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

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1
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7answers
165 views

Finding the derivative of a function.

Differentiate $$f(x) = \sin(\ln(\cos(x^2+1)))$$ My work: $u = \ln(\cos(x^2+1))$ so $f(x) = \sin u$ , $f'(x) = \cos u = \cos(\ln(\cos(x^2+1)))$. I keep getting this answer, but where am I going wrong?...
1
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1answer
141 views

Proving uniform continuity of function of two variables.

Proving uniform continuity of function:$$f(x,y)=\begin{cases} \frac{x^3-xy}{x^2+y^2}, & (x,y)\neq (0,0) \\ 0, & (x,y)=(0,0) \end{cases}$$ This is supposedly solve, but I don't understand the ...
7
votes
7answers
386 views

showing an inequality not using stirling formula

I don't know how to show that $$ \frac{k^k}{k!}\leq e^{k} $$ without using Stirling's approximation. I want to show it directly. I guess I need some inequality to achieve this but I don't know.
0
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1answer
31 views

I'm asked to compute the gradient of a scalar function

$$h(x,y)=\begin{cases} y- \frac{\sin x}{x}, & x \neq 0; \\ y-1, & x=0 \end{cases} $$ So my thoughts are: $$\textrm{grad}(h(x,y))=\left(\dfrac{x\cos x-x \sin x}{x^2},1\right), \quad x\neq0$...
1
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3answers
69 views

Given $P(x) = x^4+ax^3+bx^2+cx+d$ such $P'(0)=0.$ If $P(-1)<P(1)$ Then max. and min. of $P(x)$

Given $P(x) = x^4+ax^3+bx^2+cx+d$ such that $x=0$ is the only root of $P'(x) = 0.$ If $P(-1)<P(1)\;,$ Then in interval $\left[-1,1\right],$ $\bf{Options::}$ $(a)\;\; P(-1)$ is the minimum ...
1
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1answer
39 views

Area calculation

How could we best approach calculating the area inside $r=\cos^{2n-1}(x)+\sin(x)$, $0\leq x\leq \pi$, for $n=1,2,...$? For $n=3$ we get the following "potato/bean" graph: and for $n=51$ we get
2
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1answer
103 views

Need help with this proof, I don't understand it , could anyone clarify some of the details. System of linear Differential equations.

$$(*)X'=A(t)X - system$$ $$(*)PX(\alpha)+QX(\beta)=0.$$-border conditions, where P,Q constant square matrices $n \times n $. Let $Y(t)$ be the fundamental matrix for the system $(*)$ normed for$ t= \...
2
votes
2answers
260 views

The sum of the lengths of the hypotenuse and another side of a right angled triangle is given.

The sum of the lengths of the hypotenuse and another side of a right angled triangle is given.The area of the triangle will be maximum if the angle between them is: $(A)\frac{\pi}{6}\hspace{1cm}(B)\...
3
votes
5answers
9k views

Difference between maximum and minimum?

If I have a problem such as this: We need to enclose a field with a fence. We have 500m of fencing material and a building is on one side of the field and so won’t need any fencing. Determine the ...
2
votes
2answers
579 views

Point of Inflection or Turning Point?

I recently wrote a maths test in school, which by any standards, was exceedingly simple. The last question however, required that we find the value of "t" for which the rate of change of "h" is a ...
6
votes
2answers
248 views

Integral $\int_0^{1/\phi}x\log(x)\log(1+x)\log(1-x)\,dx$

How can we evaluate this definite integral $$I=\int_0^{1/\phi}x\log(x)\log(1+x)\log(1-x)\,dx,$$ where $\displaystyle\phi=\frac{1+\sqrt5}2$ is the golden ratio?
1
vote
1answer
55 views

Problem that involves partial derivative of temperature function

A circular piece of metal with radius $a$ has the temperature given by the following relation: for a point $(x,y)$, the temperature $T(x,y)$ is proportional to the square of the distance of this point ...
0
votes
1answer
42 views

How do I solve the following limit using L'Hospital's Rule?

I want to show that: $\lim_{y\ -> 0+}$ e^(-1/y)/y^2 = 0. Substituting directly we get an indeterminate limit of the form 0/0, so applying L'Hopital's Rule I get that the above limit is equal to: ...
4
votes
2answers
290 views

Is the limit $ \lim_{n\to \infty}\left(\sum^{n}_{r=0} \binom{n}{r}\big/{n^{r}(r+3)}\right)$ rational or irrational?

