$ are real constants and $x$ is a real number, $x\in [c_1,c_2]$. For which $x$ is this ...
0
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0answers
17 views

a problem involving binary entropy function

let $\alpha<1/2$ such that $2^{H(\alpha)}\le 2^{1-\epsilon}$,when $H$ is binary entropy function. how can i prove that then we have: $2^{n(1-\epsilon)}\ge \sum\limits_{i\le \alpha n } {n \choose ...
9
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3answers
195 views

If $u_{n+1}\le u_n+u_n^2$ and $\sum u_n$ converges, prove that $\lim\limits_{n\to +\infty}(n\cdot u_n)=0$

Given the positive sequence $\{u_n\},n\in \mathbb{N}$ that meets the conditions: $\boxed{1}$. $u_{n+1}\le u_n+u_n^2$ $\boxed{2}$. Exist the constant $\text{M} >0$ so that ...
-8
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1answer
52 views

Can somebody integrate this function for me? [on hold]

This is the function. $\frac{1}{6.08 \cdot \sqrt{2\pi}}\exp\left(-\frac{(x-10.75)^2}{2 \cdot 6.08^2}\right)$ Thanks in advance!