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Questions tagged [calculus]

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

Calculus is the branch of mathematics studying the rate of change of quantities, which can be interpreted as slopes of curves, and the lengths, areas and volumes of objects.

Calculus is divided into differential and integral calculus, which are concerned with derivatives

$$\frac{\mathrm{d}y}{\mathrm{d}x}= \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}$$

and integrals

$$\int_a^b f(x)\,\mathrm{d}x = \lim_{\Delta x \to 0} \sum_{k=0}^n f(x_k)\ \Delta x_k,$$

respectively.

134529 questions
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Evaluate the integral $\int_0^{\pi} \frac{dx}{a^2\sin^2{x}+b^2\cos^2{x}}$ where $ab\neq 0$

I’ve got some problems evaluating the integral $\int_0^{\pi} \frac{dx}{a^2\sin^2{x}+b^2\cos^2{x}}$. I’ve found a solution in my textbook as follows: The integral $\int…
闫嘉琦
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Given $f(x+y)=f(x)+f(y)+x^2y+xy^2$ ,find $f'(x)$

This question has already been asked but the solution was not satisfactory Suppose $f$ is a function satisfying the equation $$f(x+y)=f(x)+f(y)+x^2y+xy^2$$ for all real numbers $x$ and $y$ Suppose also that $\lim \limits_{x \to…
PiGamma
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Integration of $\int{\frac{e^{\arctan{x}}}{\sqrt{1 + x^2}}dx}$

I am not familiar with integral a lot but I know most of important rule. $$\int{\frac{e^{\arctan{x}}}{\sqrt{1 + x^2}}dx}$$ My method : I assume $u = \arctan{x}$ so $\displaystyle du = \frac{1}{1 + x^2}dx$ and I get this…
Amin
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Evaluate this limit

$$\lim\limits_{x \to 0}\frac{(n+1)^x +(n+2)^x +....+(2n)^x -n }{x}$$ I tried L'Hospital's Rule, I got $$\lim\limits_{x \to 0} \frac{(n+1)^x ln(n+1) +(n+2)^x ln(n+2) +...+(2n)^x ln(2n) }{1}$$ $$=ln(n+1) +ln(n+2) +.....+ln(2n)$$…
Hussien Mohamed
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Compute $\lim_{\theta\rightarrow 0}\frac{\sin{(\tan{\theta})}-\sin{(\sin{\theta})}}{\tan{(\tan{\theta})}-\tan{(\sin{\theta})}}$

I rewrote it by writing the tan as sin/cos and cross multiplying: $$\frac{\sin{(\tan{\theta})}-\sin{(\sin{\theta})}}{\tan{(\tan{\theta})}-\tan{(\sin{\theta})}}=…
Parseval
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Given $f(x)+f''(x)=2\cos x$ and $f(0)=f'(0)=0$, prove $f'(x)\sin x = f(x) \cos x + \sin^2 x$

Given $f$ is twice differentiable, $f(x)+f''(x)=2\cos x$ and $f(0)=f'(0)=0$, prove $$f'(x)\sin x = f(x) \cos x + \sin^2 x$$ and $$f'(x) \cos x + f(x) \sin x = x + \sin x \cos x.$$ I've tried having LHS be $g(x)$ in a blind attempt to…
kevinbobbkoh
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Evaluate $\int \frac{dx}{x\sqrt{4x^2+1}}$

I was evaluating $\int \frac{dx}{x\sqrt{4x^2+1}}$ using Table of Integrals. My work I was evaluating $\int \frac{dx}{x\sqrt{4x^2+1}}$ using Table of Integrals. I found in the Table of Integrals an integral that is akin to $\int…
fitzmerl duron
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Spivak Calculus $13-33$

This question is from Spivak's Calculus Chapter $ 13$ Integrals Problem $33$ (it deals with upper and lower sums which is what the $L$ and $U$ denote): I can't seem to do part (b). I tried just defining the $U(f,P)$ but can't get anything that…
helios321
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Limit Of a Sequence involving square-Root

How can I use the fact that $\sqrt[n] {n} \to 1 $ in order to calculate the limit of the sequence: $ \sqrt[n] {9n^2 + 30n + 17} $ ? Thanks
yuta
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Solve $2\tan{2x}\leq3\tan{x}.$

Problem: $$\text{Solve} \quad 2\tan{2x}\leq3\tan{x}.$$ A problem of this character will yield 5 points on an exam. However, having the correct answer does not suffice to get all the 5 points. Full stringency and mathematical accuracy, on top of a…
Parseval
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Max and min of $f(x)=\sin^2{x}+\cos{x}+2$.

My first try was to look for when $\cos{x}$ and $\sin{x}$ attain their min and max respectively, which is easy using the unit circle. So the minimum of $\sin{x}+\cos{x}$ has maximum at $\sqrt{2}$ and minimum at $-\sqrt{2}.$ But if the sine-term is…
Parseval
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How to prove a property of integrals (Spivak's Calculus - Chapter 13, Problem 16)

I'm trying to self study a bit of rigorous calculus before starting university and I would love some help with this problem. Prove that $\int_{ca}^{cb} f(x)dx=c\int_{a}^{b}f(cx)dx$ Thank you! As someone said, by the situation of this problem on…
Ignacio Encinas Rubio
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What does the equation $x^2+24xy+68y^2=0$ represent?

The equation $x^2+24xy+68y^2=0$ represents a Ellips Parabola Hyperbola Can't be decided I know the general equations of all of these geometric figures, but I can't rework the given equation to match any of them. Completing the square w.r.t. $x$, I…
Parseval
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Spivak - Calculus. Chapter 1 Problem 21 - what is the point of the theorem?

I proved it but I don't understand what was the point of proving it. I can rephrase this theorem: If you have a rectangle with sides lengths $x_0$ and $y_0$ and you want to lengthen or shorten each side in such way that area of the rectangle…
CrabMan
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Maximizing the Integral

Find the interval $[a,b]$ for which the value of the integral $\int_{a}^{b} (2+x-x^2)dx$ is maximized. To solve this problem, I believe I need to the largest interval over which the integrand is nonnegative. To that end, $2+x-x^2 \ge 0$ if and…
user193319
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