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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Differentiablility of $f(x)=x^m\sin(1/x^n)$

My attempt: I have to choose one from option A and option D. Option B can be eliminated by taking $m=1, n=2$. option C can also be eliminated by taking $m=4, n=3$. Please help me to choose one from A and D.
ketan
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An integral condition

Let $f$ be a non-decreasing and continuous function on $[0,1]$, such that $\int_0^1f(x)dx=2\int_0^1xf(x)dx$. Given that $f(1)=10.5$. Find the value of $f(0)+f(0.5)$.
user213422
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Related rates word problem

I am trying to figure this out but the book is not very specific with its terms so I am not sure what is actually meant by the author. "Boyle's Law states that when a sample of gas is compressed at constant temperature, the pressure P and volume V…
user138246
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The position of a particle moving along a line is given by $ 2t^3 -24t^2+90t + 7$ for $t >0$

For what values of $t$ is the speed of the particle increasing? I tried to find the first derivative and I get $$6t^2-48t+90 = 0$$ $$ t^2-8t+15 = 0$$ Which is giving me $ t>5$ and $0 < t < 3$, but the book gives a different answer
Sam
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Try to prove $\lim_{n \to \infty}n(\ln 2-A_n) = \frac{1}{4}$

$$A_n=\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{2n}$$ Try to prove $$\lim_{n \to \infty}n(\ln 2-A_n) = \frac{1}{4}$$ I try to decompose $\ln 2$ as $$\ln(2n)-\ln(n)=\ln\left(1+\frac{1}{2n-1}\right)+\dots+\ln\left(1+\frac{1}{n}\right)\;,$$ but I…
89085731
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The integral form of inner product .

I want to know how scientists know that the inner product of f and g equal to integration from $$\int_a^b f(x)g(x)\ dx.$$
nari
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Show $f(x) \geq \sin(x)$ if $f(0)=0$, $f'(0)=1$ and $f''(x)+f(x)\geq 0$

Let $f:[0,\pi] \to \mathbb{R}$ be a twice differentiable function. Show $f(x) \geq \sin(x) \forall x \in [0,\pi]$ if $f(0)=0$, $f'(0)=1$ and $f''(x)+f(x)\geq 0$ $\forall x \in [0,\pi]$. I have tried making the taylor series. Let $g(x)…
k99731
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Related Rates Problem Involving Airplanes

I took a test yesterday, and would like to know how to answer this specific question on the exam: One airplane flew over an airport at the rate of $300$ mi/hr. Ten minutes later another airplane flew over the airport at $240$ mi/hr. If the first…
Audrey
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One integral, two solutions?

I ran into an interesting integral problem: the indefinite integral of $\int \frac{dx}{a^2 x^2 - b^2}$. I can do a hyperbolic trig substitution and get that the result is $- \frac{1}{ab}\operatorname{arctanh}\frac{ax}{b}$ and Mathematica agrees with…
Ron
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Derivative of $x-\sqrt { x } $

Compute $f'(x)$ using the limit definition $$f(x)=x-\sqrt { x } $$ Steps I took: $$f'(x)=\lim _{ h\rightarrow 0 }{ \frac { x+h-\sqrt { x+h } -(x-\sqrt { x } ) }{ h } } \quad $$ $$f'(x)=\lim _{ h\rightarrow 0 }{ \frac { x+h-\sqrt { x+h } -x+\sqrt {…
Cherry_Developer
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A basic question about integration

I have read that if $f$ is integrable, it is bounded. But consider $f=x^{-\frac 12}$, it is integrable in $[0,1]$, but it is not bounded. Can you explain？
89085731
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Using the definition to show that $D_{x}(e^x)=e^x$.

Our applied calculus text defines $e$ by $e=\displaystyle\lim_{h\to0}(1+h)^{1/h}$, and then gives the following argument to show that $D_{x}(e^x)=e^x$: If $f(x)=e^x$, then…
user84413
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"Taylor series" and "proof of Euler's formula"

I have learned that we can prove Euler's formula by using Taylor series, as shown on wiki: Euler's Formula. I have a question. As wiki says: In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are…
Dongguo
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The roots of a certain recursively-defined family of polynomials are all real

Let $P_0=1 \,\text{and}\,P_1=x+1$ and we have $$P_{n+2}=P_{n+1}+xP_n\,\,n=0,1,2,...$$ Show that for all $n\in \mathbb{N}$, $P_n(x)$ has no complex root?
Ramand
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{$f_n(x)$} is defined by $f_{n+1}(x)=\int_{0}^{x}f_n(t)dt$, Prove: $\sum_{1}^{\infty}f_n(x)$ converges and is differentiable in $(0,1)$.

Given $f_1(x)$ continuous function in $[0,1]$ and differentiable in $(0,1)$. The series {$f_n(x)$} is defined by $f_{n+1}(x)=\int_{0}^{x}f_n(t)dt$ for $x\in [0,1]$. I need to prove that $\sum_{1}^{\infty}f_n(x)$ converges and is differentiable in…
Jozef
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