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

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0
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0answers
18 views

Computing the sixth derivative of $F(x) = \int_1^x\sin^3(1-t)\mathrm dt$

Compute the sixth derivative at $x_0 = 1$ of $$F(x) = \int_1^x\sin^3(1-t)\mathrm dt$$ It's from a multiple choice test. I was able to narrow down the choices to $0$ and $60$. I guessed $0$ and ...
0
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0answers
10 views

Is this theorem about integration with substitution wrong?

A theorem in my book states: If $g$ is differentiable, f is continuous, and F is an antiderivative of f, then : $\int f[g(x)]g'(x)dx=F[g(x)]+C$ The reason I am asking if this is correct, ...
-1
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1answer
22 views

When does the integral converges?

For what $\alpha, \beta$ the integral $$\int_0^\frac{\pi}{2} \frac{(\frac{\pi}{2} - x)^\alpha}{(\cos x)^\beta} dx$$ converges? So first I've approved (using WolframAlpha) that $\frac{\pi}{2} - x ...
0
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0answers
27 views

How can I prove this integral?

I have to use the identity $b^4-a^4=(b-a)(b^3+b^2a+ba^2+a^3)$ to prove that: $\int_b^ax^3dx=\frac{b^4-a^4}{4}$. I know that you can just do $F(b)-F(a)$ and since the integral of $x^3$ is ...
0
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0answers
18 views

Non-smooth function which is arbitrarily often partially differentiable?

Are there functions $f:U\subseteq{\bf R}^n\rightarrow{\bf R}$ with $U$ non-empty open such that all partial derivatives $\partial^{i(0)}\partial^{i(1)}\dots\partial^{i(m-1)}f(x)$ with $m\in{\bf ...
0
votes
1answer
27 views

Getting ready for Calculus?

So I wanted to start a Masters program but they require that I have Calculus III. I want to take that course at the university, but I need to be ready for it. As I look at Khan Academy and do some ...
3
votes
3answers
35 views

Inequalities proven by real analysis or induction.

Let $t\in [-1,1]$. Prove that $(1+t)^p+(1-t)^p\ge2$ when $p\ge 1$ and that $(1+t)^p+(1-t)^p \le 2$ where $0 \le p\le 1$. I am not sure how I should solve it. I tried induction at first and it was ...
0
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4answers
50 views

Convergence of $\int_0 ^\infty \frac {dx}{\sqrt {1+x^3}}$

Convergence of $\int_0 ^\infty \dfrac {dx}{\sqrt {1+x^3}}$ Attempt: $\lim_{x \rightarrow \infty} \dfrac {x^{\frac{3}{2}}}{\sqrt {1+x^3}} =1$ Hence, $\dfrac {1}{x^{\frac{3}{2}}}$ and $\dfrac ...
0
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2answers
27 views

calculate $\int_{0}^{2\pi}\frac{1-\sin(t)}{2-\cos(t)}dt$

I need to calculate $\int_{\gamma} \frac{1-\sin(z)}{2-\cos (z)}dz$ where $\gamma$ is the upper hemisphere of the circle with center $\pi$ and radius $\pi$, with a positive direction. The original ...
0
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5answers
48 views

Evaluate the Limit $\lim_{x\to 0} {\left((e^x - (1+x)) \over x^n\right)}$

Evaluate the Limit: $$\lim_{x\to 0} {\left((e^x - (1+x)) \over x^n\right)}$$ I am trying to understand how to do this. I have to use series expansion and not L'Hospital. Any help would be great. ...
0
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0answers
31 views

$S_{1}\iff S_{2}$ in complex numbers

Let : $$a_0 , a_1 , a_2 \in \mathbb{C} \text{ and } :b_0 , b_1 , b_2 \in \mathbb{C}$$ : Show the following equivalence : $$\begin{cases} ( 1 + a_0 ) ( 1 + a_1 ) ( 1 + a_2 ) &=& ( 1 + b_0 ) ( ...
0
votes
0answers
18 views

Function with the opposite definition of Dirichlet function?

I just happened onto the Dirichlet function today that states: $D(x)= \begin{cases} 0 & \text{if $x$ is irrational,}\\ 1 & \text{if $x$ is rational} \end{cases}$ which shows that points can ...
0
votes
1answer
26 views

If a power series converges uniformly on $\mathbb{R}$ then it must be to $0$?

