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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Problem with denominator in transformation

I can't understand where the 2 comes from in the following transformation. $$\frac{\partial}{\partial k}f(k)=\frac{\partial}{\partial k}k^{0.5}=\frac{1}{2k^{0.5}}$$ Any help would be appreciated.
user170810
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How do I find the relative extrema of a function in spherical coordinates?

I want to find the relative extrema for the following function. $f(\theta,\phi)=AR\cos\theta\sin\phi + BR\sin\theta\sin\phi + CR\cos\phi $ $A,B,C,R$ are constants In a function $g(x,y)$ using cartesian coordinates, you can find critical points by…
Pris
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Simple integration

I'm currently learning calc 2 and feel I'm making a very silly, obvious mistake with solving the integral $\int\frac{-5\sqrt[3]{x^2}}3 dx$ I guess I'm making an algebraic mistake somewhere, this is how I went about solving…
anastasia
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Weird Calculus problem

Let $f : (0,+\infty) \rightarrow (0,+\infty)$ and $g$ being a derivative of $f$ such as $g(x)=f^2(x)+f(x), \forall x>0$. Prove that $\lim_{x \to \infty} \frac{f(x)}{x} = 1/2$
John
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$f'(x)=0$ implies $f$ constant, although finite or countable exceptions

I am reading a text and I do not know why this follows: I $f$ is a continuous function and $f'(x)=0$ for every $x \in \mathbb{R}$ except for a finite set $E$ or a countable set $E$, then $f$ must be constant. Does anybody know an example for such…
monoid
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Help with epsilon-delta proof of continuity

I just asked a specific homework question on this topic, but I want a more general explanation for how to go about proving continuity with this method. I can't even wrap my head around what the proof is really saying, let alone figure out the steps…
Logan Serman
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Analysis Question from Berkeley Problems in Mathematics

I'm wondering if the following is correct. The original question(1.1.21, Fa96) asks to prove that \begin{align*} f''(x) = \lim_{h\rightarrow0}\dfrac{f(x+h) - 2 f(x) +f(x-h) }{h^2} \end{align*} I thought it was straightforward and proposed the…
user135562
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if $\frac{1}{(1-x^4)(1-x^3)(1-x^2)}=\sum_{n=0}^{\infty}a_{n}x^n$,find $a_{n}$

Let $$\dfrac{1}{(1-x^4)(1-x^3)(1-x^2)}=\sum_{n=0}^{\infty}a_{n}x^n$$ Find the closed form …
math110
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Function with a removable discontinuity

Is a function with a removable discontinuity considered continuous? Take for example $\frac{x^2-4}{x-2}$. It reduces to $x+2$.
user135247
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Related Rates Galore!

Water from one tank is being drained into another tank at a rate of 3 m3/min. The first tank is an inverted circular cone with height 3 m and radius 2 m. The second tank is a circular cylindrical tank with height 4 m and radius 2 m. a) How fast is…
flyinghigh
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Compute $\int_1^e \frac{dx}{x(x+(\ln x)^2)}$

My friend asked me how to integrate the following: $$\int_1^e \frac{dx}{x(x+(\ln x)^2)}$$ How am I going to solve this?Any help is greatly appreciated. Thanks.
Philip Benj
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True/ False differential equation

Are the statements in Problems 46-54 true or false? If $F(x)$ is an antiderivative of $f(x)$, then $y=F(x)$ is a solution to the differential equation $\frac{dy}{dx}=f(x)$. If $y=F(x)$ is a solution to the differential equation…
user157908
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Mellin transform of a shifted function

I have an application where it would be very useful to take the Mellin transform of a shifted function. Specifically \begin{equation} M(f(y-x))(y \rightarrow s) = \int_{y=0}^{\infty} f\left(y-x\right) y^{s-1}\, dy. \end{equation} Assuming $x>0$,…
Lee
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