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

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0
votes
2answers
45 views

Find the function $h(x) = g(2g^{-1}(x))$

Show that the function $g(x) = x^4 + x^3 + 1$ is one-to-one on [0, 2]. In addition, for the function $h(x) = g(2g^{-1}(x))$, find h′(3). For the first part, I manage to prove that g(x) is increasing ...
2
votes
1answer
43 views

I need help showing this inequality

Let $f: \mathbb{R}\rightarrow \mathbb{R}$ be a twice differentiable function such that $f'>0$, $f''<0$, and $f(0)=0$. I need to show, that for every $x>0$: $\frac{f(x)}{f'(x)}>x$ Thanks ...
3
votes
2answers
91 views

Series Word Problem

So the questions is: A ball dropped from a height of 13 feet begins to bounce. Each time it strikes the ground, it returns to $\frac 34$ of its previous height. What is the total distance traveled by ...
0
votes
4answers
114 views

Limit: $\lim_{x\to 0} \frac {x-\sin x}{x-\tan x}$

Here's the question: $$\lim_{x\to 0} \frac {x-\sin x}{x-\tan x}$$ I've used l'Hospitals to get $$\lim_{x\to 0} \frac {1-\cos x}{1-\sec^2x}$$ I then tried to use it again, resulting in $= ...
2
votes
1answer
86 views

Alternating series, does the series converge or diverge?

Does the series $\sum _{n=0}^{\infty }{\frac { \left( -1 \right) ^{n}}{\sqrt [3]{n+1}}}$ converge or diverge? The series can be written as $\sum _{n=1}^{\infty }{\frac { \left( -1 \right) ...
1
vote
1answer
51 views

Is the function $\frac{1}{\sqrt{|x_1|}}$integrable on the unit sphere $S^{n-1}\subset\mathbb{R}^n$?

Is the function $\frac{1}{\sqrt{|x_1|}}$integrable on the unit sphere $S^{n-1}\subset\mathbb{R}^n$? That is, is the integral $$\int_{S^{n-1}}\frac{1}{\sqrt{|x_1|}}d\sigma(x)$$ finite? Where $\sigma$ ...
6
votes
1answer
55 views

Alternating Series and Convergence

The question is: Approximate the value of the series within an error of at most $10^{-4}$. $$ \sum_{n=1}^\infty \frac{(-1)^{(n+1)}}{(n+79)(n+73)} $$ According to $$|S_N - S| ≤ a_{N+1}$$ what is ...
1
vote
1answer
27 views

How do I do Linearization at a point that lies on a curve?

I keep applying the formula to the info given but I keep getting lost/weird answers. Can someone please help me? I know $L(x)=f(a)+f'(a)(x-a)$ question Y(x) satisfies $x^2y^2 + xy = 6$. Point (x,y) ...
0
votes
2answers
186 views

Help to evaluate $\int\ t^2 \sin\beta t\, dt$

$\int\ t^2 \sin\beta t\, dt$ How do I treat $\sin\beta t$ ?
0
votes
1answer
1k views

Volume of a curve rotated around the y-axis

"Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the curves $y = 3+2x-x^{2}$ and $y+x=3$ about the y-axis. Below is a graph of the bounded ...
1
vote
0answers
64 views

Number of zeros of Wronskian

Is there some relation between the number of zeros of a Wronskian and properties of given functions? Having Wronskian (e.g. $2$ x $2$) $$W(x)=\left|\begin{array}{c}f_1(x) & f_2(x)\\f'_1(x) & ...
1
vote
0answers
38 views

Prove $\lim \limits_{u \to \infty} (a(u) − u) = 0$

Question: If the normal line to the curve y = ln x at the point (u, ln u) intersects the x-axis at the point (a(u), 0), prove that $\lim \limits_{u \to \infty} (a(u) − u) = 0$. I tried doing the ...
1
vote
3answers
97 views

Proof that $\lim_{n\to\infty} (n+1)^{1/n} = 1$

How would I go about showing that $$ \lim_{n\to\infty} (n+1)^{1/n} = 1 $$ without using L'Hopital's rule? Through writing a MATLAB code, I confirmed that it is $1$ - I just need to formally prove ...
0
votes
1answer
19 views

Count if there are more smaller or larger number in a result made of multiple Xs?

Firstly, I am not a math pro or something. So far I was able to use math for my needs, but now I am at my wits' ends. Here is the deal. These things are known to me: ...
1
vote
1answer
35 views

Inverse Laplace transform of R(s)

Inverse Laplace transform of I have split it into Bu I am facing difficulty to find Inverse Laplace of First part Thanks in advance
0
votes
1answer
58 views

Convergence of the infinite series $2^x\ln(1+1/3^x)$

The Q: determine whether the series converges or not $$\sum_{k=1}^\infty 2^k\ln(1+1/3^k) $$ So far I figured out that the function is positive and decreasing on [1,infinity). I decided to try ...
2
votes
3answers
63 views

Is the graph of $r^2 = 4$ a circle with radius $2$?

