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

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2
votes
2answers
47 views

Using $\int_{0}^{\frac{\pi}{2}} \sin^{n}dx$ to show an equality

We know that $I_n = \displaystyle \int_{0}^{\frac{\pi}{2}} \sin^{2n+1}(x)dx = \dfrac{2^{2n}(n!)^2}{(2n+1)!}$ and $I_n = \displaystyle \int_{0}^{\frac{\pi}{2}} \sin^{2n}(x)dx = ...
2
votes
2answers
42 views

Calculate a limit changing the variable

I have the following limit to calculate: $$\lim_{x\to 0^+} \frac{e^{-\frac{1}{x}}}{x}$$ I have to use: $$y=\frac{1}{x}$$
2
votes
1answer
90 views

Proving that an equation has a solution

If $a$ and $b$ are positive numbers, prove that the equation $$\frac{a}{x^3+2x^2-1}+\frac{b}{x^3+x-2}=0$$ has at least one solution in the interval $(-1,1)$ My solution is as follows, I ...
2
votes
1answer
32 views

Derivative when change variable

This may be a silly question but I am confused with some notation of a derivative function ater changing variable. For example, if I have $f(x) = x^2$. Let $u(x) = 1-e^x$ and $F(x) = f(u(x))$, then if ...
2
votes
1answer
26 views

Prove an inequality involving a norm

We define the following inner product on intergrable, $2\pi$ periodic functions from $\mathbb{R}$ to $\mathbb{C}$: $$\langle f,g\rangle = \frac{1}{2\pi} \int_{-\pi}^\pi f(t)\overline{g(t)}\ dt$$ I ...
2
votes
1answer
74 views

Prove that the length of a curve is given by integral

I have found the following problem in "Introduction to Analysis" by Rosenlicht. I am not sure if my solution is correct and I highlighted my uncertainties. First we have to show that the set ...
2
votes
1answer
34 views

Finding $w_1,w_2,b$ such that $w_2 \sigma(w_1 x + b) \approx x$

Let $\sigma(z) = 1/(1+e^{-z})$. How can I find $w_1, w_2, b$ such that $w_2 \sigma(w_1 x + b) \approx x$ for $x \in [0,1]$? The hint provided was to rewrite $x = \frac{1}{2}+\Delta$, assume $w_1$ is ...
2
votes
3answers
53 views

Suppose $p=x^3+ax^2+bx+c$ and it has both a local max and min, why is the midpoint of the line segment connecting local values a point of inflection?

How can you do a $\Rightarrow$ proof? So I get that $p^{\prime}$ has two local extreme values if the discriminant $a^2-3b\gt 0 \iff b\lt \frac {a^2}{3}$. And calculating its POI I get ...
2
votes
1answer
25 views

Solve $y'=y$, $y(0)=1$ using method of successive approximations, obtaining the power series expansions of the solution

Solve $y'=y$, $y(0)=1$ using method of successive approximations, obtaining the power series expansions of the solution. Solving the above initial value problem is equivalent to solving (for ...
2
votes
1answer
79 views

$\lim_{x\to 0}\frac{2+\cos x}{x^3\sin x}-\frac{3}{x^4}=\frac{1}{60}$ without using L Hospital rule or series expansion

Prove that $\lim_{x\to 0}\frac{2+\cos x}{x^3\sin x}-\frac{3}{x^4}=\frac{1}{60}$ without using L Hospital rule or series expansion. I tried $\lim_{x\to 0}\frac{2+\cos x}{x^3\sin ...
2
votes
4answers
34 views

Problem with solving a specific limit

So, here is the limit $\lim_{n\to\infty}\left(\frac{3n^{2}+4n-5}{3n^{2}-7n+9}\right)^{n}$ I'm not really sure how I should approach this limit. Would really appreciate if someone could say the in ...
2
votes
1answer
47 views

Double Integral with a Delta Function

Consider the integral $$\int_0^b\int_0^a\delta(x-y)f(x,y)dxdy$$ where $b>a$. I know that we need to integrate over the larger range first (i.e do the $y$ integral) and then do the remaining ...
2
votes
1answer
31 views

Optimization problem involving rectangles [Calculus 1]

Spent WAY too long trying to figure this out and I just don't know what I am doing wrong. "A rectangular region is to be fenced using 5100 feet of fencing. If the rectangular region is to be ...
2
votes
3answers
44 views

Question about varying density of a sphere to find its mass.

