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

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2answers
68 views

Proof: $f(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$ has a limit of $\lim_{x \rightarrow \infty}$ $a>0$

I was give this to prove: This is the proof I have provided. Please bear with me, I have never written a proper proof so I am going with my gut on writing this proof. I am sorry if this makes some ...
0
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2answers
50 views

How to find bounds for $x$ and $y$ for this triple integral?

I want to find the volume of the region enclosed by $z=x^2+y^2$ and $z=x+y$. How can I find the bounds for $x$ and $y$?
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2answers
505 views

What is Leibnitz rule for a double integral?

My question is similar to this one here but a bit different. I want to know the result of the following operation. I'm familiar with Leibnitz' rule but can't see how to extend it to this case. If it ...
0
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1answer
26 views

Positivity of mixed polynomial

I have posted a related question before (link), but I didn't really get a completely satisfactory answer, and also I believe that I was able to simplify the problem a little. Therefore, I hope that it ...
2
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1answer
91 views

Implicit derivative and related rates

I'm having a bit of trouble with this assignment. I had to miss a couple class, and I'm afraid the teacher talked about this. I have the notes, but they are not of much help and don't talk about ...
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5answers
87 views

when $f(x)^n$ is a degree of $n$ why is useful to think $\sqrt{f(x)^n}$ as $n/2$?

I have come across this question when doing problems from "Schaum's 3000 Solved Calculus Problems". I was trying to solve $$\lim_{x\rightarrow+\infty}\frac{4x-1}{\sqrt{x^2+2}}$$ and I couldn't so I ...
1
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1answer
71 views

Intergal Calculation

Can someone suggest me how to solve this integal $$ I = \int_0^\infty {\frac{{\sqrt {ax + b} }}{x}} {e^{ - \mu x}}dx $$ with $a,b,\mu$ are postive real numbers.
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1answer
50 views

Can you use indefinite integration to prove equivalence of two functions?

Is it always the case that if: $$ \int f(x) dx = F_1(x) + C $$ and $$ \int f(x) dx = F_2(x) + C $$ then $$ F_1(x) = F_2(x) $$ and why? Is it a legitimate way to prove the equivalence of two ...
2
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3answers
728 views

If $f'(x) = \sin{\frac{\pi e^x}{2}}$ and $f(0)= 1$, what is $f(2)$?

If $f'(x) = \sin{\dfrac{\pi e^x}{2}}$ and $f(0)= 1$, then what will be $f(2)$? This is what I tried to find the antiderivative of $f'(x)$ with u-substitution, $$ \begin{align} u &=\frac{\pi ...
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2answers
60 views

A little confused. Indefinite integral help.

$\displaystyle\int\tan(5x+1)\,\mathrm{d}x$ Okay, so Im having trouble with this problem. I get the answer as $-\frac15\ln(\cos(5x+1))+C$ every time I do this question but It says it is incorrect.
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2answers
40 views

Help with calculating another integral.

Not sure what identity I would be using here.
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5answers
88 views

Need help evaluating the definite integral.

$$ \int_0^6 \frac{dx}{x^2+36} $$ This question is killing me. Ive done it 5 different times and not one answer was right.
1
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2answers
173 views

Is instantaneous rate of change rigorously defined for a curve that is not a function?

Algebraic Definition Consider the following definition for the instantaneous rate of change, $m$, for some value $x$: $$m = \lim_{h \to 0}\frac{f(x+h) - f(x)}{h}$$ The above definition (usually ...
3
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2answers
836 views

Integral of Natural Logs

I had this problem on an integral test today. I tried using u substitution but to no avail. Integral: $\int (1+\ln(x))x\cdot \ln(x)dx$.
3
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2answers
159 views

Does $\int_{1}^{\infty} \frac{\mathrm dx}{x^{\alpha}-1} $ converge?

I did a quick search here but couldn't find a similar problem (it's probably out there somewhere...) I'm stuck with this rather simple improper integral: $\int_{1}^{\infty} \frac{1}{x^{\alpha}-1} ...
2
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2answers
266 views

Minimizing arc length of $y=\ln(\cos x)$

Suppose you know that the difference between $a$ and $b$ is $2$. How can we find the values of $a$ and $b$ (with $-2\leq a,b\leq 2$) which minimizes the arc length of the curve $y=\ln(\cos x)$ from ...
2
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3answers
508 views

Having a tough time finding the area between two curves.

Find the area between $y=9\sin x\,$ and $\,y = 10\cos x\,$ over the interval $[0, \pi]$. Having a tough time with this problem. Im good with finding the area between two curves but the sin and ...
3
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2answers
805 views

I need assistance in integrating $ \frac{x \sin x}{1+(\cos x)^2}$

Find the integral $$ \int_0^{\pi} \frac{x \sin x}{1+(\cos x)^2}$$
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3answers
91 views

What allows one to use a change of variables?

