Mathematics Stack Exchange
Mathematics Stack Exchange

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
7
votes
3 answers

Find the maximum value of a function of $x$ for a given range of values of $x$

What is the most efficient and reasonable way to find the maximum value of a function of $x$ within a given range of $x$ values? For example, given the function $f(x)=3\sin(x)+0.01x^2$, how can I find the maximum between $x=0$ and $x=37$, inclusive.…
Mario
  • 173
7
votes
3 answers

Proving roots with Mean Value Theorem

Stewart wants me to prove stuff but I have no idea how to. a) Show that a polynomial of degree 3 has at most three real roots. b) Show that a polynomial of degree n has at most n real roots.
user138246
7
votes
1 answer

integration with indicator function

I was trying to solve the following simple integration involving indicator function $I_{(a,b]}$ in a journal article. Here are the equations (in LaTeX notation): $$ f(u) = \int_{0}^{1} (I_{(0,s]}(u) - s)\; ds\tag{1} $$ $$ g(u,v) = \int_{0}^{1}…
Wayan
  • 71
7
votes
3 answers

Computing $f(x)$ if $\int_1^{xy} f(t) dt = y \int_1^x f(t) dt + x\int_1^y f(t) dt$ and $f(1) = 3$

This is a problem from Apostol, Calculus, Volume I, Chapter 6.9 (p. 238), that I was hoping someone could help with: A function $f$, continuous on the positive real axis, has the property that $$\int_1^{xy} f(t) dt = y \int_1^x f(t) dt + x \int_1^y…
user23784
7
votes
2 answers

Expressions for the second derivative

Suppose that $f$ has continuous second derivatives. How do I show that $$\frac{f(x+h) + f(x-h) - 2f(x)}{h^2}$$ and $$2\frac{f(x+h) - f(x) - f'(x)h}{h^2}$$ both tend to $f''(x)$ as $h \rightarrow 0$? For the first expression, I can rewrite it…
user0192351
  • 73
7
votes
1 answer

Integrating reciprocals of functions with known antiderivatives

If $$\int_{}^{} f(x)\,dx$$ is known, is there a way to directly find $$\int_{}^{} \frac{1}{f(x)}\,dx$$
AHumbleCoder
  • 71
7
votes
3 answers

$ \int_{0}^{2} (2x - x^2)^n dx $ recurrence relation

Given $$ I_n = \int_{0}^{2} (2x - x^2)^n dx $$ Compute $I_2$ I simply expanded it into $$ \int_0^2 4x^2 - 4x^3 + x^4 dx $$ and computed it. Show that $$ (2n+1)I_n = 2nI_{n-1} $$ I first tried doing integration by parts by writing it as $…
andreas.vitikan
  • 1,140
7
votes
2 answers

Showing y≈x for small x if y=log(x+1)

Given: $y=\log(1+x)$ Show that $y≈x$ if $x$ gets small (less than 1). I don't think we're supposed to use Taylor series (because they were never formally introduced in class), but I do think we have to differentiate and show that the derivative of…
Matthew
  • 81
7
votes
1 answer

How would you answer this integral using integration by parts?

$$\int \frac {xe^{2x}}{(1+2x)^2}dx$$ I've tried setting $u=e^2$ and $dv=\frac {x}{(1+2x)^2}$ but I'm getting a really long partial answer like: $$\int \frac {xe^2}{(1+2x)^2}dx = e^{2x}\left[\frac {\ln (1+2x)}{4}+ \frac {1}{4(1+2x)^2}\right] -…
mopy
  • 1,825
7
votes
2 answers

Maximization problem

I've been trying to solve the following problem from Stewart's Calculus Textbook for a while without any success. My answer makes sense, but I'm looking for a way to solve it analytically. The problem concerns a pulley that is attached to the…
Jashin_212
  • 255
7
votes
1 answer

Simple Partial Fractions Question

For practice, I am integrating, $$\int \frac{x}{3x^2 + 8x -3}dx$$ So, I can then factor it as, $$\int \frac{x}{(3x-1)(x+3)}dx$$ By partial fractions, I decompose $$\frac{x}{(3x-1)(x+3)}= \frac{A}{3x-1} + \frac{B}{x+3}$$ For finding $A$, I multiply…
Samuel Reid
  • 5,072
6
votes
1 answer

Indefinite Integral of a Function multiplied by Heaviside

I want to do the following integral: $$\int^\ x*H(x-a) dx $$ In mathematica I get that $$\int^\ x*H(x-a) dx = \frac{1}2*(x-a)*(x+a)*H(x-a)$$ where H(x-a) Is the heaviside function. But by hand I can't find the same result.What I'm trying to do is…
Aquele Mesmo
  • 115
6
votes
2 answers

Why can we only talk about derivatives on an open interval?

For instance, in my calculus class, all theorems are in the following form: For example, Rolle's theorem: If $f(x)$ is continuous on $[a,b]$, differentiable on $(a,b)$ ... (etc) My question is, when presented with a closed interval, why must we talk…
Jason
  • 3,563
6
votes
5 answers

Limit $\frac{0}{0}$ which tends to $\frac{\pi}{2}$

I'm trying to evaluate the following limit: $$\lim_{x\rightarrow\pi/2}\frac{\cos(x)}{(1-\sin(x))^{2/3}}$$ The limit has the form $\frac{0}{0}$, I've tried using L'Hopital's rule but I can't resolve it. Any idea?
Alejandro
  • 703
6
votes
2 answers

Intuition on Mean Value Theorem

The following is my opinion, please correct it if I'm wrong or not good explanation: if we get $a, b$, then there must be a constant changing-rate $m$ of $(a, f(a))$ and $(b, f(b))$, if the function go this straight line, then everything point has…
Xingdong
  • 2,091
1 2 3
…
99 100