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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What is $f(x)$ if $f(x) +xf(-x)=x+1$?

I am trying to find the functional equation of $f(x)$, where $f$ satisfies $$ f(x) +xf(-x)=x+1.\tag{1} $$ There is no further information. I attempted inverting the equation $f^{-1}(f(x)) = x = f^{-1}(x(1-f(-x))+1)$ but it doesn't seem to lead…
HelloWorld
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Divergent integrals $\int_{a}^{\infty}\frac{dx}{f(x)}$ and $\int_{a}^{\infty}\frac{dx}{f(x)+b}$ for positive $f$ and $b$

Assume that $f\colon [a,\infty)\to(0,\infty)$ is continuous ($a\in\mathbb{R}$) and such that $$\int_{a}^{\infty}\frac{dx}{f(x)}=+\infty$$ and let $b>0$. Can we claim that $$\int_{a}^{\infty}\frac{dx}{f(x)+b}=+\infty\quad ?$$ If $f$ is nondecreasing,…
Rado
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Piecewise function find f' and f"

Suppose $$ f(x) = \left\{\begin{array}{cr} e^{-1/x^2} & x \neq 0\\ 0 & x = 0 \end{array} \right. $$ Show that $f^\prime(0)$ exists and is equal to $0$, also verify that $f^{\prime\prime}$ exists and is equal to $0$. Do I solve this by finding the…
user71317
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Express as an infinite sum $ \int^1_0 \frac{\ln(x)}{1-x^2}\,dx $

I have this problem from my calc II class. The question asks to express as an infinite sum $$ \int^1_0 \frac{\ln(x)}{1-x^2}\,dx $$ So far I've tried to rewrite the denominator as a power series and then multiply it by the numerator which gives $$…
Bob Cobb
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How do you evaluate continuity for a g(x) function with given conditions?

Given $g(x)$ such that $ \frac{af(x)}{(x^2-5x+6)}+4x$ if $x<2$ and 2 if $ x=2$ $\frac{bf(x)}{sin(x-2)}+a$ if $ x>2$ Find the values for continuity with the given conditions: Condition 1: $$\lim_{x\rightarrow 2}\frac{f(x)}{x-2}=6$$ And condition…
Charlie Van Basten Øydne
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Is the function $F(x)=\int_{0}^{x} f(t) dt$ differentiable at 0?

Let $$ f(x)=\left\{\begin{matrix} \cos\frac{1}{x} & x\neq0\\ 0, & x=0 \end{matrix}\right.$$ I think the answer is not differentiable at 0 because of the chain rule, but I don't know how to prove it.
learning31
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Solving the limit of integrals $\lim\limits_{q \to 0}\int_0^1{1\over{qx^3+1}} \, \operatorname{d}\!x$

how do I solve this one? $$\lim_{q \to 0}\int_0^1{1\over{qx^3+1}} \, \operatorname{d}\!x$$ I tried substituting $t=qx^3+1$ which didn't work, and re-writing it as $1-{qx^3\over{qx^3+1}}$ and then substituting, but I didn't manage to get on. Thanks…
ohad
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If $f(x)>0$, and $f''(x)\leq 0$ for $x>0$, show that $f'(x)\geq 0$ for $x>0$.

Assume on the contrary, $f'(x)\geq 0$, and since $f''(x)\leq 0$, then $f(x)$ is concave down and decreasing, so $f(x)$ will eventually less than $0$, contradict to $f(x)>0$, but how can I prove that $f(x)$ will eventually less than $0$ formally?…
user533661
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if $\sum_{i=1}^{n}\sum_{j=1}^{n}|1-x_{i}x_{j}|=\sum_{i=1}^{n}\sum_{j=1}^{n}|x_{i}-x_{j}|$ show that $\sum_{i=1}^{n}x_{i}=n$

let $n>1$ is give postive integers,and $x_{i}>0,i=1,2,\cdots,n$,and such $$\sum_{i=1}^{n}\sum_{j=1}^{n}|1-x_{i}x_{j}|=\sum_{i=1}^{n}\sum_{j=1}^{n}|x_{i}-x_{j}|$$ show that $$\sum_{i=1}^{n}x_{i}=n$$ when $n=2$, since…
math110
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Partial Derivative On Contour Map

This is just a simple contour I download from the textbook. The original question is what is f(2,1), fx(2,1), and fy(2,1). I understand all of them with basic partial derivative knowledge. However, when I see what is fxx(2,1) or fyy(2,1). I cannot…
Zhecheng Li
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Why do we need an open interval to define local maxima?

We say that $f(x)$ has a local maximum at $\tilde{x}$ if for every $x$ in some open interval $(\tilde{x}-\delta,\tilde{x}+\delta)$, $\delta>0$ we have $f(\tilde{x})>f(x)$. Why do we need an open interval? Why can't a closed interval be good enough?
Shirin
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Prove there exists a $c \in (a,b)$ such that $\frac{f'(c)}{f(c)} = \frac{1}{a-c}+\frac{1}{b-c}.$

Let $f$ be continuous on $[a,b]$, differentiable on $(a,b)$ and positive for all $x \in(a,b).$ Prove that there exists $c\in(a,b)$ such that $$\frac{f'(c)}{f(c)} = \frac{1}{a-c}+\frac{1}{b-c}.$$ This seems like just an application of the mean value…
lvxvl
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Arc length of the graph of a non-differentiable function

While studying for a calculus test I came upon the definition of arc length for the graph of a differentiable function $f(x): \mathbb{R}\to\mathbb{R}$ in an interval $[a,b]$ to be defined as $\int_a^b\sqrt{1+f'(x)^2} dx$. This got me thinking about…
J. Doe
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Determine $f^{(5)}(0)$ without computing any derivatives?

Determine $f^{(5)}(0)$ without computing any derivatives? How do i approach a problem like this? I think i use maclaurin/taylor series but i'm not completely sure? $f(x)=x\ln(1+x)$ $f(x)=x\cos x$
lawlipop
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Intuition why the derivative of $e^x$ is itself

Is there an intuitive reason why the constant $e$ to the power of $x$ has a derivative that equals the value of the function? I know that this is the result of differentiating, and I've seen several proofs of how you work out the derivative, I was…
Martin
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