I find myself often using a change of variables or coordinates to solve various integration problems. Some are easy to justify for me such as integrating over a circle in Cartesian vs polar ...
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5answers
104 views

An inverse of the function $e^x$

How can I prove that the function $L(x)=\int_1^xdt/t$ which is definte on $(0,\infty)$ is an invers of $\exp(x)$. Should I work on $L(x)\circ e^x=e^x\circ L(x)=x$. I am stuck.... Thanks.
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1answer
62 views

Where does the linearity of differential operators come from?

I just quoted the linearity of a differential operator, namely d/dz, in a proof and I was wondering where the root of this linearity lies. All of the differential operators which I have encountered ...
2
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2answers
1k views

Bringing a limit inside of an infinite sum

Is uniform convergence justification for bringing a limit inside of an infinite sum? I was trying to evaluate $\displaystyle \lim_{n \to \infty} n \int_{0}^{1} \ln(1+x^{n}) \ dx $ and found that it ...
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4answers
170 views

Proving ${(n+2)!}< \left(n(\sqrt{2}-1)+\sqrt{2}\right)^{n+2}$

While simplifying an inequality, this inequality was derived: $${(n+2)!}< \left(n(\sqrt{2}-1)+\sqrt{2}\right)^{n+2},\quad\quad\quad\quad n\in \mathbb{N}$$ Do you have any idea to prove it? It is ...
0
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1answer
601 views

Finding the 4th order Taylor expansion of $g(t)= t^3 + 2t^2 + 2t + 1$

Given the function $$g(t) = t^3 + 2t^2 + 2t + 1$$ I would like to find the 4th order expansion of $g(t)$ at $t=t_1$. So far, I have performed the differentiation of $g$, up to $g'''(t)$ w.r.t. $t$, ...
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3answers
55 views

Find approximation of $y= {x^2}$

I have a function $\large{f(x)=\sqrt{x} \space \space \space x \forall \geq 0}$ I am looking for a quadratic approx. to $f(x)$ at $x=9$. So far, I know that the quadratic approx. at $x = x_0$ ...
2
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2answers
247 views

How to get rid of this exponentiation

I would like to remove exponentiation in this equation $y=x^t$ and use only multiplication or division. I have x, t and ln(x). Is it possible?
0
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1answer
25 views

Quantify product-appropriateness score

I'm looking for a way to quantify and explain the scenario below to my managers. I'm really good at understanding issues, but unfortunately no so good at communicating them to others. I have written ...
7
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1answer
279 views

On Magnitudes of Sums of Roots of Unity and a Simple Trigonometric Inequality

The Problem Let $r,q,m$ be positive integers such that $4 \leq r$ and $1<m,q\leq r/2$. Is it the case that $$\left | \sum_{k=0}^{q-1} \zeta^{km}\right | < \left |\sum_{k=0}^{q-1} ...
5
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3answers
382 views

Non-equivalence of D'Alembert's and Cauchy's criterion?

Is there a simple example where D'Alembert's and Cauchy's criterion (the root test) for convergence of infinite series don't agree, i.e. one of them proves inconclusive? Can you explain why that ...
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1answer
80 views

Shell theorem for a general potential

I have read that the inverse square potential outside a spherical shell is the same as that due to a point mass/charge at the origin of the same mass/charge, and that in general, for the Yukawa ...
3
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2answers
2k views

Generalized mean value theorem

I know and understand the mean value theorem. But at the moment I don't have the intuition to understand the generalized mean value theorem If $f$ and $g$ are continuous on the closed interval ...
7
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2answers
4k views

Difference between “undefined” and “does not exist”

What is the difference between the terms "undefined" and "does not exist", especially in the context of differential calculus? Most calculus materials state, for example, that $\frac{d}{dx}{|x|}$ ...
10
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2answers
185 views

Evaluate $\lim_{x\to1^-}\left(\sum_{n=0}^{\infty}\left(x^{(2^n)}\right)-\log_2\frac{1}{1-x}\right)$

Evaluate$$\lim_{x\to1^-}\left(\sum_{n=0}^{\infty}\left(x^{(2^n)}\right)-\log_2\frac{1}{1-x}\right)$$ Difficult problem. Been thinking about it for a few hours now. Pretty sure it's beyond my ...
1
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1answer
109 views

Derivative at a point (Linear approximation at point)--what is the valid range for approximation?

Derivative at a point $x=c$ for $f(x)$ gives linear approximation (approaching a tangent) at that point. But what is the range around $x=c$ for this approximation is valid? What books say is only ...
1
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1answer
46 views

Can this theorem be extended to sequences and other functions?

