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.

134529 questions
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Tangent line to an infinitely differentiable curve

We have a curve $A$, which consists of all points $(x,y) \in \mathbb{R}^2$ which satisfy $$9x + 27y - \dfrac{10}{81} (x+y)^3 = 0$$ You're given that the curve $A$, sufficiently close to $(0,0)$, is the graph of an infinitely differentiable…
AbbasM
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Graph of the Dirichlet Function

we know, the Dirichlet function as: $$ f(x) = \begin{cases} 1, &\text{if } x \text{ is irrational; and } \\ 0, &\text{if }x \text{ is rational}. \end{cases} $$ R.A. Silverman in his book Modern Calculus says that this wild function cannot be plotted…
Mikasa
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What is the average temperature of the surface of this planet?

A spherical, $3$-dimensional planet has center at $(0, 0, 0)$ and radius $20$. At any point of the surface of this planet, the temperature is $T(x, y, z) = (x + y)^2 + (y - z)^2$ degrees. What is the average temperature of the surface of this…
user230283
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Interesting tangents problems

My question is simple. I'm teaching Calculus 1 and I'm looking for counterintuitive problems to find tangents. I want to show to my students the importance of the limit definition of tangents. Thanks EDIT (Clarification) I'm looking for tangents…
user42912
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Derivative of $\ln(x\sqrt{x^2-1})$

I am trying to find the derivative of $\ln(x\sqrt{x^2-1})$ but I can not get what the book gets. I get $$\frac{1}{x \sqrt{x^2-1}} \cdot \sqrt{x^2-1} + x\cdot\frac{1}{2}(x^2-1)^\frac{-1}{2}\cdot2x$$ which I reduce…
user138246
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All real solution of the equation $2^x+3^x+6^x = x^2$

Find all real solution of the equation $2^x+3^x+6^x = x^2$ $\bf{My\; Try::}$ Let $$f(x) = 2^x+3^x+6^x-x^2\;,$$ Now Using first Derivative $$f'(x) = 2^x\cdot \ln 2+3^x\cdot \ln 3+6^x\cdot \ln 3-2x$$ Now for $x<0\;,$ We get $f'(x)>0,$ So function…
juantheron
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Differentiating $y=x^{-3/2}$

I'd like to differentiate $y=x^{\large\frac{-3}{2}}$. So, $$y + dy = (x + dx)^{\large\frac{-3}{2}} = x^{\large\frac{-3}{2}}\Big(1 + \frac{dx}{x}\Big)^{\large\frac{-3}{2}}$$ is at least a start. How do I calculate the parentheses?
dash
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Differentiable at endpoints?

If $f$ is continuous on [$a,b$] then the area function $A(x)=\int_{a}^{x} f(t)dt,$ for $a \le x \le b$, is continuous on [$a,b$] and differntiable on $(a,b)$. My question is: why is it not differntiable on [$a,b$]? Why is it an open interval and not…
Jason
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If a statement is true for every element in a sequence, would it be true at limit when $n \to \infty$?

The answer is obviously No, with an example: $\frac{1}{n} > 0$ for all n, and $lim \frac{1}{n} = 0$ (I can prove this using delta-epsilon method or just draw a picture). But I can't wrap my head around this logic. If it is positive for ALL integer n…
user262959
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What is the significance behind taylor series?

Why does taylor series have ample amount of importance in calculus? I like to know some insights behind taylor series.
EHMJ
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If $\frac{dy}{dx}\frac{dx}{dy} = 1$, does $\frac{d^2 y}{dx^2} \frac{d^2 x}{dy^2} = 1$?

I know $\frac{dy}{dx}\frac{dx}{dy} = 1$ because the chain rule says $1 = \frac{dy}{dy} = \frac{dy}{dx}\frac{dx}{dy}$. But does $\frac{d^2 y}{dx^2} \frac{d^2 x}{dy^2} = 1$? Or would that be too good to be true?
Jessica
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Choosing a continuous function satisfying the mean value theorem

The mean value theorem tell us that if $f:[a,b]\to \mathbb{R}$ is continuous and differentiable at $(a,b)$ then there is some $c\in(a,b)$ such that $f(b)-f(a)=f'(c)(b-a)$. We can apply the mean value theorem for each $t\in (a,b)$ to get a $c_t$…
Zero
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Finding equation of straight line that is tangent to $y = 2^x$.

Problem: Find an equation of the straight line that is tangent to $y= 2^x$ and that passes through the point $(1,0)$. Attempt: Let $(a, 2^a)$ be the point of tangency. Now we have that $y' = 2^x \ln(2)$, which evaluated at the tangency point becomes…
Kamil
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Evaluate the limit $\lim_{x \to 0} \left(\frac{1}{x^{2}}-\frac{1}{\tan^{2}x}\right)$

Evaluate the limit $$\lim_{x \to 0}\left( \frac{1}{x^{2}}-\frac{1}{\tan^{2}x}\right)$$ My attempt So we have $$\frac{1}{x^{2}}-\frac{\cos^{2}x}{\sin^{2}x}$$ $$=\frac{\sin^2 x-x^2\cos^2 x}{x^2\sin^2 x}$$ $$=\frac{x^2}{\sin^2 x}\cdot\frac{\sin…
HighSchool15
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Compare $e^2$ and $7$ without using calculator

Which is bigger? $e^2$ or $7$? Any tricks? Don't know quite how to approach those kind of things.
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