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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Limit of $(1+\frac{1}{n^{3\alpha}})^{n^{5}}$

I was trying to solve the $\lim\limits_{n \to \infty}(1+\frac{1}{n^{3\alpha}})^{n^{5}}$ where I have to say for which $\alpha$ parameter the limit is finite. I tried to sobstitute $t=n^{3\alpha}$: $(1+\frac{1}{t})^{{t}^{\frac{5}{3\alpha}}}$ and I…
Tarlo_x
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Finding the points on a curve, closest to a specific point

Find the point(s) on the curve $y^3=x^2$ closest to the point $P=(0,4).$ I understand that there is a way to solve this, using the distance formula, however this turns out to seem rather complicated. I am also aware that there is a calculus method…
user341952
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Is this Series Convergent and does it also Absolutely Converge?

$$\sum\limits_{n=1}^{\infty}a_{n}\ \text{where}\ a_{n} = \left\{ \begin{array}{ll} \frac{1}{n} & n\mbox{ is even} \\ \frac{-1}{n^2} & n\mbox{ is not even} \end{array} \right.$$ I need to check if this series converges and also converges…
Shanon
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Does the derivative with respect to something have to be a variable?

When you take the derivative of an expression with respect to x, does x have to be a variable, or is it allowed to be a polynomial, a term, a vector, or anything else? It doesn't seem to make sense to me if x is not a variable.
wrongusername
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Proving :$ \frac{\textrm{d}}{\textrm{d}x}\int^{g(x)}_{h(x)}f(t)\textrm{d}t =f(g(x))g'(x)-f(h(x))h'(x). $

How to prove that : $$ \frac{\textrm{d}}{\textrm{d}x}\int^{g(x)}_{h(x)}f(t)\textrm{d}t =f(g(x))g'(x)-f(h(x))h'(x). $$
Frank
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Error in "Calculus, a complete course"?

My calculus book says that if $x\le-1$ then: $$\sec^{-1}{x}=\pi-\sin^{-1}\frac{\sqrt{x^2-1}}{x}$$ I have limited experience with mathematics, but my calculator disagreed with the above statement. Shouldn't it…
GambitSquared
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Determine the local maximum and local minimum points for the function $f(x) = 2x^3 + 3x^2 - 12x + 3$.

Determine the local maximum and local minimum points for the function $f(x) = 2x^3 + 3x^2 - 12x + 3$. I know that the local maximum and local minimum points largest and smallest value of the function, but I do not understand how to find the points…
Kate.K
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Why can we use the following integration method?

In order to integrate the following function, our professor taught us the following method: $\dfrac{x^4+1}{x^3+x^2} = \dfrac{x^4+1}{x^2(x+1)}=\dfrac{A}{x^2}+\dfrac{B}{x}+\dfrac{C}{x+1}$ I don't understand why we can do that, in reality,…
aribaldi
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sum of binomial series

How do I solve sum of binomial series which is as follows: $$\frac{1}{3}\sum^{\infty}_{x=0}\begin{pmatrix}x\\y\end{pmatrix}\frac{1}{3}^x$$ I think it would be pretty easy to sum from $y=0$ to $x$, but I have no idea how to sum from $x=0$ to…
user1292919
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Suppose $f(x)$ is a nonnegative convex function in $[0,1]$, prove an inequality.

Suppose $f(x)$ is a nonnegative convex function in $[0,1]$. Prove: $$\displaystyle \int_0^1f^2(x)\,\mathrm dx\leqslant\frac43\left(\int_0^1f(x)\,\mathrm dx\right)^2$$ I have tried Cauchy Mean Value Theorem: Construct $\displaystyle…
Shine Mic
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If $P(x)=2013x^{2012}-2012x^{2011}-16x+8$,then $P(x)=0$ for $x\in\left[0,8^{\frac{1}{2011}}\right]$ has

If $P(x)=2013x^{2012}-2012x^{2011}-16x+8$, then $P(x)=0$ for $x\in\left[0,8^{\frac{1}{2011}}\right]$ has exactly one real root. no real root. at least one and at most two real roots. at least two real…
user1557
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Show that if $f'$ is strictly increasing, then $\frac{f(x)}{x}$ is increasing over $(0,\infty)$

Suppose $f$ is a differentiable function over $\mathbb{R}$ satisfying $f(0) = 0$. Show that if $f'$ is strictly increasing, then $\frac{f(x)}{x}$ is increasing over $(0,\infty)$. Attempt: Since $f'$ is increasing we know that $f'' > 0$ for all…
user19405892
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Prove that if $|f|$ is differentiable at $a$ and $f$ is continuous at $a$, then $f$ is differentiable at $a$

Prove that if $|f|$ is differentiable at $a$ and $f$ is continuous at $a$, then $f$ is differentiable at $a$. We are given $\displaystyle \lim_{x \to a}\dfrac{|f(x)|-|f(a)|}{x-a} = c$ and we know $f$ is continuous at $a$. We must prove that…
user19405892
  • 15,592
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Need help understanding proof for $g(x) = f(cx) \implies g'(x) = c\cdot f'(cx)$

So I used the definition of a limit on $g(x)$ to get: $$g'(x) = \lim_{h \to 0} \frac{g(x+h)-g(x)}{h}$$ then subsituted $f(cx)$: $$g'(x) = \lim_{h \to 0} \frac{f(c(x+h))-f(cx)}{h}$$ and then my textbook says to do this: $$f(x) = \lim_{h \to 0}…
Anmol Bhullar
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Is this strict inequality of integrals true?

Let $f,g$ be integrable functions on $[a,b] \subset \mathbb{R} $. If it is known that $ f(x) \leq g(x) \ \forall x \in [a,b] $, then $ \int^{a}_{b} f(x) dx \leq \int^{a}_{b} g(x) dx $. I was wondering about the following, as it would also seem to be…
user308485
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