I have a question about the process to find the mass of a sphere with a varying radial density in respect to the radius. It's something really simple, but I would like someone to explain it me. Say ...
2
votes
2answers
48 views

Find $\sum\limits_{n=1}^{\infty}\frac{n^2}{2^n}$ using the function $f(x)=\frac{1+x}{(1-x)^3}$

Find $\sum\limits_{n=1}^{\infty}\frac{n^2}{2^n}$ using the function $f(x)=\frac{1+x}{(1-x)^3}$ Power series representation of $f(x)$ is $\sum\limits_{n=1}^{\infty}n^2x^{n-1}$. Question: Why is ...
2
votes
2answers
59 views

Derivative of 1-norm

Let $A\in \mathcal{R}^{n\times n} $ be a real matrix and $x_0,x\in \mathcal{R}^{n\times 1} $. What is the gradient of the following function at $x$ assuming $Ax \neq x_0$: $\|Ax-x_0\|_1$
2
votes
2answers
20 views

Second Derivative Of A Parametric Function

If $ y = 2t^3 + t^2 + 3$ $ x = t^2 + 2t + 1 $ then what is $d^2y \over dx^2$ for t = 1? This is the question. What I tried is that, I first took the derivative using the rule $dy/dt \over dx/dt$, ...
2
votes
2answers
55 views

Linearizing a function involving an integral about a point

Find the linearization of $$g(x)= \int_0^{\cot(x)} \frac{dt}{t^2 + 1}$$ at $x=\frac{\pi}{2}$. I know to find linearization I first plugin the $x$ values into my function $g(x)$: $g(\pi/2)$. Then ...
2
votes
1answer
55 views

Area of segment of folium of descartes

I was looking through past exams for a multi course I'm taking and came across this problem which I'm not sure how to approach. Find the area of the two regions formed by the part of the folium of ...
2
votes
2answers
42 views

Does $\frac{f'(x+h)- \frac{f(x+h)-f(x)}{h}}{h}$ converge uniformly to the second derivative?

I have arrived at an expression $$\frac{f'(x+h)- \frac{f(x+h)-f(x)}{h}}{h}$$ for a compactly supported function $f \in C_C^{\infty}(\mathbb{R}).$ Now I was asking myself, whether we have uniform ...
2
votes
1answer
50 views

Lagrange multipliers with two constraints.

I have been working on the following problem. I need to find a minimum of the following function: $$\ 6x-y^2+xz+60=0 $$ subject to the following constraints: $$ z-x+y=0 \\ x^2+y^2+z^2=36 $$ I ...
2
votes
2answers
28 views

indefinite integral by substitution

I know we are supposed to post as far as we get on a problem in order not to waste people's time, and I understand the reason for that. But I really couldn't get very far on this problem. I am not ...
2
votes
1answer
72 views

Integral of $(\log x)^2e^x$

How to calculate $\int_0^\infty e^x(\log x)^2$. I have tried putting $x=\log(t)$ but can't reach the solution.
2
votes
1answer
35 views

Actuarial problem. Amortization Problem in a 25 year period at 5% [duplicate]

From The problem statment: What is the monthly payment for a $800,000 mortgage for the first 119 payments that is due in 10 years, has a 25 year amortization, at 5% interest? What is the amount ...
2
votes
3answers
33 views

Find the double integral by changing to polar coordinates [duplicate]

So I have the following double integral $$\int_{-2}^{0} \int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}} \sqrt{x^2+y^2} dydx$$ If I integrate with respect to y first I get: ...
2
votes
2answers
61 views