Let $f(x) = \sum a_n x^n$. Let's assume that $f(x)$ has a radius $R=\infty$ and $f(x)$ converges uniformly. Now, obviously $f(0) = 0$. Meaning, $f(x)$ pointwise converging at $x=0$. Since we assumed ...
0
votes
1answer
15 views

Verifying transport equation solution

I have just started PDE's and I have the transport equation $u_t + au_x = 0$ which has the general solution $u(x,t) = f(x - at)$ In a book I'm reading it says this can be verified by substitution ...
0
votes
1answer
8 views

What is the connection between slant/oblique asymptote to the polynomial part of the function and polynomial division?

What is the connection between slant/oblique asymptote calculation to the polynomial part of the function and polynomial division? To find the slant asymptote $y=mx+n$ we can can calculate it in two ...
4
votes
2answers
39 views

$\lim_{n \rightarrow \infty} \frac {1^{a+1}+2^{a+1}+\cdots+n^{a+1}}{n.(1^{a }+2^{a }+\cdots+n^{a })} $

The value of $$\lim_{n=\infty} \dfrac {1^{a+1}+2^{a+1}+\cdots+n^{a+1}}{n.(1^{a }+2^{a }+\cdots+n^{a })} $$ Attempt: $S = \lim_{n \rightarrow \infty} \sum_{n=0} ^\infty \dfrac {k^{a+1}} {n.( 1^{a ...
1
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1answer
44 views

Computing a strange integral

Prove that $(-1)^n \int_{-1}^1 (x^2 - 1)^ndx = \frac{2^{2n+1}(n!)^2}{(2n+1)!}$ This one has me stumped. I've tried the obvious (using binomial theorem and then integrating termwise, or computing the ...
1
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1answer
33 views

Help with troublesome limit [on hold]

I need help in computing the following limit: $$\lim_{x\to 0} \frac{a}{x}~\exp \left(-\frac{a^2(\log(bx))^2}{2}\right)$$
2
votes
2answers
44 views

How to solve $\int{\frac{1}{\sqrt{3-2x-x^2}}\,dx}$?

$$\int{\frac{1}{\sqrt{3-2x-x^2}}\,dx}$$ I tried to do it by substitution with no sucess. Anyone can solve it?
-4
votes
2answers
34 views

How to evaluate $\lim_{x\rightarrow +\infty } \frac{\ln(a+be^x)}{\sqrt{a+bx^2}} =? $

Suppose $b>0$ $\lim_{x\rightarrow +\infty } \frac{\ln(a+be^x)}{\sqrt{a+bx^2}} =? $ I know that this is an indetermination of the form $\infty / \infty$ . I tried to use L'Hôpital, but I didn't ...
2
votes
3answers
40 views

How to evaluate $ \lim_{x\rightarrow +\infty } \sqrt[x]{a^x+b^x} = ? $

If a>0 and b>0, $ \lim_{x\rightarrow +\infty } \sqrt[x]{a^x+b^x} = ? $ What I was trying to do: Suppose a>b. Then, for sufficiently large values of x, $ a^x >> b^x $; so $\sqrt[x]{a^x+b^x} ...
1
vote
3answers
90 views

Anyone can integrate $e^{-\frac{x^2}{3}}$ by hands?

I just used wolfram integral calculator and the result is weird, there is something called error function. $$ \int_{-\infty}^\infty e^{-\frac{x^2}{3}}\,\mathrm dx $$ Hint says that change of variable ...
2
votes
2answers
29 views

Which of the following statements are true $(NBHM - 2015)$?

Let $X =\{f \in C[-5, 5] : f(-5)= f(5) = 0 \} $ There exist $ f \in X$ such that $ f \equiv 2$ on $[-1, 0]$ and $ f \equiv 3$ on $[1 , 2] \cup [3 , 4]$ For every $ f \in X$, there exist ...
1
vote
1answer
33 views

Can a divergent alternating series by rearrangement of terms be made to converge to a value?

