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
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Related Rates Calculus Trigonometric Problem

I've been stuck on this related rates problem for a while now and I just can't figure out how to even approach it. The problem goes something like this: The above diagram shows two objects moving at different speeds. Both objects are 0.5 miles from…
Joe Tec
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Why can we "separate" variables?

Possible Duplicate: What am I doing when I separate the variables of a differential equation? My school textbook has a section on differential equations. One of the tricks used is the…
user60469
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A question on the limit $\lim \limits_{n \rightarrow \infty} n \sum \limits_{j=1}^{n} \frac{\cos(\frac{n}{j})f(\frac{n}{j})}{j^2}$

I stumbled upon this question from a Calculus exam: Let $f \in C^1(\mathbb{R})$ be monotonically decreasing such that $\lim \limits_{x \rightarrow \infty} f(x) = 0$. Prove that the limit $$\lim \limits_{n \rightarrow \infty} n \sum \limits_{j=1}^{n}…
Yuval Gamzon
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How to compare $2^{\pi}$ and $\pi^2$ using calculus

How to compare $2^{\pi}$ and $\pi^2$ using calculus I guess $$f(x)=\frac{\ln x}{x}$$ wont help here since $2 \lt e \lt \pi$
Umesh shankar
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$\int\dfrac{dx}{x^2-a^2}$

For evaluating $\int \dfrac{dx}{x^2 - a^2}$, how can we make the substitution $x= a\sec \theta $ because $\sec \theta$ can be 1 and then that would give 1/0 form. So how can we do that and why does it work? Why not use $a\tan \theta$? And: $a^2…
Archer
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What is the relative maximum or minimum and the point of inflection?

$f(x) = ax^3+bx^2+cx +d$, determine a, b, c, and d such that the graph of $f$ has a extreme in $(0,3)$ and a point of inflection in $(-1,1)$. When is a quadratic I know that the formula $V=(\frac{-b}{2a},\frac{-\triangle}{4a})$ gives the maximum…
Ju Grigori
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Trigonometric integral with cosinus

I cannot solve the equation below: I know what I typed is wrong byt I can't understand where it went sour. $$\int \frac{1}{\cos(5x)}…
Dovendyr
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Is this possible

If $r_1, r_2, t_1,$ and $t_2$ are real numbers and if $\left|r_{1}\right|<\left|r_{2}\right|$, $\left|t_{1}\right|<\left|t_{2}\right|$ and…
lisana
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Evaluating $\int_{0}^{\pi\over 2}{\mathrm dx\over \cos(x)+\cos^2(x)}\ln\left({1+a\cos(x)\over 1-a\cos(x)}\right)$

I am trying to evaluate this integral, where $a<1$ $$\int_{0}^{\pi\over 2}{\mathrm dx\over \cos(x)+\cos^2(x)}\ln\left({1+a\cos(x)\over 1-a\cos(x)}\right)$$ It is look obvious to enforce a substitution of $u=cos(x)$ because of it commonly appeared in…
user550936
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Showing that $ae^x=1+x+\frac{x^2}{2}$ has exactly one real root (part 2)

I made this question here Show $a e^x=1+x+\frac{x^2}{2}$ has exactly one real root but I wrote the equation wrong. I reproduce the post with the correct equation: I'm struggling with the following problem: Show that the equation $a…
user55286
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Infinitely differentiable function with compact support

I already know that the function $$ f(x) = \begin{cases} \exp(- \frac{1}{x^2}), \quad x > 0 \\ 0 , \quad x \leq 0 \end{cases} $$ is infinitely differentiable throughout $\mathbb R$. The only real problem, of course, lies in showing that…
Jose L. Lykón
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How to evaluate $\int_0^{2\pi} \frac{d\theta}{A+B\cos\theta}$?

I'm having a trouble with this integral expression: $$\int_0^{2\pi} \frac{d\theta}{A+B \cos\theta}$$ I've done this substitution: $t= \tan(\theta/2)$ and get: $\displaystyle \cos\theta= \frac{1-t^2}{1+t^2}$ and $\displaystyle…
Ada
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Showing that $f(x)$ is increasing on $(0,+\infty)$

I am collecting some easy problems for my students and now I am facing to the following problem: Prove that the function $$f(x)=\left(1+\frac{1}{x}\right)^x$$ is increasing in $(0,+\infty)$. Undoubtedly, they will solve it by using the…
Mikasa
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How would you prove this idea?

Is it true that if the slope of the secant line between A and B is less than the slope of the secant line between B and C, then the slope of the secant line between A and B is less than the slope of A to C? Is this true and if so, how would you…
James
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