How can I prove that the result of the following limit is rational/irrational?$$ \lim_{n\to \infty}\left(\sum^{n}_{r=0} \frac{\binom{n}{r}}{n^{r}(r+3)}\right)$$ Would solving this limit satisfy? How ...
3
votes
1answer
323 views

Limit Question involving infinite multiplication?

$$\lim_{n\to ∞} \prod_{i=1}^n ({n+i\over n})^{1\over n} $$ For this question I tried using the substitution $$ u = {1\over n}$$ So as "n" goes to infinity, "u" goes towards zero. So the limit ...
1
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1answer
36 views

Understanding central difference formula for computing numerical gradient

More can be found here: http://www.math.ohiou.edu/courses/math3600/lecture27.pdf. I'm having trouble understanding what happens to the $h$ in this example where the central difference error is ...
2
votes
3answers
115 views

Differentiability of this picewise function

$$f(x,y) = \left\{\begin{array}{cc} \frac{xy}{x^2+y^2} & (x,y)\neq(0,0) \\ f(x,y) = 0 & (x,y)=(0,0) \end{array}\right.$$ In order to verify if this function is differentiable, I tried to ...
1
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1answer
57 views

Is there a 2D analogue of Thomae's function?

Thomae's function, $$ f(x)= \begin{cases} 0 & \text{$x$ is irrational}\\ \frac{1}{p} & \text{$x = \frac{p}{q}$ where $\gcd(p,q) = 1$} \end{cases}, $$ is an example of a function from $(0,1)\...
6
votes
3answers
317 views

Lagrange Multipliers Example

Minimize $$f(x,y) = x^2+y^2$$ subject to the constraint $xy=3$. I know the formula for Lagrange multipliers to be $\nabla f = \lambda \nabla g$ so we get a system of equations like this $$\begin{...
1
vote
1answer
102 views

To evaluate $\int_{0}^\pi \frac {\sin^2 x}{a^2-2ab \cos x + b^2}dx$? [duplicate]

How do we evaluate $\int_{0}^\pi \dfrac {\sin^2 x}{a^2-2ab \cos x + b^2}dx$ ? I tried substitution and some other methods , but its not working ; please help .
1
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5answers
103 views

Solving a double integral using change of variables.

$$\int ^{1}_{0} \int^{1}_{y}e^{-x^{2}}\,dx\,dy$$ To solve this I know one must use change of variables, but the problem is that I do not know how to approach the actual change. Just thinking out ...
1
vote
1answer
164 views

Find the nodes and coefficients of Gauss-Lobatto Quadrature with $n=4$

I am stucked at this problem: Gauss-Lobatto quadrature is defined as: $\int_{-1}^1 f(x)dx\approx w_1 f(-1)+w_n f(1) + \Sigma_{k=2}^{n-1}w_k f(x_k)$ ($2\leq n\in\Bbb{N}$) Where the nodes $x_k$ ...
0
votes
3answers
118 views

A simple looking integration : $\left(\frac{x^3}{1+x^5}\right)$

One of my friends gave me this problem about a week back and since then, I have been toiling to get a solution to this problem, but I just get stuck at some step. Can someone please tell me the steps ...
0
votes
1answer
95 views

$\displaystyle\int_{0}^{\pi} \frac{\sin^2 x}{a^2 - 2ab\cos x +b^2} \,dx$ [duplicate]

An integration: $$\int_{0}^{\pi} \frac{\sin^2 x}{a^2 - 2ab\cos x +b^2} \,dx$$ I am stuck with this definite integral. Will putting $\,\cos x = z\,$ help out here?
0
votes
1answer
59 views

Are Riemann integrals necessarily convergent sums?