If $r^2 = 4$, taking the square root of both sides will give me $r = 2$, so its graph is a circle with radius $2$. Is this correct? I just wanted to make sure because $r^2$ might imply another graph.
1
vote
2answers
40 views

absolute and conditional convergence

Hi! I am working on some online homework problems on absolute and conditional convergence for my calc2 class. I am really struggling with this one problem though. I set S= the summation of n=1 to ...
1
vote
3answers
75 views

What is the equivalent polar equation of $x^2 + (y-1)^2 = 1$?

It's a question in the textbook that I have and I am having a hard time understanding it. How am I supposed to get the polar equation with this format?
1
vote
1answer
642 views

What is the cartesian equation of $r = 1 + r \sin(\theta)?$

There are no values given for $r$, or $\theta$. How do I derive the cartesian equation for this? It's a question from a textbook I have.
2
votes
0answers
32 views

Implicit differentiation and rules

I'm supposed to write the rules used for some differentiable functions. I got all of them correct except for the last one which is $d(x^c)$. I put in $cx^{c-1}$ because I thought it was the power ...
1
vote
2answers
24 views

Implicit differentiation at a given point

I was given this problem on my online homework, but I'm getting stuck when trying to get the $dy/dx$ term on one side. The question is: "For the equation given below, evaluate $dy/dx$ at the point ...
2
votes
1answer
37 views

Series Problems

I was hoping someone could assist me with this problem. I'm confused as to what they are asking me for? I found that S is equal to 1/4 but what do they want for S sub n?
0
votes
2answers
69 views

How do I find $(f^{-1})'(a)$? [closed]

if $$f(x) = 3x^3 + 3x^2 + 6x + 9 $$ $$a = 9$$ and also $$f(x) = 2x^3 + 3\sin x + 3\cos x$$ $$a = 3$$ I know I have to find the inverse but I think I’m getting overly complicated answers and my ...
0
votes
2answers
60 views

Is there exist the integral of the consine integral?

Is there exist integral of the consine integral? http://www.wolframalpha.com/widgets/view.jsp?id=dc816cd78d306d7bda61f6facf5f17f7 When i input $\int ci(x) dx$ into the calculator, it says no results ...
3
votes
1answer
1k views

Finding the derivatives of inverse functions at given point of c

Hoping someone can help me the understand the steps to solve a problem like this. I'm guessing it involves the formula: $\frac{d}{dx}f^{-1}(f(x))=1/f'(x)$. Am I right in this assumption? I would post ...
1
vote
1answer
56 views

When people could use taylor's theorem to evaluate the integral and when cannot?

When people could use taylor's theorem to evaluate the integral and when cannot? Give one or two counter example please. There must be something that the taylor's theorem is incapable with otherwise ...
0
votes
1answer
88 views

How do I properly set up this optimization equation?

So I've been the given the task to fully optimize any packaging. I chose a DS game box. So first I took the measurements of the cartridge itself ($3.5 \text{ cm} \times 3.3 \text{ cm} \times 0.38 ...
0
votes
5answers
116 views

Find $\lim \limits_{x\to 8} {\frac{64-x^2}{8-x}}$

I'm making a bunch of limits exercises and I found me stuck with this limit: $$\begin{align} \lim_{x\to 8} {\frac{64-x^2}{8-x}} \end{align}$$ It gives me an indetermination of $$\begin{align} ...
1
vote
2answers
36 views

Function composition and differentiability

This problem asks for an example of functions $f$ and $g$ such that $g$ takes on all values, $f \circ g$ and $g$ are differentiable, but $f$ is not differentiable. I'm having trouble jumping straight ...
0
votes
3answers
60 views

Finding interval of convergence for series

Find the interval of convergence and radius of convergence for the series: $$ \sum_{n= 0}^\infty \frac{x^n}{3^n} $$ I'm not sure if I'm correct, but would the interval of convergence be $(-3,3)$ ...
0
votes
1answer
57 views

Indefinite Integral Question - What kind of substitution?

I've been trying to solve this integral for the past two hours, but haven't gotten anywhere: $$ \int \frac {dx}{2\sqrt{x-4}+x} $$ I've tried various kinds of substitutions to no avail. Even just ...
0
votes
1answer
34 views

$L^p$ norm of the harmonic polynomial restricted to $\mathbb{S}^2$

Let $\phi(x)=(x_1+ix_2)^k$, $x\in\mathbb{R}^3$. I want to know the asymptotic property of its $L^p$ ($1\leq p< \infty$) norm when restricted to $\mathbb{S}^2$, i.e., I want to know for which ...
-1
votes
1answer
84 views

Darboux sums. find an integral . [closed]

If $$ f(x)=\cos^{17}{x}$$ I need to prove that $$\int\limits_0^\pi{\cos^{17}{x}}=0$$ with Darboux sums. Thanks
3
votes
1answer
37 views

