Mathematics Stack Exchange
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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.

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A Geometrical Interpretation of the line integral of $f$ along $C$ with respect to $x$ and $y$.

I'm studying the line integral of a function along a curve $C$ with respect to $x$. Is the assertion as the following figure indicated true or false? I have read the questions Interpreting Line Integrals with Respect to $x$ or $y$ and Interpretation…
bfhaha
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Quadratic approximation of $9^{1/3}$

Find a quadratic approximation of the cube root of $9$ by using the equation $9=8(1+\frac 18)$, and estimate the difference between the exact value and the approximation. How am I supposed to find the quadratic approximation of this formula? If…
Sarah
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Maximize the area of a triangle inscribed in a semicircle.

https://i.stack.imgur.com/YZ679.png Consider the semicircle with radius 1, the diameter is AB. Let C be a point on the semicircle and D the projection of C onto AB. Maximize the area of the triangle BDC. My attempt so far, I'm new at these problems…
HighSchool15
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Find the length of the loop of the given curve: $x=3t-t^3$ $y=3t^2$

I used the arc length formula (where you take the integral of square root of x' squared + y' squared $\int \sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}dt$) to get $t^3 + 3t + C$ which seems to be the wrong answer. Not sure what I…
Baki Hanma
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Why doesn't the fundamental theorem of calculus depend on the lower bound?

I found this question and answer: Fundamental Theorem of Calculus: Why Doesn't the Integral Depend on Lower Bound? . Would anyone be able to explain it words? I don't get the connection between the specific integral property mentioned in the answer…
John
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Inequality of continuous functions

Let $u, v$ be continuous functions on $[a,b]$ and let $c>0$. Suppose that for all $x \in [a,b]$ we have the following inequality: $$|u(x)-v(x)| \leq c \int_a^x |u(t)-v(t)| dt$$ Show that $u(x)=v(x)$ for all $x \in [a,b]$ My first thought was to…
jggarita
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Is it a good practice to write this integral in this form?

I'm trying to compute the following integral: $$\int e^{3x}\cos2x \;dx$$ Now I'm about to use the integration by parts. Suppose that I do not know what is the integral of $\displaystyle \int \cos2x\; dx$. Is it a good practice to write it like…
Red Banana
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Proving $e^x \sin x$ is not uniformly continuous on $[0,\infty)$

I have to prove that $e^x \sin x$ is not uniformly continuous on $[0,\infty)$ using the mean value theorem. I proved that $\sin x + \cos x \geq 1$ for all $x\in [2 \pi k, 2\pi k + \pi/2], k\in\mathbb{N}$, and I see that $f'(x)=e^x (\sin x + \cos…
user249733
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Find the shortest distance between $y=x+10$ and $y=6\sqrt{x}$

This is a Max/min problem, I'm trying calculate the shortest distance between the 2 using pythagoras theorem and diffrenciate it in order to calculate the mininmum of the Red line below: I'm having trouble putting the equation together...am I going…
Modrisco
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What is the purpose of the second variable in Taylors remainder theorem?

In textbooks and online tutorials I see that the remainder is calculated by using a new unknown variable on the same interval. For example we take the Taylor polynomial $T_n(a)$ but find the remainder $R(x)$ with a new variable $z$ inside it. See…
Dave
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How can I prove this theorem about integrals?

be f a integrable function in [a,b], proof that exists c in (a,b) such that $\int_a^cf(t)dt=\int_c ^bf(t)dt$. I think that use the Fundamental theorem of calculus can help to proof that. Whath I did is this: $\int_a^bf(t)dt=F(b)-F(a)$ for…
User 2014
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Big O Notation Proof

[THIS IS HOMEWORK, Please do not post solutions, just help me understand] I know this is kind of a CS related question, however I was told this might be the right place to post the question. I'm trying to prove that $2(\log_{2}{6})^{n}=O(3^{n})$ so…
JanosAudron
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How to integrate $\int_{-1}^1 \tan^{-1}\left(\frac{1}{\sqrt{1-x^2}}\right )\,dx$?

Evaluate $$\int_{-1}^1 \tan^{-1}\bigg (\dfrac{1}{\sqrt{1-x^2}}\bigg ) dx $$ Could somebody please help integrate this without using Differentiation under the Integral Sign?
User1234
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Finding roots of a function with mean value theorem

I am suppose to show that the equation $x^3 - 15x + c = 0$ has at most one root in the interval [-2,2] I have sort of memorized the mean value theorem but I don't really understand how it is applicable to this.
user138246
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Progression from indefinite integral to definite integral - $\int_{0}^{2\pi}\frac{1}{5-3\cos x} dx$

I'm trying to evaluate the following integral: $$\int_{0}^{2\pi}\frac{1}{5-3\cos x} dx$$ We can evaluate indefinite one first - $\int\frac{1}{5-3\cos x}dx = \frac{1}{2}\tan^{-1}(2\tan(\frac{x}{2})) + C$. The problem is that…
qiubit
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