For example, does the Riemann sum of $\sum_2^n$ $\frac{1}{k} \frac{1}{n}$ converge to the integral $\int_2^n \frac{1}{x}dx$? Does the 1/n factor help the sum of 1/k converge? It doesn't look obvious,...
7
votes
4answers
458 views

Solve quadric equation system

How to solve this analytically(not a numerical solution)? For given real and symmetric matrices $A_1,A_2,A_3,A_4\in\mathbb{R}^{4\times4}$ find $0\neq x\in\mathbb{R}^4$ $$x^TA_1x=0$$ $$x^TA_2x=0$$ $$...
4
votes
1answer
217 views

Complex Conjugate of Integral

I want to know that the equality $$ \overline{\int_{\mathbb R} f(x)dx} = \int_{\mathbb R} \overline{f(x)}dx $$ holds, if the both integral converges. Here $f:\mathbb R \ni x \mapsto f(x)\in \mathbb C $...
1
vote
1answer
73 views

Integral inequality

Let $f:[0,1]\longrightarrow \mathbb{R}$ be a continuous function such that $$\int_{0}^{1}f(x)dx=0.$$ Is the following inequality true? $$2\left(\int_{0}^{1}xf(x)dx\right)^2\leq \int_{0}^{1}(1-x^2)f^2(...
1
vote
1answer
107 views

Calculating the volume of a surfboard

I'm building a website for a client in which customers can customise the shape of their board (curvature, length, width, thickness, and so forth) and the client has asked if we can calculate the ...
1
vote
2answers
110 views

Finding $\lim_{(x,y)\rightarrow (0,0)} \frac{\tan(x^2+y^2)}{\arctan(\frac{1}{x^2+y^2})} $

I'm having trouble understanding how the $\displaystyle\lim_{(x,y)\rightarrow (0,0)} \frac{\tan(x^2+y^2)}{\arctan(\frac{1}{x^2+y^2})}$. I used the product law to set it up as $\displaystyle\frac{\...
1
vote
2answers
57 views

Find $\sum_{n=1}^{\infty}a_nx^n$ given $a_0=3, \ 3na_n+3(n-1)a_{n-1}=2a_{n-1}$

Given $\ a_0=3$, $\,3na_n+3(n-1)a_{n-1}=2a_{n-1}$, find $\ f = \sum_{n=1}^{\infty}a_nx^n$. I have proved that when $\ \left\lvert x \right\rvert<1$, this exponential series function is convergent. ...
4
votes
4answers
288 views

Showing $\lim_{n \to +\infty} \log(n!)/(n\log n) = 1$ without using Stirling approximation

As a passage of a bigger limit I have to show that $$ \lim_{ n \to \infty } \frac{\log(n!)}{n\log(n)} = 1. $$ I think it could be done using Stirling approximation, but I'm wondering if there's a way ...
1
vote
3answers
306 views

Help with Spivak's Calculus: Chapter 1 problem 21

I've been stuck on this problem for over a day, and the answerbook simply says "see chapter 5" for problems 20,21, and 22. But I want to complete the problem without using knowledge given later in the ...
2
votes
2answers
43 views

Elementary Differential Equations

I'm currently studying Elementary differential equations, and I came across a confusion that I had that I think arises from notation, but I would like to clarify with someone. The example problem said ...
2
votes
1answer
99 views

Division in Summations

Suppose $a_n=\dfrac{2^n}{n(n+2)}$ and $b_n=\dfrac{3^n}{5n+18}$. I need to find the value of: $$\displaystyle\sum_{n=1}^{\infty}\dfrac{a_n}{b_n}$$ I think this problem is meant for me to compute ...
2
votes
3answers
81 views

How to simplify elegantly $\arcsin(2t-1)+2\arctan\left(\sqrt{\frac{1-t}{t}}\right)$?

I currently try to simplify the following trigonometric expression: $$ \arcsin(2t-1)+2\arctan\left(\sqrt{\frac{1-t}{t}}\right) $$ where $t\in(0;1]$. I know that $\arctan(x)=\arcsin\left(\frac{x}{\...
1
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0answers
82 views

differentiating a smooth function defined by an integral

Suppose we define a function by the integral $$ f(x) = \int_{-\infty}^{\infty}g(x,y) dy $$Suppose we know that $f(x)$ is smooth. Does this mean that necessarily $$ \frac{d}{dx}\int_{-\infty}^{\infty}g(...
0
votes
0answers
29 views

Finding the $h'(x,y,z)$ if $h= p \circ q $ $p(x,y,z)=(x \sin y, x \cos y, z+y ), q(x,y,z)=(x^2,x+y,2e^z)$

I just want someone to check my work basically. Providing thoughts and insight, into possible mistakes: Finding the $$h'(x,y,z)$$ if $$h= p \circ q ,\ \ p(x,y,z)=(x \sin y, x \cos y, z+y ), \ \ q(x,y,...
5
votes
4answers
210 views

“multiplicative inverse in the modulo of the larger number” what does that mean?

while I was reading this artical I have read the following paragraph: The interesting thing is that if two numbers have a $\gcd$ of $1$, then the smaller of the two numbers has a multiplicative ...
1
vote
1answer
180 views

Help with Spivak's Calculus: Chapter 1 Problem 22

I've tried a lot of things, but I can't seem to get anywhere with this problem. I'm hoping the solution is simple but that I'm just missing it. The problem is as follows: Prove that if $y_0 \neq ...
0
votes
2answers
44 views

Find the rate of change of the frequency when D, L, σ and T are varied singly.

