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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Trigonometic Substitution VS Hyperbolic substitution

The following tables were taken from University of Pennsylvania's page about Calculus: Trigonometric Substitution Hyperbolic Substitution As you can see, the forms $1+x^2$ and $x^2-1$ are repeated in the tables. How does one know when a…
krismath
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Integrating $x^2e^{-x}$ using Feynman's trick?

In the second episode of season $8$ of "The Big Bang Theory," which aired yesterday night, it is stated that one can integrate $x^2e^{-x}$ by using Feynman's trick of differentiating under the integral. Is this actually true, and if so, how to do…
Nishant
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How to prove $\sin(1/x)$ is not uniformly continuous

How do I go about proving $f(x)=\sin(1/x)$ is not uniformly continuous? (Or: different question, but same intention* how do I prove that $x\sin(x)$ is not uniformly continuous) *I'm trying to grasp how one would prove $f$ is not uniformly continuous…
MathMathCookie
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Computing the exact value of $\sum_{n=1}^\infty \left(\frac{2n+3}{3n+2}\right)^n$

I found this problem in my textbook, and I know that it converges, but I wanted to know if there was a way to find the exact value of the convergence (similar to what Euler did with the sum of reciprocal squares). I tried to rewrite the sum as a…
transcenmental
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Finding the fraction $\frac{a^5+b^5+c^5+d^5}{a^6+b^6+c^6+d^6}$ when knowing the sums $a+b+c+d$ to $a^4+b^4+c^4+d^4$

How can I solve this question with out find a,b,c,d $$a+b+c+d=2$$ $$a^2+b^2+c^2+d^2=30$$ $$a^3+b^3+c^3+d^3=44$$ $$a^4+b^4+c^4+d^4=354$$ so :$$\frac{a^5+b^5+c^5+d^5}{a^6+b^6+c^6+d^6}=?$$ If the qusetion impossible to solve withot find a,b,c,d then…
mnsh
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Why is $\int\limits_0^1 (1-x^7)^{1/5} - (1-x^5)^{1/7} dx=0$?

When I tried to approximate $$\int_{0}^{1} (1-x^7)^{1/5}-(1-x^5)^{1/7}\ dx$$ I kept getting answers that were really close to $0$, so I think it might be true. But why? When I ask Mathematica, I get a bunch of symbols I don't understand!
Larry Wang
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Integrating both sides of an equation: what does it really mean?

The operation of integrating some equation up to some time $t$ from, say, $0$ or on a small interval is very common. But what does it really mean? Adding +2 to both sides of an equation is rather straightforward, but integrating something is not all…
user561840
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Proving that multiplication of convex function is convex

Here's a homework question I'm struggling with: Prove/disprove the next statement: Let $f,g$ two convex functions, then $h(x)=f(x) \cdot g(x)$ is also convex So, we know that $h'(x)=f'(x) \cdot g(x) + f(x) \cdot g'(x)$. We also know that…
yotamoo
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Where do the higher order terms in Taylor series come from?

I can see the first order approximation for a Taylor series: if we want to approximate $f(x)$ near $x_0$, then it's close to the line with slope $f'(x_0)$ that intersects $f(x_0)$, giving $f(x) \approx f(x_0) + f'(x_0)(x-x_0)$ However, I don't see…
user98012325
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How to prove that continuous functions are Riemann-integrable?

In other words, how to prove A continuous function over a closed interval is Riemann-integrable. That is, if a function $f$ is continuous on an interval $[a, b]$, then its definite integral over $[a, b]$ exists. Edit: The Definite Integral as a…
NonalcoholicBeer
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Evaluation of $\int\frac{(1+x^2)(2+x^2)}{(x\cos x+\sin x)^4}dx$

Evaluation of $$\int\frac{(1+x^2)(2+x^2)}{(x\cos x+\sin x)^4}dx$$ $\bf{My\; Try::}$ We can write $$x\cos x+\sin x= \sqrt{1+x^2}\left\{\frac{x}{\sqrt{1+x^2}}\cdot \cos x+\frac{1}{\sqrt{1+x^2}}\cdot \sin x\right\}$$ So we get $$(x\cos x+\sin x) =…
juantheron
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Infinitely differentiable

How can one find if a function $f$ is infinitely differentiable?
bobobobo
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What does smooth curve mean?

In this problem, I know that the hypothesis of Green's theorem must ensure that the simple closed curve is smooth, but what is smooth? Could you give a definition and an intuitive explanation?
Jichao
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Prove that $g(x)=\frac{\ln(S_n (x))}{\ln(S_{n-1}(x))}$ is increasing in $x$, where $S_{n}(x)=\sum_{m=0}^{n}\frac{x^m}{m!}$

I want to prove that the function $$g(x)=\frac{\ln(S_n (x))}{\ln(S_{n-1}(x))},\,x >0$$ is increasing in $x$ for all $n$, where $ S_n(x)= \sum_{m=0}^{n}\frac{x^m}{m!}$. Differentiating gives something messy that I have not been able to prove it is…
clark
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What is an example of an application of a higher order derivative ($y^{(n)}$, $n\geq 4$)?

Can you suggest some useful things we can do with higher order derivatives? A fellow student in my Calculus and Analytic Geometry I class asked what some applications of higher order (e.g., ( $y^{(n)}$, $n\geq 4$) ) derivatives might be. Our…
FreeTrader
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