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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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Finding an addition formula without trigonometry

I'm trying to understand better the following addition formula: $$\int_0^a \frac{\mathrm{d}x}{\sqrt{1-x^2}} + \int_0^b \frac{\mathrm{d}x}{\sqrt{1-x^2}} = \int_0^{a\sqrt{1-b^2}+b\sqrt{1-a^2}} \frac{\mathrm{d}x}{\sqrt{1-x^2}}$$ The term…
anon
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If $ I = \int_{0}^{1}x^{1004}\cdot (1-x)^{1004}dx$ and $J=\int_{0}^{1}x^{1004}\cdot \left(1-x^{2010}\right)^{1004}dx\;,$ Then $I/J$

If $\displaystyle I = \int_{0}^{1}x^{1004}\cdot (1-x)^{1004}dx$ and $\displaystyle J=\int_{0}^{1}x^{1004}\cdot \left(1-x^{2010}\right)^{1004}dx\;,$ Then Relation between $I$ and $J.$ $\bf{My\; Try::}$ Given $$\displaystyle J =…
juantheron
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Evaluation of $\displaystyle \int_{0}^{1}\left(1-x^3+x^5-x^8+x^{10}-x^{13}+\ldots\right)dx$

Evaluation of $\displaystyle \int_{0}^{1}\left(1-x^3+x^5-x^8+x^{10}-x^{13}+\ldots\right)dx$ $\bf{My\; try::}$ We can write $\displaystyle 1-x^3+x^5-x^8+x^{10}-x^{13}+\ldots$ as $$\displaystyle (1-x^3)\cdot (1+x^5+x^{10}+\ldots ) =…
juantheron
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Euler's constant greater than 0 for all values of n?

If Euler's constant is described as the limit as n approaches infinity of the following: $$t_n = 1 + \frac 12 + \frac 13 \cdots + \frac1n -\ln(n)$$ How can one prove that $t_n$ is greater than $0$ for all values of $n$? Thanks!
user270893
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What is the simplest way to show that $\cos(r \pi)$ is irrational if $r$ is rational and $r \in (0,1/2)\setminus\{1/3\}$?

What is the simplest way to show that $\cos(r \pi)$ is irrational if $r$ is rational and $\displaystyle r \in \left(0,\frac{1}{2} \right)\setminus \left\{\frac{1}{3} \right\}$? I proved it using the following sequence $x_1 = \cos(r \pi)$; $x_{k} = 2…
Physicsworks
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For what positive value of $c$ does the equation $\log(x)=cx^4$ have exactly one real root?

For what positive value of $c$ does the equation $\log(x)=cx^4$ have exactly one real root? I think I should find a way to apply IMV and Rolle's theorem to $f(x) = \log(x) - cx^4$. I think I should first find a range of values for $c$ such that the…
user246836
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prove that $\sin (\tan x)\geq x\;\forall x\in \left[0,\frac{\pi}{4}\right]$

Using the relation $2(1-\cos x)
juantheron
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Can someone walk me through this differential problem?

I'm having a little difficulty understanding how to do the .05 using differentials. I'm just hoping someone can walk me through, step by step, and explain why they are during it that way. $$\sqrt[3] {27.05}$$ Edit Apologies to all, I wasn't very…
Math_Phase
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Calculating the $k$th digit of $\pi$

I'm new to math.stackexchange so apologies in advance for any blunders: I am trying to calculate $\pi$ using the following technique here. Considering the above link says: The discovery of this formula came as a surprise. For centuries it had been…
Bendy
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Intuition for gradient when you only have one variable?

I am learning about gradient. I understand how gradient is a vector that represents the sum of the rates of change for each component variable of a function. I am able to follow the Khan Academy video showing the gradient of f(x,y). I am also able…
bernie2436
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Prove that there exists some $c\in(-3,3)$ such that$ \ \ g(c) \cdot g''(c)<0$.

$f(x)$ is a differentiable function and $g(x)$ is a double differentiable function such that $|f(x)|\leqslant 1$ and $f'(x)=g(x)$. If $$f(0)^2+g(0)^2=9$$ then prove that there exists some $c\in(-3,3)$ such that$ \ \ g(c) \cdot…
MathMan
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Wrong solution for $\int \frac{1 - e^x}{e^x}\, dx$

The solution book gives this as an answer: $$\int \frac{1-e^x}{e^x}dx = \int \frac{1}{e^x}-\frac{e^x}{e^x}dx = -e^{-x} + C$$ I would think it would be solved this way: $$\int \frac{1-e^x}{e^x} dx = \int \frac{1}{e^x} -1 \; dx = -e^{-x}-x + C$$
yiyi
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Exercise 26 from Apostol's Calculus (p. 209, parts (c) & (d)))

This is a problem from Apostol's Calculus (p. 209 Ex. 26 (c) & (d)). The problem is to find a function $f$ with a continuous second derivative $f''$ satisfying the following conditions: (c) $f''(x) > 0 \quad \text{for every } x, \qquad f'(0) = 1,…
user23784
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Spivak - Chapter 7 Question 5 - Could f(x) be a function other than a constant function?

I'm self studying, and I was wondering if there were anything else that could be said about the following: Suppose that $f$ is continuous on $[a,b]$ and that $f(x)$ is always rational. What can be said about $f$? I said that $f$ could be a constant…
rabidapple
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Finding infinite limit of hyperbolic trig functions

I am trying to do random problems in my book and I do not know what to do for this one. I am suppose to find the limit as x approaches infinity of $\tanh x$ I really do not know what to do I know the problem is $ \frac{(e^x - e^{-x})/2}{(e^x +…
user138246
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