Partial limits and sets of indices

Let $\{a_n\}$, and $A_1...A_n \subseteq \mathbb{N}$, such that $A_1\cup A_2... \cup A_n= \mathbb{N}$. We denote $P_k$ as the set of all partial limits (= limit of a subsequence) with indices $\in$ ...
0
votes
1answer
46 views

some integral inequality

I need help to prove the following integral inequality. f is continuous function in [-1,1]. $$ \int_{-1}^{1}f^{2}(x)dx \geq \frac{1}{2}\left(\int_{-1}^{1}f(x)dx\right)^2 + ...
0
votes
2answers
44 views

Doubt about Taylor's polynomial to approximate f(x)

Use the degree 2 Taylor polynomial of $f(x) =$ $\sqrt[3]{1728 + x}$ to approximate $\sqrt[3]{1731}$ and give a bound for the error. To obtain the degree 2 Taylor's polynomial, I computed the second ...
0
votes
1answer
34 views

Calculate the Derivative of a univariable integral at a point $4$

Considering the function below: the objective is to calculate $F'(4)$ (the derivative of $F(x)$ in the point $4$)? we know that: and that: so if I try to replace $x$ by $4$ in $F'(x)$ I get ...
13
votes
2answers
236 views

Need help with $\int_0^\infty\frac{e^{-x}}{\sqrt[3]2+\cos x}dx$

Please help me to evaluate this integral: $$\int_0^\infty\frac{e^{-x}}{\sqrt[3]2+\cos x}dx$$
0
votes
1answer
91 views

Taylor Approximation

I have got this question Suppose that $f$ is twice differentiable at every $x\in\mathbb{R}$ and that for every $x\in\mathbb{R}$ $$f''(x) + f(x) = 0.$$ Show that if $f(0) = 0, f'(0) = 0, $ and ...
0
votes
1answer
45 views

$D=\{(x,y)\mid x\ge y,\,y \ge -2,x+y \le 1\}$ calculate $\iint_D (2x-3y)\,dx\,dy$

Draw the domain $D=\{(x,y) \mid x\ge y,y \ge -2, x+y \le 1\}$ and calculate the integral $$\iint_D (2x-3y)\,dx\,dy.$$ I have this problem solved but I don't understand how to calculate the limits and ...
3
votes
1answer
32 views

Show there must be another “partial limit”

Let $P$ the set of all partial limits of the sequence $\{a_n\}$ (partial limit = limit of a subsequence). It's given that $\{0,2\} \subseteq P$ and $\forall n\in \mathbb{N}. \left| {a_{n+1}-a_n} ...
0
votes
1answer
35 views

$-2(\sin x+2\cos 2x)=0$

I am finding the 2nd derivative critical values for graphing a trig function. So far I have it simplified to $$-2(\sin x+2\cos 2x)=0$$ What values for x make this equal zero? And is there a ...
0
votes
1answer
31 views

Integration by parts in a bit abstract form

Given some function $f: \mathbb{R} \rightarrow \mathbb{R}$, I would like to express: $$ \int_{z_{min}}^{z_{max}} z f(z) \mathrm{d}z $$ in the terms of $\int_{z_{min}}^{z_{max}} f(z) \mathrm{d}z$ (I ...
0
votes
1answer
54 views

Specify a function that majorizes $\frac{2}{\pi}\sqrt{1-x^2}, -1\leq x \leq 1$ [closed]

Specify a function that majorizes $\frac{2}{\pi}\sqrt{1-x^2}, -1\leq x \leq 1$ Could anyone please help me? I dont have a clue how to start.
3
votes
0answers
85 views

Prove the converse of convolution theorem

I am trying to prove the converse of convolution theorem: $$ \mathscr{F}[f(x)g(x)]=\frac{1}{\sqrt{2\pi}}\,\widetilde{f}(\omega)*\widetilde{g}(\omega)$$ I try to apply the definition of convolution ...
4
votes
4answers
91 views

L'Hospital's rule problem

$$\lim_{x\to 0^+}(x^{x}-1)\ln(x)$$ I need to solve this by L´Hopital´s rule: this is an indetermination of the type $0 \cdot \infty$: $$\lim_{x\to 0^+}(x^{x}-1)\ln(x)=\lim_{x\to 0^+}{(x^{x}-1)\over ...
3
votes
4answers
109 views

Optimization with a constrained function

Okay so I understand how to find points of extrema when for example, We have $3x^2 + 2y^2 + 6z^2$ subject to the constaint $x+y+z=1$. I followed the method of the Lagrange multiplier and resulted in ...
0
votes
2answers
90 views

Proving two sequences converge to the same limit $a_{n+1}\frac{a_n+b_n}{2} \ , \ b_{n+1}=\frac {2a_nb_n}{a_n+b_n} $

$\text{We have two sequences}$ $(a_n), (b_n)$ where $0<b_1<a_1$ and: $$a_{n+1}=\frac{a_n+b_n}{2} \ , \ b_{n+1}=\frac {2a_nb_n}{a_n+b_n} $$ Prove both sequences converge to the same ...
0
votes
2answers
41 views

Memorylessness and its square

If we have that $T$ is a memoryless random variable, how do we know if $T^2$ is one too? I am supposed to investigate the cases $T: \Omega \rightarrow \mathbb{R_{\ge 0}}$ and $T: \Omega \rightarrow ...