Riemann discovered that a conditionally convergent series, through rearrangement of it's terms, can be made to converge to any value. But, if $S$ is a divergent alternating series, through ...
4
votes
2answers
23 views

Investigate the convergence of $\int_1^\infty \frac{\cos x \ln x}{x\sqrt{x^2-1}}$

Investigate the convergence of $$\int_1^\infty \frac{\cos x \ln x}{x\sqrt{x^2-1}}$$ so first of all let's split the integral to: $$I_1 = \int_1^2 \frac{\cos x \ln x}{x\sqrt{x^2-1}}, I_2 = ...
-2
votes
0answers
20 views

Find area of surface $x^2+y^2+z^2=4$ which lies directly above cardioid $p=1-\cos t$ [on hold]

It doesn't appear like a hard problem since the overall formula would be: $$S=\int_0^\pi\int_0^{1-\cos\theta}{\frac 2{\sqrt[2]{4-\rho^2}}}\, \rho\, d\rho d\theta,$$ but then I get stuck on solving the ...
0
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0answers
22 views

Need help solving this integral

$\int_1^\infty du$$\int_{-2}^2 dv(u-v)e^{-u}$ Do I just evaluate the integrals separately and then multiply the answers together?
0
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4answers
56 views

Demonstrate that this equation has three real roots if fullfills this condition

I have this: demonstrate that this equation $$ {x}^{3}+p\,x+q=0 $$ has one real root if "p" is positive and has three real roots if $$ \frac{{q}^{2}}{4}+\frac{{p}^{3}}{27}<0 $$ I did ...
2
votes
0answers
20 views

Check my answer - complex analysis, using residue and rouche's theorem

I was asked the following questions and I am unsure of my solutions, any advice would be appreciated, maybe there is a better way of doing this. Question: We are given $f(z)=2z-\sinh (z)$ defined on ...
1
vote
1answer
35 views

Convergence of $\sum_{n=1}^\infty n^s(\sqrt {n+1} - 2 \sqrt n + \sqrt {n-1})$

Convergence of $$\sum_{n=1}^\infty n^s(\sqrt {n+1} - 2 \sqrt n + \sqrt {n-1})$$ Attempt: $\sum_{n=1}^\infty n^s(\sqrt {n+1} - 2 \sqrt n + \sqrt {n-1}) \sim \sum_{n=1}^\infty n^s( \sqrt n )$ As ...
0
votes
1answer
28 views

Discontinuity of a function

Take $f(x)=\frac{1}{\sqrt{x}}$ on $(0;1)$ and 0 everywhere else on $\mathbb{R}$ take also an enumeration $\{r_n\}_{n\in\mathbb{N}}$ of the rationals on $(0;1)$. And define $g(x):=\sum_1^{\infty} ...
3
votes
4answers
54 views

How do I find the error of nth iteration in Newton's Raphson's method without knowing the exact root

In our calculus class, we were introduced to the numerical approximation of root by Newton Raphson method. The question was to calculate the root of a function up to nth decimal places. Assuming that ...
0
votes
0answers
16 views

Parametrize given curves

I'm given the following curves: $x = y^2 + 1$, $z = x + 5$ I'm eventually trying to find the unit tangent vector, so I need to find the r vector. Could I just assign $y = t$, and then have $<t^2 ...
12
votes
5answers
488 views

Studying math all day and really young [on hold]

I am very young and want to learn algebra and calculus for fun. What should I keep in mind when I start learning? I am going to try the textbooks I have borrowed out: Dummit and Foote and Spivak's ...
3
votes
1answer
67 views

Prove that if $a_n>0$ and $\sum a_n$ converges then $\sum (\frac {b_n}{a_n})$ converges

Let $\{a_n\},\{b_n\}$ be two sequences such that for each $n$ we have $e^{ a_n }= a_n + e^{b_n }$. Show that if $a_n>0$ and $\sum a_n$ converges $\implies \sum \left(\frac {b_n}{a_n}\right)$ ...
0
votes
0answers
14 views

Derivative of a helicoidal trajectory

I am given the position vector: $r(t)=\frac{1}{3}\begin{pmatrix}R\space cos(\omega t)\\R\space sin(\omega t)\\v_0t\end{pmatrix}$ $R$=Radius $v_0$= veclocity in the z-direction I want to calculate: ...
2
votes
2answers
57 views

Does Intermediate Value Theorem $\rightarrow $ continuous?

i try to understand Intermediate Value Theorem and wonder if the theorem works for the opposite side. I mean, if we know that $\forall c\:\:\:f\left(a\right)\le \:c\le \:f\left(b\right)\:,\:\exists ...
0
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0answers
35 views

Understanding Cauchy's mean value theorem

We studied in class today about the Cauchy's mean value theorem, but in somewhat more complicated version, and i find it difficult to prove. here the theorem: Let $f,\ ...
0
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2answers
27 views

Checking when an $a$-dependent function is continuous, differentiable.

