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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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What exactly do these integral functions mean?

$$F(x) = \int{f(x)}\,dx$$ $$G(x) = \int_0^x{g(z)}\,dz$$ I am confused about the exact meaning about these functions. The second function is clear to me, $G(x)$ is just the area under the graph of $g(x)$ from $0$ to some $x$. But the first function…
Jason
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Evaluation of the limit, $\lim \limits_{x\rightarrow\infty} \left(\frac{20x}{20x+4}\right)^{8x}$, using only elementary methods

I was assisting a TA for an introductory calculus class with the following limit, $$\lim_{x \rightarrow \infty} \left(\frac{20x}{20x+4}\right)^{8x}$$ and I came to simple solution which involved evaluating the "reciprocal" limit $$\lim_{z…
Alex Troesch
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Differentiate.$ y = \frac{7x}{ 6 − \cot x}$

I have no idea what I have done wrong. Please criticise.
Cetshwayo
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Looking for Calculus exercise book with really good and complete solutions

I have the "Calculus 3-d Edition, Michael Spivak". The book itself is really nice, it explains the stuff very well. However, not all sample problems in the book have their solutions in the "answers" sections. Many are left out. Unfortunately, it is…
azerIO
  • 153
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Computing integrals in terms of $\pi$

My question is from Apostol's Vol. 1: One-variable calculus with introduction to linear algebra textbook. Page 94. Exercise 17. We have defined $\pi$ to be the area of a unit circular disk. In Example 3 of Section 2.3, we proved that…
George Apriashvili
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Linear approximation $y= \ln(1+x)$ for small x

How can I show with linear approximation that $y \approx x$ for small x? I know the rule $$f(x) \approx f(a) + f^{\prime}(a) (x-a),$$ but I don't know how to put it to use in this case.
Theodorus
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When does a line integral equal an ordinary integral?

I was working through a proof, and it changed a line integral into a normal integral. It was analogous to the below, where $\cos(\theta)\,dl=dr$ and $C$ is a path from $b$ to $a$: $$\int_C F(r)\cos(\theta)\,dl=\int^{r_a}_{r_b} F(r)\,dr$$ Why is this…
Quantum spaghettification
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How to modify the conditions so that the theorem is true.

Let $f(x)$ and $g(x)$ be two increasing and differentiable functions from $\Bbb{R}$ to $\Bbb{R}$. If $f'(x)>g'(x)$ $\forall x$ then there exists an interval $[a,\infty)$ for some real number $a$ for which $f(x)>g(x)$. This theorem is false, as…
curious
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Confused by Calc II question regarding derivative of rational integrals

So here's the question: If $f$ is a quadratic function such that $f(0) = 1$ and $\int \frac{f(x)}{x^2(x+1)^3}\,dx$ is a rational function, find the value of $f’(0)$. What I've done so far is try to solve the integral using partial fractions…
Leslie
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Prove that $(a^{-1})^{-1}=a$

How does one prove that if $a \ne 0$, then $(a^{-1})^{-1}=a$? My friend (I'm trying to help her) has in her class notes: $a+(-a)=0$ and $(-a)+a=0$ implies that $a=-(-a)$ by the uniqueness theorem. Why does $a+(-a)=0$ and $(-a)+a=0$ imply…
Jeff
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Uniform continuity allows pushing limit inside integral?

Let's say I have a (uniformly) continuous functions $f:[a,b] \to \mathbb{R}$ and an arbitrary function $h:\mathbb{R}^2 \to [a,b]$ such that $$ \lim_{t\to 0} ~h(t,x) = h(0,x) = x $$ for all $x$. I would like to be able to conclude…
nullUser
  • 27,877
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Calculating limits for indeterminate forms

I am asked to calculate the limit as $x\to0$ of: $$ \frac{e^x+e^{-x}-2}{1-\cos(3x)} $$ I believe this is an "infinity/infinity" problem where i could directly apply L'Hopital's rule. Is this right? how would this limit be found?
Allison Rodriguez
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Is this a valid way of taking the derivative.

$\def\lnx{\ln x}\def\lny{\ln y}$ The problem is find $f'(x)$ of $f(x)=x^{2\lnx}$ Here's my approach: Let $$y=x^{2\lnx}$$ $$\lny=\lnx^{2\lnx}$$ $$\lny=2\lnx\cdot\lnx$$ $$\lny=2(\lnx)^{2}$$ $${d\over dx}\lny = {d\over dx}2(\lnx)^{2}$$ $${1\over…
Nolohice
  • 465
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contractive sequence?

A sequence $\left({a_{n}}\right)_{n\in\mathbb{N}}$ is contractive iff there exists a constant $c$, with $0
Grigorios Kostakos
  • 273
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The proof use intermediate value theorem?

Let $x_1, \dots, x_n$ distinct points in $[\alpha, \beta]$, and $y_1, \dots, y_n$ reals with same sign. Assume that $f: [\alpha, \beta] \rightarrow \mathbb{R}$ is continuous. Then, prove that exists $\lambda \in (\alpha, \beta)$ ( or conform @arovai…
Student
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