I'm reading Calculus made easy to learn the notation (I know derivatives with the limit/prime style) and also some integral calculus which I haven't seen at school yet. You can check it here: https://...
1
vote
1answer
63 views

Find the derivative of the following by definition: $f(x,y)=(x^3, xy^2-y^2)$

$$f(x,y)=(x^3, xy^2-y^2)$$ So with these types of functions the derivative is $f'(x,y)=\begin{pmatrix} \frac{\partial f_1}{\partial x} & \frac{\partial f_1}{\partial y} \\ \frac{\...
0
votes
3answers
1k views

Proving a function has real roots

I am not interested in finding roots but interested in proving that the function has real roots. Suppose a function $f(x) = x^2 - 1$ This function obviously has real roots. $x = {-1, 1}$ How could ...
0
votes
1answer
53 views

Divergence theorem to prove a relation.

I found this problem, I wanna do. Let $f$ be continuesly differentiable, let $r=\sqrt{x^2+y^2+z^2}$ Prove $\iint \limits _S \vec f (r) \cdot \vec n \ \Bbb dS = \iiint \limits _B \vec {f'(r)} \cdot \...
0
votes
1answer
86 views

Finding the partial derivatives of $f(x,y,z)= \int_{0}^{x}t^tdt + x ^{\sin(y^z)}$ and the first derivative.

$$f(x,y,z)= \int_{0}^{x}t^tdt + x ^{\sin(y^z)}$$ The derivative would be $f'(x,y,z)(h^1,h^2,h^3)= \frac{\partial f}{\partial x}h^1+\frac{\partial f}{\partial y}h^2+\frac{\partial f}{\partial z}h^3.$ ...
-1
votes
3answers
91 views

Solve for $a$ and $b$ in a limit

Find $a$ and $b$ such that $$ \lim_{x \to 0}\frac{\sqrt{ax+b}-2}{x}=1.$$ I'm not sure how to solve for two variables given that I only have one equation.
0
votes
0answers
55 views

Riemann sum normalization

I have a continuous function which satisfies the following relation, $$\int_0^a \int_0^a f(x,y)\sin (x) \, dx \, dy = 1$$ However, I have some experimental results in a matrix $g(x,y)$ at points $x=...
1
vote
1answer
53 views

How would I find the second derivative of the bilinear $B(x,y)=Ax \times y$ where $A=\begin{pmatrix} 1 & 2&3 \\ 0 & -1 & 2\\ -1 & 2 & 4 \end{pmatrix}$

$$B(x,y)=Ax \times y \text{ where } A=\begin{pmatrix} 1 & 2&3 \\ 0 & -1 & 2\\ -1 & 2 & 4 \end{pmatrix}$$ Second derivative is obviously the first derivative of the first ...
3
votes
2answers
165 views

Integrating $ \frac{{ \int_{0}^{\infty} e^{-x^2}\, dx}}{{\int_{0}^{\infty} e^{-x^2} \cos (2x) \, dx}}$

I need help calculating the following integrals. For the top integral we can use the jacobin, right? But how do I calculate the bottom one?: $$ \frac{{ \int_{0}^{\infty} e^{-x^2}\, dx}}{{\int_{0}^{\...
0
votes
1answer
72 views

If $f(x)$ is diferentiable in $[0,1]$ and $f(0)=f(1)=0$, then find $\alpha$ such that $f'(c)+\alpha f(c)=0$

Attempt: Let $g(x)=e^{\alpha x}f(x)$ $g'(x)=\alpha e^{\alpha x}f(x)+e^{\alpha x}f'(x)$ $g'(c)=\alpha e^{\alpha c}f(c)+e^{\alpha c}f'(c)$ Now, $g(0)=f(0)$ and $g(1)=e^{\alpha }f(1)=e^{\alpha }g(0)$...