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
3 answers

>Prove that $\frac d {dx} x^n=nx^{n-1}$ for all $n \in \mathbb R$.

Prove that $\frac d {dx} x^n=nx^{n-1}$ for all $n \in \mathbb R$. I saw some proof of $\frac d {dx} x^n=nx^{n-1}$ using binomial theorem, which is only available for $n \in\mathbb N$. Do anyone have the proof of $\frac d {dx} x^n=nx^{n-1}$ for all…
JSCB
  • 13,456
  • 15
  • 59
  • 123
4
votes
2 answers

How to solve $\lim_{x\rightarrow \infty} \frac{x+\sin x}{3x+\cos x}$ using L'Hospital's rule

This question is pretty easy if we solve it without using L'-Hospital's Rule. We can say that as $x$ goes to $\infty$, $\sin x$ and $\cos x$ still oscillate between $-1$ and $1$. So, not much effect of them. So, we are left with $\frac{x}{3x}=…
Simran
  • 447
  • 4
  • 13
4
votes
2 answers

A question about free fall, velocity, and the height of an object.

A falling stone is at a certain instant $100$ feet above the ground. Two seconds later it is only $16$ feet above the ground. a) If it was thrown downward with an initial speed of $5$ ft/sec, from what height was it thrown? b) If it was thrown…
Jinzu
  • 819
4
votes
3 answers

$a \leq b$, is $a^b \leq b^a$ correct?

I have just thought about an interview question, it was maybe asked previously, but I thought about it myself. Consider, $a \leq b$, is $a^b \leq b^a$ correct? Justify. I thought about solving it in the following manner, but don't end with a…
QFi
  • 1,215
4
votes
2 answers

Evaluating a linear integral (Calculus I)

I'm given: $$\int_{5}^{-2} [3f(x) + 1]\,dx$$ with the additional information that: $$\int_{0}^{5} f(x)\,dx = 10$$ and $$\int_{0}^{-2} f(x)\,dx = -4$$ My layman mind looks at it as, since the sum of the two function pieces = 6, then the integral is…
Nate222
  • 315
4
votes
1 answer

Need Help With Path Integrals.

Find the mass of a wire formed by the intersection of the sphere $$ x^2 + y^2 + z^2 = 1 $$and the plane $$ x + y + z = 0 $$if the density at $$(x, y, z)$$ is given by $$p(x, y, z) = x^2$$ per unit length of wire. I really have no idea how to start…
Fernand
  • 41
4
votes
5 answers

How is the partial sum of a geometric sequence calculated?

I have been asked to calculate a general formula for $1 + b + 2b^2 + 3b^3 + ... + Nb^N$ I think that a part of this is a geometric sequence, and I have rewritten this as $f(x) = 1 + \sum_{i=1}^n i\cdot x^i$ (I couldn't figure out a way to make the…
Joon
  • 157
  • 1
  • 6
4
votes
1 answer

How to solve this challenging limit?

I believe more than one answer may be correct. I thought about this for a while and I feel like there is a simplification that makes it look a whole lot easier but I can't figure out what it is. I noticed that every term within the limit is…
Tidronicus
  • 51
4
votes
2 answers

Good ways to integrate $\int_0^1 x^{k+1} (1-x)^k dx$?

Let $k$ be a natural number. What are some good ways to integrate $\int_0^1 x^{k+1} (1-x)^k dx$ and $\int_0^1 x^{k+2} (1-x)^k dx$? I wanted to use some symbolic math software but I don't have one, so I have to learn a way to do it manually. Thanks!
Tim
  • 47,382
4
votes
1 answer

Find function from Fourier Series

Problem statement: Let us assume that the Fourier series of an unknown function $f(x)$ is $$ \sum ^{\infty} _{n=1} \frac{\sin(nx)}{n}. $$ Find the function $f(x)$ to which it belongs. Attempt at solution: The first thing I did was plot for $x \in…
jommi
  • 85
4
votes
2 answers

How can I calculate the limit of this?

What is the limit? $$\lim_{n\rightarrow\infty}\dfrac{3}{(4^n+5^n)^{\frac{1}{n}}}$$ I don't get this limit. Really, I don't know if it has limit.
Henfe
  • 1,208
4
votes
2 answers

How to solve $x \log^2x - c=0$?

I'm optimizing parameters to an algorithm to minimize its run time. Substituting some variables to clean up the presentation, I basically need to solve $x \log^2x - c=0$, but I forget how to solve an equation of this form. How can I do this?
Joe
  • 631
4
votes
4 answers

$f(x) = (x-2)(x-4)(x-6) +2$ then $f$ has all real roots between $0$ and $6$. True or false?

$f(x) = (x-2)(x-4)(x-6) +2$ then $f$ has all real roots between $0$ and $6$ $($ true or false$)?$ Here $f(0) = -46$ and $f(6) = 2$ since function is continuous so it must have at least one root between $0$ and $6$, but how to check if it has all its…
Mathaddict
  • 2,300
4
votes
2 answers

Spivak's Calculus: Chapter 12, Problem 26

Suppose that $f(x) > 0$ for all $x$, and that $f$ is decreasing. Prove that there is a continuous decreasing function $g$ such that $0 < g(x) \le f(x)$ for all $x$. To be quite honest, I have no idea how to approach this problem. (I also have no…
Arpon
  • 1,246
4
votes
0 answers

How to determine if a parametric curve is smooth?

The algorithm for identifying whether or not a parametric curve is smooth contains a first step of finding out when dx/dt and dy/dt is simultaneously 0. What are the other steps? My textbook points at that there are functions where dx/dt and dy/dt…
Hung Trinh
  • 53
  • 3
1 2 3
…
99 100