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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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Why is this equation of what appears to be a circle not a function? How do show this algebraically.

I know this is a circle (right?) and so for every x, there are 2 values of y (so y is not a function of x), but how do I algebraically show this. This is the question: If I square root both sides, I get: $$y = \sqrt{4 - x^2} $$ But that seems to…
Jwan622
  • 5,704
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Is there a definite integral for which the Riemann sum can be calculated but for which there is no closed-form antiderivative?

Some definite integrals, such as $\int_0^\infty e^{-x^2}\,dx$, are known despite the fact that there is no closed-form antiderivative. However, the method I know of calculating this particular integral (square it, and integrate over the first…
Cheerful Parsnip
  • 27,278
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Is the proof of $\lim_{\theta\to 0} \frac{\sin \theta}{\theta}=1$ in some high school textbooks circular?

I was taught the following proof in high school. By constructing triangles with $0<\theta<\pi/2$ and a circle with radius $r$ and by comparing the areas, we have $$\frac{1}{2}r^2\sin\theta\cos\theta \le…
velut luna
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Mathematical function that converges towards $7$?

My friends and I are finishing High School in Denmark. We have to do a math poster for some school activity, where the poster needs to have something to do with the number $7$. So my question is: does someone know a cool mathematical function that…
Oscar Arellano
  • 159
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Proving that the maximum of two convex functions is also convex

Here's a homework question I'm struggling with: Let $f,g$ two convex functions. Prove that $h(x)=\max\{f(x),g(x)\}$ is also convex I don't know where to begin. The only thing I had in mind was was to try proving that if a function is convex on two…
yotamoo
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Finding the angle between two line equations

I need to find the angle between two lines in $y = mx + b$ form and they are: \begin{equation*} y = 4x + 2~\text{and}~y = -x + 3 \end{equation*} I have no idea how to solve this and if you could please consider that I'm in Grade 12 Calculus and…
John
  • 347
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Approaching to zero, but not equal to zero, then why do the points get overlapped?

It is my question when I was in Senior High School. Up to now, I have no idea about the correct explanation. I just accept it by faith :D Here is the question: If the symbol $\Delta x \to 0$ does not mean $\Delta x =0$, how can the points $A$ and…
Display Name
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if $2^x+5^y=2^y+5^x=\frac{7}{10}$

let $x,y$ such $$2^x+5^y=2^y+5^x=\dfrac{7}{10}$$ prove or disprove $x=y=-1$ is the only solution for the system. My try: since $$2^x-2^y=5^x-5^y$$ But How can prove or disprove $x=y$?
user94270
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Limit of $\sin (a^{n}\theta\pi)$ as $ n \to \infty$ where $a$ is an integer greater than $2$

In Hardy's Pure Mathematics, Hardy discusses the limit $$\lim_{n\to\infty}\sin (2^{n}\theta\pi)$$ and says that if this limit exists it must be zero and then $\theta$ must be a rational number whose denominator is a power of $2$. Then he asks the…
Paramanand Singh
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How do I evaluate the limit $\large\lim_{x\to \infty }\frac{\ln(x)^{\ln(x)^{\ln(x)}}}{x^x}$?

$$\large\lim_{x\to \infty }\frac{\ln(x)^{\ln(x)^{\ln(x)}}}{x^x}$$ As $x$ approaches infinity, both functions approach infinity. Therefore I should use the hopital rule, right? But it seems to complicate the answer.
Rubbles
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derivative with respect to constant.

I have been beating my head against this question for quite some time, I do not know whether it has been asked before, but I can't find any information about it! I am taking Calculus 1 course and I cannot grasp the concept of a derivative. From what…
Dmytro
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Weakest hypothesis for integration by parts

I was wondering what are the weakest hypothesis for applying integration by parts to calculate $$\int_a^b fg \, dm,$$ where $m$ denotes the Lebesgue measure on $\mathbb R$. Is it enough that $f$ be differentiable on $(a,b)$, $f' \in L^1(a,b)$ and $g…
Cantor
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6 answers

Prove $\sin(\pi/2)=1$ using Taylor series

Prove $\sin(\pi/2)=1$ using the Taylor series definition of $\sin x$, $$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots$$ It seems rather messy to substitute in $\pi/2$ for $x$. So we have $$\sin(\pi/2)=\sum_{n=0}^{\infty}…
hawaii99
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Leibniz rule derivation

How is Leibniz Integral rule derived? $$\frac {\mathrm{d}}{\mathrm{d}x}\left(\int_{a(x)}^{b(x)}f(x, t) \,\mathrm{d}t\right)= f(x,b(x))\frac{\mathrm{d}}{\mathrm{d}x}b(x)- f(x, a(x))\dfrac{\mathrm{d}}{\mathrm{d}x}a(x)+…
Archer
  • 6,051
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3 answers

$\ln(x^2)$ vs. $2\ln(x)$

Are the following functions equal or one is the restriction of the other? $f(x) = \ln(x^2)$ $g(x) = 2\ln(x)$ My book says that $g(x)$ is the restriction of $f(x)$ to $\mathbb{R^+}$ and I can verify that on my calculator. But that doesn't make…
Mark Read
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