# Tagged Questions

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

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### 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. ...
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### Meaning of $dx \times dy = k$

Does $dx \times dy = k$ have a mathematical meaning? What about when considering $y = y(x)$?
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### 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 ...
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### 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 ...
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### Multiplicative version of the principle of Archimedes

Any clear proof of the above theorem is greatly appreciated.
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $b=0$? ...
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### 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.
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### 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 ...
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### 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 ...
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### 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 $$\frac{1}{1-\left(\frac{\sin{2x}}{2}\right)^2}.$$
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### 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 ...
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### 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. ...
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### 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 ...
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### How to calculate Maxima and Minima? [closed]

I have to calculate Maxima and Minima of the below function $x^4-8x^2-15$
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### 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 ...
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### 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 $\forall x \in (a,b):f(x) \ge l(x)$ ...
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### 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). ...
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### 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 ...
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### 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 ...
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'...