![Questions Tagged [Calculus]](https://docuprint.cloud68.co/uploads/questions-tagged-calculus_1200.jpg)
For basic questions about limits, continuity, derivatives, integrals, and their applications, mainly of one-variable functions. For questions about convergence of sequences and series, this tag can be use with more specialized tags.
123,492
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Evaluating $\lim_{x\to 0}\frac{4\sin x-\sin 4x}{x^3}$ without L'Hopital's Rule
Could someone explain this limit without L'Hopital's Rule, please!
$$\lim_{x\to 0}\frac{4\sin x-\sin 4x}{x^3}$$
-2
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14
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Alternating infinite series problem please solve [closed]
Please hint me how to solve this.
(1-1/2+1/3-1/4+......) ^2 = 2[1/2-1/3(1+1/2) +1/4(1+1/2+1/3) -1/5(1+1/2+1/3+1/4) + ......]
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Integrating the Area of a Circle
So I understand that the circumference of a circle is represented by the expression $2πr$, and integrating it yields the area of the circle, represented by the expression $πr²$ - which also ...
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I don't know how to represent the curves in the graph below.
So I am making an Arcade game and I am trying to make the difficulty curve smooth (now the values are hard coded), but I don't know how f(x)/f(g(x)) should look like in order to achieve this.
I found ...
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1
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Need help plugging in bounds in a u-substitution
This is a really basic question but I'm just a little confused. I have this integral: $$\int_{0}^{2\pi}\cos^3tdt$$
I solved it by doing a u-sub:
$$\int_{0}^{2\pi}\cos t(1-\sin^2t)dt$$
Let $u=\sin t$, ...
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16
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Evan's statement about integral of the divergence of a vector field.
At the start of Evan;s textbook there is this statement:
How can we conclude 3 from the first equality? Why can't it be that the divergence is positive and negative over the volume in equal ...
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2
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Is there any method other than Feynman’s Integration Technique to find $ \int_{0}^{\frac{\pi}{2}} \ln \left(a \cos ^{2} x+b \sin ^{2} x+c\right) d x?$
We are going to find the formula, by Feynman’s Integration Technique, for
$$\int_{0}^{\frac{\pi}{2}} \ln \left(a \cos ^{2} x+b \sin ^{2} x+c\right) d x,$$
where $a+c$ $\textrm{ and }$ $b+c$ are ...
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35
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Let $A \subseteq \mathbb{R}^2$ and $f : A \rightarrow \mathbb{R}$. Show that A is open.
Let $A \subseteq \mathbb{R}^2$ and $f : A \rightarrow \mathbb{R}$. Also, the partial derivatives of $f$ are defined and bounded everywhere in A. Show that A is open.
I am thinking of assuming an ...
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Solve for the power series of $(1+x)^n$
I'm having a difficulty on solving this equation about power series.
I am asked to solve:
$(1+x)^n$
and I need to use this equation:
$\sum_{n=0}^{\infty} ar^n= \frac{ar^2}{1-r}$
Lastly, I need to find ...
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1
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Integral of angle between tangent and line
I have a problem like this but still haven't figured out how to solve it or what this concept is called in math.
Let's say I have a continuous and differentiable curve $S: y=f(x)$ from $A$ to $B$. $L$ ...
-3
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35
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Power series of $(1+x)^n$ [closed]
Solve for the $n$th term of the power series of $(1+x)^n$. where n=2
(include the notation)
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1
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Prove that the sequence $x_n= (\pi^1/n )-1$ converge to $0$ and proof that the sequence $x_n= \sqrt{(n)((\pi^1/n)-1)}$ converges
Please help I dont know how to start, I tried to use Bolzano's theorem:
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2
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63
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Derivative with respect to $x$ of $\frac{1}{\int^a_x{f(x)}dx}$
I would like to know the solution to: $$\frac{d}{dx}\frac{1}{\int^a_x{f(x)}dx},$$ please, where $a$ is some constant.
I know that $$\frac{d}{dx}\int^a_x{f(x')}dx' = -f(x)$$ but the inverse is less ...
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Using Green's theorem to calculate the area bounded by half of a cycloid
The area bounded by half of a cycloid $\alpha(t)=(R(t-\sin(t), R(1-\cos(t))$ where $R>0$ and $0 \leq t \leq \pi$ and the x-axis is:
I've tried to use Green's theorem to solve this, by "...
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how to prove the equation [closed]
vessel contains 2000 litres of sauce recipe which is pumped out of the vessel to be bottled at a rate of 60 litres/hr. Preservatives are added to the mixture at a rate of 80 litres/hr to which 8g of ...