$ \lim x^2 = a^2$ as $x$ goes to $a$

Prove that $\displaystyle \lim_{x \to a} x^2 = a^2$ Let $\varepsilon > 0$, and let $\delta = \min(\frac{\varepsilon}{2|a|+1}, 1)$. Suppose $x \in\mathbb{R} - \left\{a\right\} $ and $|x-a| < ...
2
votes
4answers
48 views

What is the function represented by the power series

What is the function represented by the below power series? $$\sum_{k=1}^{\infty} \frac{x^k}{k}$$ I know that the function will be derived from the summation of $k=0$ to $\infty$ of $x^k$ but I ...
2
votes
1answer
16 views

Finding critical points of a function.

The question reads y=x-4x/(x+1) It also asks what the critical points are and the min and max values. I know that I need to figure out the derivative and then set that equal to zero to find the ...
2
votes
2answers
40 views

Finding a dominating function to evaluate arctan integral

I want to find a dominating function to evaluate the limit of the integral $$ \int_{-\infty}^{\infty} \arctan\left(\frac{1}{n^2}(b - x)\right) - \arctan\left(\frac{1}{n^2}(a - x)\right) \; dx $$ as ...
2
votes
1answer
72 views

Find the limit of $\frac{1 - x}{e^{-x}}$ where x goes to infinity

I know the answer to this is negative infinity through simple analysis, however I am able to prove this using algebra. L'hopital's rule does not apply (or so I think) due to ...
2
votes
1answer
44 views

Is my intuition correct about vector field?

Let us take a $\mathbb{R}^2 $ coordinate system and in it let us create a vector field of acceleration/force - the vector field will be: $$\vec{r(x,y)} = 0 \boldsymbol{\hat{\textbf{i}}} -9.8 ...
2
votes
1answer
51 views

Sum about sine function

Is it possible to calculate the following in closed form? $$\sum _{k=1}^{\infty } \left(\frac{2 x}{k}-\frac{\sin \left(\frac{2 \pi n x}{k}\right)}{\pi n}\right)$$ It does not seem convergent for any ...
2
votes
1answer
32 views

Intersection of three period functions

Let $f(x)=\frac{1}{2}-|\frac{\sqrt{3}}{2}x-1/2|$ for $x\in [0,\frac{2}{\sqrt{3}}]$ and $f(x+\frac{2}{\sqrt{3}})=f(x)$ for all $x\in\mathbb{R}$. $g(x)=\frac{1}{2}-|\frac{1}{2}x-1/2|$ for $x\in [0,2]$ ...
2
votes
2answers
56 views

Prove $ \underset{x\rightarrow0}{\lim}f(ax+b)=\underset{x\rightarrow b}{\lim}f(x)$

Assume that $\underset{x\rightarrow b}{\lim}f(x)$ exists. Show that $\forall a,b\in\mathbb{R}$, if $a\neq 0$ then we have: $$ \underset{x\rightarrow0}{\lim}f(ax+b)=\underset{x\rightarrow ...
2
votes
1answer
27 views

Compute the length of a parametric curve.

It seems like I am not using the good process to compute the length of a given parametric curve. I am not sure if it's inside my calculations or if the steps I use are not correct. The equation of ...
2
votes
1answer
35 views

Shape Of A Blimp.

Was playing around with solids of revolution, the shape given by rotating $y=\sqrt{\sin x}$ about the $x-$axis seems to resemble a blimp. The only thing I can find out about the natural shape of ...
2
votes
3answers
28 views

Let $A\subset R^{n}$ . Then $A$ is disconnected iff there exists a continuous and surjective functon $f:A\to${0,1}

Let $A\subset R^{n}$ . Then $A$ is disconnected iff there exists a continuous and surjective function $f:A\to${0,1} How can I prove this? To prove $\rightarrow$, I know that if $A$ is disconnected, ...
2
votes
1answer
37 views