If $\lim_{n \to \infty} |a_n| = 0$, then $\lim_{n \to \infty} a_n = 0$ I read the proof by Squeeze Theorem, and it doesn't seem like the limit can only be $0$. I wonder if I can extend it to ...
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2answers
169 views

Question about convergence of series if $\{n a_n\} \to 0$

This is part of Rudin's PMA Exercise 3.14 (d). If I understand correctly, it would be helpful to prove the following: Let $a_n$ be some sequence. Assume that $\lim_{n\to\infty} na_n = 0$. Prove ...
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2answers
97 views

smoothness of multivariable functions

What does it mean for a function $\mathbb{R}^n \to \mathbb{R}^m$ to be smooth? I see this in books, but typically we only talk about smoothness when the target set is $\mathbb{R}$.
1
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1answer
125 views

Optimization, two numbers

The sum of two nonnegative numbers is 36. Find the numbers if A) the difference of their square roots is to be as large as possible. B) the sum of their square roots is to be as large as possible. ...
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4answers
89 views

rewrite $2ie^{i\pi}+i^3$

i am asked to rewrite $2ie^{i\pi}+i^3$ into $x+iy$ form. i just tried all what i know so far, but couldnot come to solution. i said: $2ie^{i\pi}+i^3=2ie^{i\pi}-i$ but further i am stuck really. i am ...
2
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4answers
146 views

How do I integrate $\int \frac{x+1}{2\sqrt{x+1}} \mathrm dx$?

I know how to do the bottom part, but I can't figure it out how to get $x+1$ on top $$\int \frac{x+1}{2\sqrt{x+1}} \mathrm dx$$
2
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4answers
603 views

Best practice book for calculus

I tried with every inch in me to not ask a question such as this but I just couldn't resist asking this. What is the best Calculus practice book? I tried looking around but couldn't find a ...
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2answers
44 views

Limits, chain help please?

We have the chain $f: \mathbb{N} \rightarrow \mathbb{R}$ where $f(n)= \frac{2}{7}, \frac{5}{12} \cdots \frac{3n-1}{5n+2}$ and $f(1) =0$, so I have to find the limit and study the convergence. If the ...
2
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1answer
134 views

Maclaurin expansion of $\sqrt{\cos 2x}$ and $\tan^2 x$ up to degree 4

Find the Maclaurin expansion $\sqrt{\cos(2x)}$ and $\tan^2x$ up to degree $4$. I tried differentiation but it gives me something really horrible.
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2answers
182 views

How to integrate a probability density function with coefficient?

I want to ask how we can integrate the following function, $f(T)$, which is near Gaussian. $$ \large \intop_{-k}^k \frac{2a}{x^{2}} \cdot {e^ {-(\frac{a}{x}-u)^{2}/b^2}} dx$$ Thanks,
0
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1answer
146 views

What is the derivative of $f(x)=(\sin^{3}(5x))^{\frac{1}{4}}$?

What is the derivative of the following function? $$f(x)=(\sin^{3}(5x))^{\frac{1}{4}}$$ So I did the chain rule and I got $$(\frac{1}{4})((\sin^{3}(5x))^{-3/4})(3\sin^2(5x))(\cos(5x))(5).$$ Does ...
0
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2answers
3k views

What is the average rate of change over this interval?

A boat at anchor is bobbing up and down in the sea. The vertical distance, $y$, in meters, between the sea floor and the boat is given as a function of time, $t$, in hours, by $$6 + \operatorname{sin} ...
5
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2answers
539 views

What does the long '|' mean? [duplicate]

I have often seen the long '|' symbol with a subscript expression afterward. What does this mean in mathematics? Here is an example I found from Wikipedia: $$\large\left.\frac{dy}{dx}\right|_{x=c} ...
2
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3answers
67 views

How to find this limit?

I have the limit $$\lim_{x\to0}\frac{\pi/2-\arccos{(x^2)}-x^2}{x^6}.$$ Can somebody please explain to me why this limit exists (according to wolfram alpha it is $1/6$)? Using the standard properties ...
2
votes
2answers
65 views

How to integrate $\frac{1}{1 + a^2 \tan^2x}$?

Can you please help me out with evaluating this integral? $$\int_0^{2\pi} \frac{1}{1 + a^2 \tan^2x}dx$$
0
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1answer
362 views

Proof of convergence of a telescoping series

Show that the telescoping series below converges if and only if the $\lim_{j\to\infty} c_j$ is defined and finite. $$\sum_{j=1}^{\infty} c_j - c_{j+1}$$ Not really sure where to start for ...