For some $a\in \Bbb{R}$ define a function $f_{a}(x) = \begin{cases} {x^{a}\cos{1\over x}}, & \text{if $x$ $\ne$ 0} \\[2ex] 0, & \text{if $x=0$} \end{cases}$. Hints firstly are preferred. b. ...
0
votes
1answer
19 views

Derive inverse Laplace Transform using two given trigonometric transforms (5.2-13)

I am not certain how to begin this problem. Someone please point me in the right direction. Problem Using the two given formulas ($1$ and $2$ below) show that: ...
2
votes
2answers
38 views

Antiderivative of $\frac{\sqrt{4-x}}{x\sqrt{x}}$

I need help to find the antiderivative of the function $\displaystyle x \, \mapsto \, \frac{\sqrt{4-x}}{x\sqrt{x}}$ on $]0,4[$. I have tried the change of variables $u = \sqrt{4-x}$ but it didn't ...
0
votes
1answer
29 views

Determine when $f_{a}(x)$ is bounded.

For some $a\in \Bbb{R}$ define a function $f_{a}(x) = \begin{cases} {x^{a}\cos{1\over x}}, & \text{if $x$ $\ne$ 0} \\[2ex] 0, & \text{if $x=0$} \end{cases}$ What should $a$ be in order for ...
0
votes
2answers
35 views

Finding the limit of the following expression

How can I evaluate: $$\lim_{n \rightarrow ∞} \bigg \{ \frac{n+3}{n+1}\bigg\}^n\quad ?$$ I know the answer is $e^2$, as this is a practice problem from a textbook. However I cannot understand how ...
0
votes
0answers
17 views

Equivalence between Hermitian positive definite and norm [on hold]

How to prove the following two are equivalent: $A$ is a Hermitian positive definite, $\sqrt{x^TAX}$ is norm. Hermitian means that $A^T=\overline{A}$. Positive definite means that ...
0
votes
1answer
36 views

Antiderivative of an even function

I'm faced with an issue in terms of antiderivatives of even and odd functions. Define $f \in C[-a,a]$ where $a>0$. Let $f$ be an even function on $[-a,a]$. We wish to show that $$\int_{-a}^a ...
1
vote
2answers
14 views

Find the general expression from the antiderivative

I am having trouble computing the original function. Question states: Let $f$ be a differentiable, positive function, such that $$f'(x)=x*f(x)$$ for all real numbers x. A) Find the general ...
0
votes
1answer
17 views

Determine average rate of change of function [duplicate]

How to determine the average rate of change of $f(x)= x^5-3x^4$ on the interval $[-2,4]$
0
votes
0answers
7 views

Need Help with this conical container word problem [on hold]

A conical container (r= 7 ft, h = 28 ft) is filled to (h=24 ft) of a liquid weighing 62.4 ft/lb^3. How much work will it take to pump the contents to the rim? r = radius h = height
0
votes
3answers
55 views

Calculus: continuous

Q: if $f$ is continuos on $[0,1]$ with $0\leq f(x) \leq 1$ for all $x \in [0,1]$, prove that there exists $C \in [0,1]$ such that $f(c)=c$. I don't understand why the following proof works: ...
1
vote
2answers
24 views

If the sequence $\{{1\over n^k}\}$ where $n\in \mathbb{N}$ is convergent, then $k\geq 0$ and the limit $0$ for all $k>0$.

If the sequence $\{{1\over n^k}\}$ where $n\in \mathbb{N}$ is convergent, then $k\geq 0$ and the limit $0$ for all $k>0$. What I have: Assume that $k<0$, need to show that this contradicts the ...