Geometric Sequences in Manufacturing

I came up with this problem: The manufacturer of a company plans to produce and sell $8000$ units per year. Each year, 10% of the units become inoperative. So basically, I need a definition for this ...
2
votes
1answer
68 views

Find the general solution to $y'' + 4y' + 4y = x^{-2}e^{-2x}$

Find the general solution to $y'' + 4y' + 4y = x^{-2}e^{-2x}$ My attempt: For $y''+4y'+4y=0$ The auxiliary equation is : $m^2+4m+4=0$ $(m+2)^2=0$ $y_p = ...
2
votes
2answers
25 views

Area between undefined curves

Let $R$ be the region enclosed by the $x$-axis and $g(x)$. If the area of region enclosed by $f(x)=x^{2}-x$ and $g(x)=ax-x^{2}$ is $9$, calculate the area of $R$. I've found that the $x$-intercept of ...
2
votes
1answer
30 views

Is an analytic function equal to its Taylor series at endpoints?

Is a real analytic function equal to its Taylor series at the endpoints of the interval of convergence, provided the series converges a the endpoint? I.e. is a power series continuous at the endpoints ...
2
votes
1answer
50 views

Solving a differential equation by separating variables

How can one solve the following differential equation by the technique of separation of variables? $$\frac{1}{x^2}\frac{dy}{dx}=y^5\ \ \ \text{ when }, \ y(0)=-1$$
2
votes
2answers
42 views

Is my integration of $\sqrt{4z^2 - 4z + 2}$ correct?

I'm trying to $$ \int \sqrt{4z^2 - 4z + 2}\ dz $$ the integrand I first rewrite to (completing the square) $$ 4\left(z^2 - z +\frac{1}{4} - \frac{1}{4}\right) + 2 = 4\left(z - \frac{1}{2}\right)^2 ...
2
votes
2answers
89 views

How to get the area between these $2$ functions?

I have a function: $(a)$ $r = 4\cos(2\theta)$ $(b)$ $r = 4\sin(2\theta)$. I need at least a set up for the integral that will yield the area inside the rose (a) but outside the rose $(b).$ I cant ...
2
votes
1answer
24 views

Function series - pointwise/ uniform converges?

Let the function series:$$S(x) = \sum_{n=1}^\infty \frac{x^2}{(1+x^2)^n}$$ study it's converges (point-wise/uniform) in $\mathbb{R}$. So we have: $$S(x) = x^2 \sum_{n=1}^\infty ...
2
votes
3answers
63 views

Prove $\lim_{x\to 7/4^+}\tfrac{3x}{4x-7}=\infty$ by definition

$$\lim_{x\to \frac{7}{4}^+}\frac{3x}{4x-7}=\infty $$ I want to prove that for every $M>0$ exists $\delta$ for which $ 0<x-\frac{7}{4}<\delta $ such that $ f(x)>M $ What I tried: ...
2
votes
1answer
62 views

$f: [0, 1] \to \mathbb{R}$ with $f = 0$ a.e. on $[0, 1]$ and range $\mathbb{R}$? [closed]

How do I construct $f: [0, 1] \to \mathbb{R}$ with $f = 0$ almost everywhere on $[0, 1]$ and range $\mathbb{R}$?
2
votes
2answers
78 views

derivative on both sides

I am reading about feedback topologies and having some problems about math. I am not a math student so I need your help. Could you explain if the operation of taking derivative of both sides correct ...
2
votes
4answers
509 views

Is this a correct/good way to think interpret differentials for the beginning calculus student?

I was reading the answers to this question, and I came across the following answer which seems intuitive, but too good to be true: Typically, the $\frac{dy}{dx}$ notation is used to denote the ...
2
votes
1answer
43 views

Vertical arrow meaning in math?

What does the symbol $\uparrow$ in this context means: Consider a function $f$ of $x$, $f: \mathbb{R}\rightarrow \mathbb{R}$ and a set $G\subseteq \mathbb{R}$. Let $1_G:= \begin{cases} 1 & ...