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

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2
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1answer
53 views

A question about a continuous function that satisfy certain limits at $\pm\infty$

I got this question: Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function such that $\lim_{x\to\infty}\frac{f(x)}{x^2}$ and $\lim_{x\to -\infty}\frac{f(x)}{x^2}$ exist and are real numbers. ...
2
votes
1answer
67 views

Meaning of $ dx \times dy = k $

Does $ dx \times dy = k $ have a mathematical meaning? What about when considering $y = y(x)$?
2
votes
3answers
118 views

Lagrange Method Problem

I am from engineering background and I am currently studying calculus. I had a question from assignment to be solved from a course on coursera but I could not do it. People have posted solution in the ...
2
votes
1answer
182 views

Integrate $\int^{\ln(2)}_0 (3e^u - e^{2u} - 2)\sin(nu)du$

I'm having trouble integrating this function $$\begin{equation} \begin{split} f(x) & = \int^1_0x(1-x)\sqrt{1+x}\sqrt{1+x}\sin(n \ln(1+x))/[(1+x)^2] = \\ & = \int^1_0x(1-x)(1+x)\sin(n\ln(1+x))...
2
votes
2answers
66 views

Let $a \in \mathbb R$. Use $\lim_{t\rightarrow 0} \frac {\log(1+at)} t = a $ to show $\lim_{t\rightarrow \infty} (1+\frac a t)^t = e^a$

Let $a \in \mathbb R$. Use the definition of the differential quotient to show: $$\lim_{t\rightarrow 0} \frac {\log(1+at)} t = a$$ In my textbook the differential quotient is defined as $$\frac {f(...
2
votes
2answers
346 views

Integral of $\frac{-2}{\sqrt{16-x^2}}$

So I am asked to find the anti-derivative of $$\frac{-2}{\sqrt{16-x^2}}$$ First step I took was to make it easier for me to visualise $$\int{-2(16-x^2)^{-\frac{1}{2}}}dx$$ I let $u = 16-x^2$ So $\...
2
votes
1answer
85 views

Examples of vector field that is continuously differentiable but not conservative?

I am just curious what would be the case in which a vector field ($\vec f :\Bbb R^2 \rightarrow \Bbb R^2$) is well-defined and continuously differentiable on a region R enclosed by a simple closed ...
2
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1answer
155 views

Integral of $e^{(a+ib)x}$

Given the function $f:\mathbb{R}\rightarrow \mathbb{C}$, such that $f(x)=e^{(a+ib)x}$, how can I compute $f'(x)$ and $\int f(x)dx$ ? Certanly, one can use the identity $e^{ibx}=\cos(bx)+i\sin(bx)$ and ...
2
votes
1answer
74 views

Given a differentiable function for every $x \geq 0$, define a differentiable function for every $x$

Given $f(x)$: $f(0)=1$ Positive for every $x \geq 0$ Differentiable for every $x \geq 0$ Let $g(x)= \begin{cases} f(x) & \text{$x \geq 0$}\\ 1/f(-x) & \text{$x \leq 0$} \end{cases}...
2
votes
1answer
140 views

arc length on circle

http://gowers.wordpress.com/2014/03/02/how-do-the-power-series-definitions-of-sin-and-cos-relate-to-their-geometrical-interpretations/#more-5401 I need an explanation of this bit on the blog: One ...
2
votes
3answers
58 views

Compact and neighborhood question

a) if $A$ is closed and $x \not \in A$, then there is a number $d>0$ such that $|x-y|\geq d$ for all $y \in A$ b) if $A$ is closed and $B$ is compact, $A \cap B=\emptyset$, then there is a number $...
2
votes
2answers
98 views

Help with separable differential equation? $\frac{dy}{dx} =2y^2$

I'm new to separable differential equations, and I'm stuck on this question: $\frac{dy}{dx} =2y^2$ Using the initial condition $y(2)=3$, find $y(1)$. So far I've integrated to get $\frac{dy}{dx} =\...
2
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2answers
57 views

Multiplicative version of the principle of Archimedes

Any clear proof of the above theorem is greatly appreciated.
2
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2answers
80 views

Limit of a Logarithm with Different Bases

We are to compute $$\lim_{n->\infty}{\frac{2^{\log_3 n}}{3^{\log_2 n}}}$$ Clearly the bases are reversed between the logarithm and exponents, so I can't seem to find any logarithm or exponential ...
2
votes
2answers
170 views

How does the epsilon-delta definition define a limit?

