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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computing an integral without making any substitution

We are trying to evaluate $$ \int\limits_0^1 ( \sqrt{2-x^2} - \sqrt{2x-x^2} ~) ~ dx $$ without any substitution (well, this is how this problem is supposed to be solved) Idea: We notice that if $y=f(x)$ is the integrand, then $f(1) = \sqrt{2}$ and…
ILoveMath
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Partial Fraction Decomposition Problem... half answered...

$$\int \frac{5x^3+19x^2+27x-3}{(x+3)^2(x^2+3)}dx$$ I know I will be using partial fraction decomposition on this problem, at least it seems that way. so far, what I have is…
TMBraxton
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Solving $e^{ix}=i$

I was assigned this problem: $$e^{ix}=i$$ I understand that with Euler's formula, $e^{ix}=\cos x+i\sin x$. I then set up the problem as $$i=\cos x +i\sin x$$ This means that $\cos x = 0$ and $\sin x =1$. This works for $\frac{\pi}2$. It has to be…
Burt
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$x^3+y^3=8$ ,Number of straight lines through origin which do not meet this curve is

$x^3+y^3=8$ ,Number of straight lines through origin which do not meet this curve is Sorry to say that I have no approach for this question, my Brain is totally blank right now. Plz help me.
Abhishek Kumar
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non-abstract intuition for the equality $\int^{\infty}_{- \infty} \frac{\sin(x)}{x} = \sum^{\infty}_{- \infty} \frac{\sin(n)}{n} = \pi$

I have trouble wrapping my head around this equality. I am looking for an intuitive (perhaps geometrical if possible) insight as opposed to an abstract one. I can follow proofs that compute the two sides of the equation. However,…
hahahasan
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Finding the Maximum of a Continuous Function over a Closed Interval

For function $f\left ( x \right )=4x^{3}-6x^{2},$ the maximum occurs in the interval $\left [ 1,2 \right ]$ when $x$ is equal to ___________ I got $x=0$ is maxima. Because, in that point $f''(x)<0$ But in answer, it is given though $f''(x)<0$ at…
Srestha
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Unable to solve a topology related indefinite integral

I'm trying to evaluate the integral below but I have no clue how to proceed any further. Does anyone knows how to deal with this? $$\displaystyle \lim_{n \to \infty} \int_0^1 \frac{1}{1+x+x^2+...+x^n}\, dx$$ I don't know what the primitive function…
Decaf Sux
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3Blue1Brown Calculus Video: Why are the unraveled rings trapezoids?

In this video https://www.youtube.com/watch?v=WUvTyaaNkzM&list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr, at 2:45, why is the unraveled ring a trapezoid when it should clearly be a rectangle? I was thinking that in real life, the unraveled ring actually is…
user532874
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A proof in Spivak on integration.

The theorem states If $f$ is bounded on $[a,b]$, then $f$ is integrable on $[a,b]$ $\iff$ for all $\epsilon > 0$ there is a partition $P$ of $[a,b]$ such that $$U(f,P) - L(f,P) < \epsilon$$ The proof is on page 220. I believe I have the 2nd ed of…
Lemon
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Tangent graph to a circle

I am trying to find the positive real number $b$ such that the graph of $f(x)=\ln(x+b)$ is tangent to the circle $x^2+y^2=1$. Here is my attempt: I realize that the point of tangency must be in the top half of the circle if $b$ is positive, so I…
Jithinash
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Simplifying sigma of sigma of sigma

I've come across the following expression in my computer science work (as you can probably tell by the nature of the expression). Is it possible to simplify it into some nice polynomial expressom? $$\sum_{i = 1}^{z-2} (z - 1) - i + \left ( \sum_{j…
user_hello1
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Solve $\lim _{t\to 0}\left(\int _t^{2t}\:\left(\frac{e^{2x}-1}{x^2}\right)dx\right)$

$$\lim _{t\to 0}\left(\int _t^{2t}\:\left(\frac{e^{2x}-1}{x^2}\right)dx\right)$$ So i got this question on my math exam today and I had no idea how to solve it. I looked at all my notes and tried looking for a solution online but couldn't find…
Josias Feulner
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Show a function is discontinuous using the epsilon-delta definition

I have the function $f(x) = \frac{1}{x^2}$. I want to show that it is discontinuous at $x=0$ using the epsilon delta definition. So, I need to show that for all $\epsilon > 0$, there does not exist a $\delta >0$, s.t. $|x|<\delta$ $\Rightarrow…
Paul
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Evaluating $\frac{d^2y}{dx^2}$

I want to evaluate $\frac{d^2y}{dx^2}$ from the given data: $x=a\cos^3\theta$ and $y=b\sin^3\theta$. I tried in this way- $\frac{x^2}{a^2}+\frac{y^2}{b^2}=\cos^6\theta+\sin^6\theta=1-3(\frac{xy}{ab})^\frac{2}{3}$. After that I seem to be lost and…
user64788
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Taylor expansion of $\ln(1-x)$

I was just wondering where the minus sign in the first term of the Taylor expansion of $ \ln(1-x) $ comes from? In wikipedia page and everywhere else $\ln(1-x)$ is given by $$ \ln(1-x) = -x-\dots $$ But assuming $x$ is small and expand around $1$,…
Lepnak
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