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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Solve in $\mathbb{R^{3}}$ : $\begin{cases}(1+4x^{2})y=4z^{2}\\(1+4y^{2})z=4x^{2}\\(1+4z^{2})x=4y^{2}\end{cases}$

Solve the system in $\mathbb{R^{3}}$ : $$\begin{cases}(1+4x^{2})y=4z^{2}\\(1+4y^{2})z=4x^{2}\\(1+4z^{2})x=4y^{2}\end{cases}$$ My try : By imaging I see $(\frac{1}{2},\frac{1}{2},\frac{1}{2})$ is a solution! From a first equation :…
Ellen Ellen
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+ C in integration by parts allows for major differences in answers?

Integration by Parts states that $$∫(f(x)g'(x))dx = f(x)g(x) - ∫(f'(x)g(x))dx$$ right? So when you try to find $g(x)$ when you are only given $g'(x)$, you take the indefinite integral. But the indefinite integral always includes $C$! So wouldn't it…
Ghibbbeey
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Derivative of $x$ with respect to $xy$

I know that $\dfrac{d(xy)}{dx} = y$ but what does $\dfrac{dx}{d(xy)} =\, ?$ I know this is an odd equation, but it comes from some ugly change of variables and I am stuck with it.
user1247549
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When is $\sqrt{x^2+x}-x$=o(1) (little-o notation)?

This is the question: determine when $\sqrt{x^2+x}-x$ is equal to little-$o$ of 1. The options are: a) $x \to +\infty$ b) $x \to -\infty$ If I understand correctly, $\sqrt{x^2+x}-x = o(1)$ if $$\lim_{x \to x_0}\frac{\sqrt{x^2+x}-x}{1}=0$$ Mutliply…
Belen
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A function $f$ is not differentiable at $a$ if it has no tangent at $a$.

I am a bit confused by the above statement from a textbook I am using. A tangent line is a line that touches a curve at a certain point, looking at $|x|$, it is not differentiable at 0 but isn't the x-axis tangent to this curve at 0? please help me…
mamotebang
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Related Rates (point moving along a curve)

Consider a point moving along the curve $$f(x) = \sqrt x$$. a). Find the position of the point on the curve where both coordinates of the point are changing at the same rate. b). If $\dfrac{dx}{dt}$ is $2 \text{ m/sec}$ at the point $(4,f(4))$, how…
harold232
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Find $\cos(\pi^4)$

This is what I did. $\pi^4 = 1049760000\deg \quad$ since $\pi = 180\deg$ and $1049760000$ is divisible by $360$ so this is equivalent to finding $\cos(0\deg) = 1$. But the answer is not correct. Can someone explain to me what am I missing. Thank you
user100478
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Is the answer sheet wrong?

The length of the curve determined by the equations $x=t-1$ and $y=\sqrt{t}$ from $t=0$ to $t=4$ is. I think that is is a parametric cruve and Paul's Online Notes the equation for the lenght of a parametric curve is: $$\int_a^b…
yiyi
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Showing $\lim_{x\to 0}\frac{\sin(x^2\sin\frac{1}{x})}{x}=0$

$$\lim_{x\to 0}\frac{\sin(x^2\sin\frac{1}{x})}{x}=\lim_{x\to 0 }\frac{x^2\sin\frac{1}{x}}{x}=\lim_{x\to 0} x\sin\frac{1}{x}=0$$ Is this solution right? Thank you very much!
Jacob.Lee
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Why does the answer sheet solve this problem with d/dt?

When $x=3$, the rate at which $\frac{1}{x}$ is decreasing is $k$ times the rate at which $x$ is increasing. What is the value of $k$? To find the rate at which $\frac{1}{x}$ is decreasing, find the first derivative $\frac{-1}{x^2}$ Now this is to…
yiyi
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Solving $x^2+ \ln^2(x)=1$

Solve $$x^2+ \ln^2(x)=1$$ Graph of the given equation From the graph or simply putting $x=1$ satisfy the equation one of the solution is $1$ . How I can find the other solution of the given equation $?$
Abhishek Kumar
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Determining if a sequence converges or diverges...

Determine if the sequence $(a_n)$ defined by $a_n := \frac{\cos{\left(e^{n^n}\right)}}{n^2}$ converges or diverges. If it converges, determine its limit. Okay so the solution is $0$ (so it does converge). I just have a question on where to start,…
squenshl
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Point on Sphere Closest/Farthest to Another Point

"What points on the sphere centered at the origin with a radius of 3 are closest to and farthest from the point P = (6,6,-3)?" The approach I took was to make a vector v going from the origin to P and see where it intersects the…
Mushy
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Determining range using Intermediate Value Theorem

Question: Let $$f(x) = \frac{x^6 -1}{3x -1}$$ Prove that the range of $f$ is $\Bbb R$.( Hint: use the Intermediate Value Theorem.) I thought IVT was meant to show that the function has a root? Please help, I don't know how I can use IVT to…
Amanda
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Derivative of $\ln\left(\frac{2x}{1+x}\right)$

I know that the derivative of $$f(x)=\ln(x) \ ,\ x>0$$ is just simply $$f'(x)=\frac{dx}{x}$$ But how do you find the derivative for the function: $$g(x)=\ln\left(\frac{2x}{1+x}\right)\ , \ x>0$$
user67253
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