I understand what the epsilon-delta definition is saying in regards to the distance from a point c and the distance from your limit, but I don't understand how this defines a limit. Any help is ...
2
votes
2answers
2k views

Closest distance between two quadratic curves

I'm having trouble with the following problem : "find the closest distance between $x^2+4y^2=4$ and $xy=4$" I tried to solve using the properties of ellipse and hyperbola, but the relatively tilted ...
2
votes
3answers
473 views

How to answer the question from Calculus by Michael Spivak Chapter 5 Problem 14

Prove that if $\lim\limits_{x\rightarrow0}{\frac{f(x)}x}=l$ and $b\neq 0$, then $\lim\limits_{x\rightarrow0}{\frac{f(bx)}x}=bl$. Hint: Write $\frac{f(bx)}x=b\frac{f(bx)}{bx}$ What happens if $b=0$? ...
2
votes
2answers
1k views

Find the angle of intersection of circles $x^2+y^2-6x+4=0 \ \&\ x^2+y^2-2x-2y-8=0$

Find the angle of intersection of circles $$x^2+y^2-6x+4=0 \\ x^2+y^2-2x-2y-8=0$$ my answer is : 41.14 degrees. but i'm not sure if it's right. please help me.
2
votes
1answer
73 views

When is partial differentiation commutitive

Consider a function: $$f:\mathbb R ^2\to\mathbb R $$ when does $$\dfrac {\partial f(x,t)}{\partial t\partial x}=\dfrac {\partial f(x,t)}{\partial x\partial t}$$ Thinking about it in terms of the ...
2
votes
1answer
2k views

Leibniz Notation Second Derivative Chain Rule?

I believe I understand the chain rule better from a few tutorials as the following: $$\frac{d}{dx}(f(g(x)) ) = \frac{\partial f}{\partial g}\frac{\partial g}{\partial x}$$ But how would you ...
2
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1answer
52 views

Substitution question $\int_{-1}^{1}\frac{1}{(1+x^2)^2}\,\mathrm{d}x\ne\int_{-1}^{1}\frac{-t^2}{(1+t^2)^2}\,\mathrm{d}t$

$$\int_{-1}^{1}\frac{1}{(1+x^2)^2}\,\mathrm{d}x=\frac{1}{2}+\frac{\pi}{4}$$ Using the substitution $x=\dfrac{1}{t}$ we get $$\int_{-1}^{1}\frac{-t^2}{(1+t^2)^2}\,\mathrm{d}t=\frac{1}{2}-\frac{\pi}{4}...
2
votes
2answers
113 views

Integration with trigonometric substitution

I have been stuck trying to figure out an integration problem involving trigonometric substitution. $$ \int \frac{1}{x^2\sqrt{x^2 + 9}}dx $$ So I substituted $$ x = 3\tan\theta $$ $$ dx = 3\sec^2\...
2
votes
3answers
159 views

Evaluating $\int \frac{\operatorname d \! x}{\sin^4{x}+\cos^4{x}+\sin^2{x}\cos^2{x}}$

How do you integrate $$\frac{1}{\sin^4{x}+\cos^4{x}+\sin^2{x}\cos^2{x}}$$ or simply $$\frac{1}{1-\left(\frac{\sin{2x}}{2}\right)^2}.$$
2
votes
3answers
143 views

Question on Integration by substitution

I have confused myself with another study question from Apostol "Calculus" Volume 1. It's Section 5.8 Question 21 which states: Deduce the formulas in Theorem 1.18 and Theorem 1.19 by the method of ...
2
votes
1answer
58 views

How to evaluate this integral containing trigonometric functions?

