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
5
votes
2 answers

Trigonometric problem (problem from a Swedish 12th grade ‘Student Exam’ from 1932)

The following problem is taken from a Swedish 12th grade ‘Student Exam’ from 1932. The sum of two angles are $135^\circ$ and the sum of their tangents are $5$. Calculate the angles. Is there a shorter/simpler solution than the one presented below…
mf67
  • 853
5
votes
1 answer

What is the value of $e^\frac{-1}{e}$?

As the title suggests, what is the value of $e^\frac{-1}{e}$?? I never understood how things to the power of irrational numbers are calculated. The more broad question would be what is the minimum value for the function $y=x^x$, such that $x \geq…
John Liu
  • 419
  • 3
  • 11
5
votes
1 answer

Volume of solid rotated about the x-axis

I am to find the volume of the area $R$ bounded by the curve $x=y^2+2$, $y=x-4$ and $y=0$. I have already found the points of intersection by first setting the lines equal to each other and used the quadratic formula: \begin{align*} y^2+2=y+4 …
Mampenda
  • 409
5
votes
5 answers

How to determine the image of a set?

Let $$f(x) = \dfrac{x}{1+x^2}.$$ I am trying to determine if the image $f([0,2])$ is closed or not closed. I think it's not closed but I don't know how to show it.
user39794
5
votes
1 answer

If $f$ is strictly increasing and strictly convex, is $f'(x)^2 > f(x) f''(x)$ for all $x$?

Question Suppose $f: \mathbb{R}_+ \rightarrow \mathbb{R}_+$ with $f(0) = 0$, $f'(x) > 0$ for all x, and $f''(x) > 0$ for all $x$. Is $f'(x)^2 > f(x) f''(x)$ for all $x$? My Attempts Attempt 0: This is true for the special case in which $f(x) =…
Christian Fong
  • 51
5
votes
2 answers

A dumb question on continuity and differentiability of function

Consider $$f(x)=\begin{cases} \frac {x}{e^x-1} & \text{if x $\ne$ 0}\\ c & \text{if x = 0} \end{cases}$$ I know that if $f(x)$ is continuous, then $$c = \lim_{x\to 0} \frac {x}{e^x-1} = 1$$ because $\frac {x}{e^x-1}$ is not continuous at $x=0$. Now…
Gerald
  • 99
  • 5
5
votes
0 answers

Spivak Calculus Chapter 3 Problem 22

I am stuck on understanding the solution of the (b)-part of this problem for quiet some time now. The (a)-part was very easy to solve. I still include the (a)-part, because Spivak uses it to solve the (b)-part. (a) Suppose $g=h \circ f$. Prove…
Carlevaro99
  • 271
5
votes
2 answers

Use L'Hopital's rule to evaluate $\lim_{x \to 0} \frac{9x(\cos4x-1)}{\sin8x-8x}$

$$\lim_{x \to 0} \frac{9x(\cos4x-1)}{\sin8x-8x}$$ I have done this problem a couple of times and could not get the correct answer. Here is the work I have done so far http://imgur.com/GDZjX26 . The correct answer was $\frac{27}{32}$, did I…
Kot
  • 3,273
5
votes
3 answers

How does the total derivative account for dependencies behind variables (intuitively)?

https://en.wikipedia.org/wiki/Total_derivative Suppose we have a multi-variable function $f(x,y)$ and we say that $y$ is parameterized as a function of $x$. Why does the total derivative thing account for directly subbing in the y into the…
tryst with freedom
  • 11,538
5
votes
1 answer

Triple Integrals in Spherical Coordinates

Use spherical coordinates to to find the volume of a solid bounded above by $x^2 + y^2 + z^2 = z$ and below by $z$ $=$ $\sqrt{x^2 + y^2}$
thebottle394
  • 129
5
votes
2 answers

Suppose that $f$ is continuous on $\mathbb{R}$ and $\int_{x-1}^{x}f(t)\,dt=x^{2}$. Find $f(x)$.

Suppose that $f$ is continuous on $\mathbb{R}$ and $\int_{x-1}^{x}f(t)\,dt=x^{2}$. Find $f(x)$. Using that Fundamental Theorem of Calculus, I get $f(x)-f(x-1)=2x$.
user73482
  • 51
5
votes
2 answers

Evaluate $\int_1^\infty (\log(x)/x)^{2011} \; \mathrm dx$

Evaluate $$\int_1^\infty (\log(x)/x)^{2011}\; \mathrm dx$$ I have this question in my book of problems and I'm stumped. I could use some help seeing how this works! Thanks a bunch.
Ryan Carter
  • 343
  • 3
  • 6
  • 11
5
votes
6 answers

When finding relative minimum/maximum, what is the point of using the second derivative test?

Why use the second derivative test over the first derivative test when finding maxima and minima if it's uncertain what $f''(x) = 0$ is? Why not just always use the first derivative since we need to take the first derivate either way?
Blue
  • 135
5
votes
2 answers

Find the arc length of the cardioid: r = 3-3cos θ

This is what I have so far: Using the formula $\mathrm ds = \sqrt{r^2 + \left(\frac{\mathrm dr}{\mathrm dθ}\right)^2}$ $$\frac{\mathrm dr}{\mathrm d\theta} = 3\sin\;\theta $$ $$r^2 = 9 - 18\cos\;\theta + 9\cos^2\theta$$ $$\mathrm ds = \sqrt{9 -…
Krysten
  • 779
5
votes
2 answers

Find $f ' (0)$ for the function $f(x) = g(x)/x^2$ when $x\not = 0$ and $ f(x)=0$ for $x=0$

Find $f '(0)$ for the function \begin{align} f(x)=\begin{cases} \frac{g(x)}{x^2}, & \text{if }x \not = 0\\ 0, & \text{if }x=0 \end{cases} \end{align} With \begin{align} g(0)=g'(0) = g''(0) = 0 \\ g'''(0) = 14 \end{align}
user71864
  • 51
1 2 3
…
99 100