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.

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How to evaluate $\int^\infty_0 \log^2(x)\exp(-x)\,dx$

I attempted to solve this equation using integration by part, but it leads me no where and it gets too complicated to solve. I hope some can give me a hint how to approach this integral $$\int^\infty_0 \log^2(x)\exp(-x)\,dx$$ Thanks,
user1292919
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Area of surface of revolution of $e^x$

An at first easy looking question has been giving me problems. Given is the function $f(x)=e^x$ on the interval $[0,1]$, asked are the areas of its surfaces of revolution about the $x$-axis and $y$-axis. $x$-axis: We have…
user50407
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Continuity at a point implies continuity on an interval?

If $f(x)$ is continuous at $a$, then is there a $\sigma$ such that $f(x)$ is also continuous on $(a-\sigma, a+\sigma)$? This looks very intuitive, but I don't know how to prove it.
qed
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Show that $\int_0^\infty \sin\left(x^2\right)dx$ converges, but that $\int_0^\infty \sqrt{\sin^2\left(x^2\right)}dx$ does not.

Show that $\int_0^\infty \sin\left(x^2\right)dx$ converges, but that $\int_0^\infty \sqrt{\sin^2\left(x^2\right)}dx$ does not. The first part I think I proved using triangles, but I could not prove the second part.
Henfe
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Question about derivative of $\cos(x)$.

The question is how to show the derivative of $\cos x$ is $-\sin x$ using the definition of the derivative. I do this proof in the normal way by using the sum of $\cos (x+h)$ using the trig identity, and then factoring out the $\cos x$ and using two…
user163862
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Geometrical meaning of $\operatorname{d}x$, $\operatorname{d}y$ and $\operatorname{d}y\over\operatorname{d}x$?

I am also confused about whether these are symbols or have some meaning of their own. PS- I know that $\operatorname{d}y\over\operatorname{d}x$ geometrically represents the slope. But, I've come across $\operatorname{d}x\over\operatorname{d}y$ to…
Rayyan Asif Khan
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How to recognize a sum as a Riemann Sum

Evaluate $$\frac{1}{1}+\frac{1}{2}-\frac{2}{3}+\frac{1}{4}+\frac{1}{5}-\frac{2}{6}+\frac{1}{7}+\frac{1}{8}-\frac{2}{9}+\cdots+\frac{1}{3n+1}+\frac{1}{3n+2}-\frac{2}{3n+3}+\cdots$$ answer choices: a) $\ln 2$ b) $\ln 3$ c) $e^2$ d) $\dfrac 9…
chrismc
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Derivative of an Integral whose integrand is discontinuous

Let, $g(x) =\int_{a}^{x} f(t) dt$ be an integral function. What can we say about $g'(c)$ when, a) $f$ is removable discontinuous at $c \in[a,x]$? b) $f$ is infinite discontinuous at $c\in [a,x]$? c) $f$ is jump discontinuous at $c\in [a,x]$? Here…
R004
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multiple loans multiple payers - how to snowball fairly

My brother and I both have a large sum of student loan debt. I have more than he does and my interest rates are slightly larger as well. We are both attempting to snowball our debt separately. It occurred to me that we may be able to accelerate…
stu
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If $f (x) +f'(x) = x^3+5x^2+x+2$ then find $f (x)$

If $f (x) +f'(x) = x^3+5x^2+x+2$ then find $f(x)$. $f'(x)$ is the first derivative of $f (x)$. I have no idea about this question, please help me.
Medo
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Proof of: $\int_0^\infty x^{m-1}e^{-ax} \cos bx \ dx = \frac{\Gamma(m)}{(a^{2} + b^{2})^{m/2}}\cos\left(m\tan^{-1}\left(\frac{b}{a}\right)\right)$

Where can I find a proof or how do you prove the following: $$\int_0^\infty x^{m-1}e^{-ax} \cos bx \ dx = \frac{\Gamma(m)}{(a^{2} + b^{2})^{m/2}}\cos\left(m\tan^{-1}\left(\frac{b}{a}\right)\right)$$ Edit: I think I see the identity now For a…
user5389726598465
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Please confirm that $\int_{0}^{\infty}{\ln(x)\sin(x)\cos\left(x\over \sqrt{\phi}\right)\over x} dx={1\over 2}\pi \left(\ln\phi-\gamma\right)$

Observe this integral Where $\phi={\sqrt{5}+1\over 2}$ and $\gamma=0.5772156...$ is Euler's Constant $$\int_{0}^{\infty}{\ln(x)\sin(x)\cos\left(x\over \sqrt{\phi}\right)\over x}\mathrm dx={1\over 2}\pi…
user441532
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Study of functions from $\mathbb{Q}$ to $\mathbb{Q}$

Is it possible to study functions from $\mathbb{Q}$ to $\mathbb{Q}$ with ordinary calculus ? Obviously with the limitation that $\mathbb{Q}$ is not complete. So much less limits, derivatives and integrals exist; but does it make sense a tangent in…
halfpog
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Finding the area between curves, calculus, how does my answer look?

i got a test tomorrow and before i step in I wanna make sure i understand this. How does my answer look? The prof didnt provide any solutions, just the worksheets, so im worried everything im doing is wrong.
Ricekrispie
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Find what the given sum is equal to

I am given the following sum: $$\sum_{n=0}^\infty{\frac{1}{2^{n+2}(n+2)}}$$ How do I find what is it equal to? It sort of looks like a logarithm, but not exactly, so how do I proceed to solve this problem?
Sartr
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