$$\int_0^\pi\frac{x\sin(2x)\sin\left(\frac{\pi}{2}\cos x\right)}{2x-\pi}dx$$ How to evaluate this integral? Please give me some hints so that I can complete it myself. No complete answers please. ...
2
votes
2answers
73 views

Find the limit $\lim_{n \rightarrow \infty} \frac{2 + (-1)^n}{2^{n+1} + (-1)^{n+1}} $

Find the limit of a) $\displaystyle \lim_{n \rightarrow \infty} \frac{2 + (-1)^n}{2^{n+1} + (-1)^{n+1}} $ and b) $\displaystyle \lim_{n \rightarrow \infty} \frac{a^n - b^n}{a^{n} + b^{n}} $ So im ...
2
votes
1answer
61 views

Can we apply squeeze in that way?

Claim: if $a_n\leq b_n\leq c_n$ for all $n\in \mathbb{N}$ and $\displaystyle\sum\limits_{n=0}^{\infty} a_n,\displaystyle\sum\limits_{n=0}^{\infty} c_n$ are convergent then$\displaystyle\sum\limits_{n=...
2
votes
2answers
77 views

What kind of functions can be Riemann integrable?

I have learned that every continuous, or piecewise continuous function can be Riemann integrated. But then, are there uncontinuous functions that are Riemann integrable? And if there is, can I still ...
2
votes
1answer
194 views

A beautiful identity of $\sin(x)$ [duplicate]

When I was in high school, I had proved that $$\sin^2(x)-\sin^2(y)=\sin(x-y)\sin(x+y) $$ I think it is beautiful since it resembles the identity $a^2-b^2=(a+b)(a-b)$. But I can not find it in ...
2
votes
2answers
75 views

Evaluating an integral in calculus III

Please can someone explain what happened after step 3 , in this upload image of the exercise Please make it clear how the "1/2" came in step 4 , and why we are subtracting the integrals
2
votes
2answers
1k views

Volume of revolution of cardioid

The parametric equations of a cardioid are $x=\cos\theta (1-\cos\theta)$ and $y=\sin\theta (1-\cos\theta)$, $0\le\theta\le 2\pi$. Diagram here. The region enclosed by the cardioid is rotated about the ...
2
votes
1answer
154 views

Partial fraction decomposition of $\frac{1}{x^{2n}+a^{2n}}$

I came across a formula for the partial fraction decomposition of $ \displaystyle \frac{1}{x^{2n}+a^{2n}}$. It seems correct (at least for $n=1,2$, and $3$). But how is it derived? $$\frac{1}{x^{...
2
votes
1answer
173 views

Am I doing something wrong here? Maximization of profit problem.

The cost and price functions for a new Internet search company are reasonably approximated by the following models: $C(x) = 37 + 1.42x – 0.0067x^2 + 0.00011x^3$ $p(x) = 3.7 – 0.007x$ Where $x$ ...
2
votes
2answers
82 views

Convergence/divergence of a series

Given the series The terms of a series are defined recursively by the equations $$a_1 = 2\\ a_{n+1}=\frac{5n + 1}{4n+3}a_n$$ Determine whether the summation of $a_n$ converges or diverges.
2
votes
2answers
53 views

how to calculate $a^x=x$ for $a>0$?

I'm struggling with this equation, but i think i'm pretty close to the solution, but i just can't figure it out, so i hope for some help! $$a^x=x \rightarrow a^x-x=0$$ Now i take derivative for $x$ ...
2
votes
2answers
160 views

Surface area of revolution problems without perfect squares

When calculating surface area of revolution, I often find myself in situations like this : Problem: My work: $f(x)=\frac{x^3}{5}$ $f'(x) = \frac{3x^2}{5}$ $f'(x)^2 = \frac{9x^4}{25}$ $1 + f'...
2
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1answer
1k views

Reading advice for Advanced Calculus by Loomis and Sternberg

I am a math major currently in my sophomore year. I have a sound base in one variable calculus and basic linear algebra. I am currently doing a course in multivariable calculus. I have completed ...
2
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2answers
167 views

