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
4
votes
2 answers

Product rule in calculus

This is wonderful question I came across whiles doing calculus. We all know that $$\frac{d(AB)}{dt} = B\frac{dA}{dt} + A\frac{dB}{dt}.$$ Now if $A=B$ give an example for which $$\frac{dA^2} {dt} \neq 2A\frac{dA}{at}.$$ I have tried many examples…
wright
  • 174
4
votes
2 answers

$u_{n}>u_{n+1}>0$ and $u_{2}+u_{4}+u_{8}+u_{16}.....$diverges. prove that $\sum \frac{u_{n}}{n}$ diverges.

$u_{n}>u_{n+1}>0$ and $u_{2}+u_{4}+u_{8}+u_{16}.....$diverges. prove that $\sum \frac{u_{n}}{n}$ diverges. The only thing i found is that $\left | \frac{u_{2^{n+1}}}{2u_{2^n}} \right |>1$.
V150
  • 163
4
votes
2 answers

series of two sequences

Given $$ \sum_{n=1}^\infty(a_n+b_n)$$ converges, and $$a_n \to 0$$ prove that $$a_1 + b_1 + a_2 + b_2 + \cdots$$ converges. The question's intention is to show that $a_n \to 0$ is a sufficient condition for the series to converge. My try: using the…
john_gayl
  • 391
4
votes
6 answers

The $n$'th derivative of $x^x$

I want to know the $n$'th derivative of $f(x)=x^x$. Then, I'll calculate $f(0)$ with Taylor expansion of $f(x)$ on $a=1$. Here is my answer, but it is unfinished. The derivative of $f(x)=x^x$ $$\begin{align} f'(x)&=x^x(\log x+1)\\ f''(x)&=x^x(\log…
user152132
  • 43
4
votes
2 answers

Show that $f \equiv 0$

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ continuous and $$\int_0^1 f(tx) \,dx=0, \forall t\in \mathbb{R}$$ Show that $f \equiv 0$. $$$$ $\int_0^1 f(tx) \, dx=0$ $u=tx \Rightarrow du=t \, dx$ $x=0 \rightarrow u=0, x=1 \rightarrow u=t$ So…
Mary Star
  • 13,956
4
votes
1 answer

How to evaluate $\int \sqrt{x^3+x^4} dx$

Find the integral: $\int \sqrt{x^3+x^4} dx$ I know I could use WolframAlpha, but I wonder if there is a way to calculate this integral in a nice way (some clever substitution perhaps?). Tried a lot of different ways, but don't seem to find a good…
Mateusz
  • 852
4
votes
1 answer

Find the exact value of $t$ that maximizes $\int_{t}^{t+1}\sin e^x \ dx$

The exact value is close to zero judging by the graph. By the fundamental theorem of calculus, $\frac{d}{dt}\int_{t}^{t+1}\sin e^x \ dx=\sin e^{t+1}-\sin e^{t}$. The solution will be given when we solve for $t$ such that $\sin e^{t+1}-\sin e^{t}=0$…
recmath
  • 2,738
4
votes
3 answers

Arc Length and Speed

Can anyone help me with this calculus problem? Calculate the length of the path over the given interval: $c(t)=(5t^2, 10t^3), 1≤t≤3$ I'm not sure how to figure out the equation to graph it. Any help is appreciated!
Christina
  • 325
4
votes
2 answers

How should I simplify this $\tan^{-1}$ expression?

Integral question I have to integrate this function $$I = \int_0^4\frac{20x-5x^2}{x^2+9} \mathrm{d}x$$ obtaining $$20\ln(5/3) + 15\tan^{-1}(4/3) -20.$$ However, my calculator, even after somehow simplifying it a bit, gives this:…
Deo
  • 127
4
votes
2 answers

Is there a way to derive the chain rule using the definition of the differential of a function?

Given a composite function, $ y = (f \circ g)(x) $ that is continuous and differentiable for all $x$, we know from chain rule that $$ \frac{dy}{dx} = \frac{d(f \circ g)}{dx} = \frac{df}{dg} \frac{dg}{dx}$$ So is there a way to arrive at the above…
Anthony
  • 608
4
votes
1 answer

Solid of revolution vs $\iiint$

In calc 1, I learned about rotating a curve around an axis, say $y=x^2$ around the y-axis. In calc 3, I learned about the shape of 3D objects in the context of $\iiint$ triple integrals . These concepts seem very related. No one, however, have…
jacob
  • 2,965
4
votes
1 answer

Showing that the perimeter of a circle is $2\pi r$. Isn't this a definition?

Today, I ran into a calculus class and the professor was calculating the perimeter of a circle using arc lenghts. Even though they seemed to be on the right track, there is a problem: They are using $\sin x$ and $\cos x$ functions, which are…
ThePortakal
  • 5,158
4
votes
1 answer

Am I misapplying L'Hopital's rule?

I have the function $f(x) = \dfrac{x^3}{e^x}$ and I'm trying to find its limit as x tends to negative infinity so that I can sketch the graph. I can see just looking at the function that if I were to sub in any negative number for x it will give me…
Katherine Rix
  • 919
4
votes
4 answers

Finding the limit of a function $n^3 / 3^n$

$$\lim_{n→\infty} \frac{n^3}{3^n} =0 $$ The answer is 0 but how would i go about proving that?
user136088
  • 251
4
votes
2 answers

Optimization, volume of a box

I am having trouble figuring this one out. If $1200cm^2$ of material is available to make a box with a square base and an open top, find the largest possible volume of the box. I know that I need to make a formula to represent the box in terms of…
user138246
1 2 3
…
99 100