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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Why is this telescoping sum divergent?

I'm trying to figure out whether $$\sum_{x = 1}^{\infty}\ln\bigg(\frac{x}{x+1}\bigg)$$ converges or diverges. Here is what I tried doing: \begin{align}\sum_{x = 1}^{\infty}\ln\bigg(\frac{x}{x+1}\bigg) &= \sum_{x = 1}^{\infty}\ln(x)-\ln(x+1) \\ &=…
user262291
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Maximum Value of the function $f(x)=x^n(1-x)^n$ for a natural number $n \geq1$ and $x\in[0,1]$.

I need some help with the following problem. How, I can find the, Maximum Value of the function $f(x)=x^n(1-x)^n$ for a natural number $n \geq1$ and $x\in[0,1]$.
ram
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How can I find maximum and and minimum values of $ f(x,y)=xye^{-(x+y)}$?

How can I find maximum and and minimum values of $ f(x,y)=xye^{-(x+y)}$ in the region $(x-1)^2+(y-1)^2 \leq4$ ?
Muhammad Khalifa
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How to take the derivative of a power.

So I'm trying to solve this problem: Take the derivative of $2^{t^{3}}$ This is the relevant text from my textbook which makes sense to me. The trick seems to convert anything in the form of $b^x$ to $e^{x\cdot lnb }$ because $b = e^{lnb}.$ So,…
Jwan622
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Find the maximum of the function

Find the maximum of the function: $$f(x)=\sin x+\sin\left(\frac{1}{x}\right) \quad x>0$$ My Try : $$f'(x)=\cos x-\dfrac{\cos(\frac{1}{x})}{x^2}=0 \\\cos x= \dfrac{\cos(\frac{1}{x})}{x^2} \ \ \ \\x^2\cos x=\cos\left(\frac{1}{x}\right)$$ Now what do…
Fricul38
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Checking to make sure I understand the English in this math problem.

Let $g$ be a function given by $g\left(t\right) = 100 + 20 \sin \left(\frac{\pi t}{2}\right) + 10 \cos \left(\frac{\pi t}{6}\right)$ For $0\leq t \leq 8$, is $g$ decreasing most rapidly when $ t =$. The answer should be 2.017. I am confused about…
yiyi
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Confusion regarding proof of Boundedness Theorem as given in Apostol's Calculus Volume 1

I was studying the proof of the boundedness theorem of continuous functions on a closed interval from Apostol's Calculus Volume 1. I am unable to understand a crucial step in the proof. First the theorem :- "If a function $f$ is continuous on a…
ameyask
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The hands of a clock are of length 5 inches (minute hand) and 4 inches (hour hand). How fast is the distance between them changing at 3:00?

I am studying calculus on my own. Using old text by Varberg and Purcell. Not sure if my solution is correct. (Cannot find online.) Differentiated using law of cosines but my rate of change seems quite fast. I proceeded as follows: To find an overall…
Victor Jaroslaw
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How to sketch the curve of parametric equations

I have the following parametric equations: $$x = \sin\theta,\qquad y = \cos\theta,\qquad 0 \leq \theta \leq \pi.$$ The corresponding Cartesian equation is $x^2 + y^2 = \sin^2\theta + \cos^2\theta = 1$. So it would be $x^2 + y^2 = 1$. How would I…
Krysten
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If $f$ is an even function, and $f$ is differentiable at $x=0$, prove that $f'(0)=0$?

I don't really know where to begin - intuitively I understand that the y-axis intersection must be an extrema, so the derivative is obviously 0, but I'm having difficulty writing the proof.. We haven't learned yet that the derivative of an extrema…
Danny
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Example of function that is differentiable at $0$, and has inverse function that is not continuous at $0$?

Is there any example of a function $f(x)$ differentiable at $x=0$, with an inverse function that is not continuous when $x=0$? Any help where to start, or maybe even if someone has the example of such a function would be greatly appreciated.
John Smith
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Explain this proof without words of integration by parts to me

Here is a proof of integration by parts: http://www.math.ufl.edu/~mathguy/year/S10/int_by_parts.pdf But I don't understand how it works. Specifically, I don't understand why $\int_r^s u \, dv$ equals one of the areas (and likewise for $\int_p^q v \,…
dever
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Derivative of $(a\,x)^{b\,x}$

is there any rule to differentiate the function $(a\,x)^{b\,x}$? I've got to find the derivative of $(x^2+1)^{\arctan x}$ and Wolfram|Alpha says the answer is $$\tan^{-1}(x) (x^2+1)^{\tan^{-1}(x)-1} \left(\frac{d}{dx}(x^2+1)\right)+\log(x^2+1)…
Garret Raziel
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About mean value theorem

Suppose $f''(x)$ exists on the interval $(-1,1)$,$f(0)=f'(0)=0$,and the inequality $|f''(x)|\leqslant|f(x)|+|f'(x)|$ holds on $(-1,1)$; How to prove that $f(x)=0$ on $(-\delta,\delta)$ for some $\delta>0$? Thanks for help.
C Weid
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