Integral from $0$ to $\infty$ of $\ln(x)/e^x$

Show $$\int_0^\infty \frac{\ln(x)}{e^x} = -\gamma$$ (gamma is Euler-Mascheroni constant). Can anyone please prove this result? Also $$ \int_0^\infty \frac{\left( \ln(x) \right)^2}{e^x}\mathrm dx. $...
2
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2answers
136 views

How to calculate Maxima and Minima? [closed]

I have to calculate Maxima and Minima of the below function $x^4-8x^2-15$
2
votes
2answers
536 views

Solve the equations $z^2 + (2 - 2i)z + 2i = 0 $ by completing the square

I tried solving this thing by completing the square and I always end up with something like this $(z^2 + (2 - 2i)z - 2i) + 2i + 2i = 0 $ and it doesn't seem like to me that you can factor the part in ...
2
votes
2answers
83 views

Prove that $f'(x_0)=c$

Let $f:(a,b)\rightarrow \mathbb{R}$, and $x_0 \in (a,b)$. $f$ is differentiable at $(a,b)$. Also, Let $l(x)= cx+d$, "passes" at $(x_0, f(x_0))$. Prove that if $\forall x \in (a,b):f(x) \ge l(x)$ ...
2
votes
2answers
50 views

Proof of a limit without geometry

Can anbody write a proof that $$\lim_{x\to-\infty}-e^x = 0$$ without using a geometrical representation of the function? Is this something that I should know for Calculus I? (I am teaching myself). ...
2
votes
3answers
156 views

Why can't a non-zero polynomial satisfy some equations?

I'm having a hard time visually picturing/understanding how to explain why a non-zero polynomial function cannot satisfy the equation: $f''(x)$ = $-f(x)$ So is it basically asking to explain why a ...
2
votes
2answers
412 views

Can an integral made up completely of real numbers have an imaginary answer?

The question is in the title, but I'll repeat it again: Can an integral made up completely of real numbers have an imaginary value? I understand what an integral is, so my natural inclination would be ...
2
votes
1answer
35 views

Maximum in given series.

Find the maximum term among, $1$, $2^{\frac{1}{2}}$, $3^{\frac{1}{3}}$, $4^{\frac{1}{4}}$, $...$ Now, if we take $f(x) = x^\frac{1}{x}$, and differentiate it is quite simple to see that it reaches it'...
2
votes
1answer
1k views

Solve polynomial congruence using Hensel's lemma

Solve $x^4+2x+46 \equiv 0$ $(\mod 4375 )$ for x. . My attempt: $x^4+2x+46 \equiv 0$ $(\mod 5^47 )$ breaks down to a Chinese Remainder Problem with the 2 following congruence's': (1) $x^4+2x+46 \...
2
votes
4answers
1k views

Rolle's theorem prove polynomial has only 1 root

Prove that $x^3-x-4=0$ has exactly one real root: This is my working so far: suppose $f(x) = x^3-x-4$ has $2$ roots : $a,b$ $f(a) = f(b) = 0$ $f'(x)=3x^2-1$ $f'(x)$ exists on $(a,b)$ so $f$ is ...
2
votes
1answer
77 views

How to integrate this?

$$\int{xsec^2xdx}$$ The solution is unclear to me. I see $u=x$ $du=dx$ and $dv=sec^2x$ $v=tanx$ I know how to integrate with u-substitution, but, am not sure how to put this all together. Is ...
2
votes
1answer
148 views

How to evaluate this integral $\int_0^{\frac{\pi}{2}}(\ln(\tan x))^2dx$?

How to evaluate this integral ? $$\int_0^{\frac{\pi}{2}}(\ln(\tan x))^2dx$$ I changed it to $$\int_{-\infty}^{\infty}\frac{x^2e^x}{e^{2x}+1}dx$$ Thanks in advance.
2
votes
3answers
75 views

$\int \frac{1}{x\sqrt{x^2-5}} \operatorname d\!x$ by substitution?

I need to evaluate the integral $$\int \frac{1}{x\sqrt{x^2-5}} dx$$ The integral should be solved by substitution. I tried substituting $u=x^2-5$, but did not come with an answer.The correct answer ...