For basic questions about limits, derivatives, integrals, and applications, mainly of one-variable functions.

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6
votes
3answers
123 views

What is $\dfrac{dr}{d\theta}$?

Suppose we have an equation of a polar curve with usual notation $r=f(\theta).$ I am curious about the geometric meaning of $$\dfrac{dr}{d\theta}=f'(\theta).$$ Also I would like to know the relations ...
-6
votes
1answer
64 views

how to calculate this limit? without l'hospitalet rule or taylor's or any use of derivatives [closed]

How can i solve this limit? $\displaystyle\lim_{x \to \infty} x^2\ln\left(\dfrac{x^2}{x^2+1}\right)$ I can't use l'hospital rule or taylors or derivatives Just limit modification
0
votes
1answer
60 views

Surface area using integration.

I am stuck on the following problem: Find the surface area of the solid formed be revolving $$y=\frac{a}{2}\left(e^\frac{x}{a}+e^\frac{{-x}}{a}\right)$$ about the $x$ axis from $x\in[-a,a]$. Can ...
5
votes
1answer
75 views

showing that an inequality holds

I am trying to figure out how to show that for $n\geq 3$, $$(2^n-1)^{\frac{n}{2(n-1)}}\geq (2^{n-1}-1)^{\frac{n-1}{2(n-2)}}+1.$$ I've tried basic algebra and induction, but the inductive hypothesis ...
3
votes
2answers
35 views

relation between $\frac{\partial(x,0)}{\partial x}$ and $\left.\frac{\partial(x,t)}{\partial x}\right|_{t=0}$

if $u(x,t)$ differentiable function and i only have $u(x,0)$, then is it right $\frac{\partial(x,0)}{\partial x} = \left.\frac{\partial(x,t)}{\partial x}\right|_{t=0}$ or can i derive $u(x,0)$ to $x$ ...
0
votes
1answer
95 views

definite integrals (very interesting question) [closed]

If $u = \displaystyle\int_{0}^{\pi/4}\left(\dfrac{\cos x}{\sin x +\cos x}\right)^2\,dx$ and $v = \displaystyle\int_{0}^{\pi/4}\left(\dfrac{\sin x +\cos x}{\cos x}\right)^2\,dx$ then find $\dfrac{v}{u}$...
4
votes
1answer
64 views

Differentiation of every order and Taylor series

Let $f(x)$ be a function defined in $(-1,1)$ with derivatives of all orders at zero equal to zero; that is: $f'(0)=0 , f''(0)=0 , f'''(0)=0 ...$ If there exists $c>0$ such that: $Sup|f^{(n)}(x)| &...
3
votes
2answers
85 views

How to demostrate that $\int_{\{a\}} f(x) d \mu= \mu(\{a\}) f(a)$?

I don´t know how to demostrate that: $$\int_{\{a\}} f(x) d \mu= \mu(\{a\}) f(a)$$ Note: I have read on a book that "The Lebesgue Integral of a constant function on a measurable set will be that ...
1
vote
1answer
33 views

For what $k$ does $\lim_{x \to-\infty} \frac{5^{kx}-1}{5^{-2x} + 1}$ exist?

For what values of $k$ does this limit exist? $$\lim_{x \to-\infty} \frac{5^{kx}-1}{5^{-2x} + 1}$$ Progress: I worked it by dividing everything by $5^{-2x}$ and now have $5^{x(k+2)}$ since the ...
0
votes
0answers
73 views

If all the critical values are shown in the chart below…

This is a strange question: If all the critical values of f(x) are shown below in the chart, which of the following could be values for f'(x) at x=1.5,2.5 and 3.5? CHART: x: 1, 2, 3, 4 f'(x): 0.5, ...
0
votes
1answer
32 views

Given curve is $y=x^2-1$, and $A(0,y_{1}),B(1,y_{2})$. Determine point $M$ between $A$ and $B$ so the area $AMB$ has maximum value.

I have found the equation for line between $A$ and $B$: $$y=x-1$$ Equation for tangent is: $$y=x-\frac{5}{4}$$ Coordinates of point $M(\frac{1}{2},\frac{-3}{4})$ Because the area $AMB$ is ...
0
votes
1answer
27 views

Finding/approximating possible antiderivative given d/dx at multiple points

Suppose I am given multiple x values and the derivative of f(x) at each point. Ex: d/d(0) = 0, d/d(2) = 2, d/d(3) = 3. How do I find a function with these derivatives?
1
vote
1answer
26 views

Interval of convergence? (Relatively simple question)

What is the interval of convergence of the power series: $\dfrac{(-1)^{(n-1)}x^n}{n^3}$ I know it should be |x| < 1, but does that mean the interval of convergence is $(1,-1)$ or $(-1,1]$ or $[-1,...
1
vote
1answer
26 views

Maximum for two-dimensional function (function of two variable) using calculus

$P(q_1, q_2) = 1000(q_1+q_2) - q_1^2 - q_2^2 - 2q_1\cdot q_2 - 100\cdot(q_1+q_2)$ Here's what I did: Partial derivative of the function with respect to $q_1$; $$\frac{\partial P}{\partial q_1}(q_1, ...
1
vote
2answers
61 views

$ I = \int_0^1 ((y')^2-y^2)dx $ and $I_1= \int_0^1 (y' + y \tan x)^2dx$

Let $$ I = \int_0^1 ((y')^2-y^2)dx $$ and $$ I_1= \int_0^1 (y' + y \tan x)^2dx$$ where y is a given function of $x$ satisfying $y=0$ at $x=1$. Show that $I - I_1 = 0$ and deduce that $I \geq 0$. ...
2
votes
1answer
26 views

Prove that $\frac{f(x)}{x^n}=\frac{f^{(n)}(\theta x)}{n!},0<\theta <1$ if $f^{'}(0)=…=f^{(n-1)}(0)=0$ using Cauchy's mean value theorem

I don't know how to apply theorem on the problem. By this theorem, if two functions $f$ and $g$ are defined on $[a,b]$ continuous on $[a,b]$, differentiable on $(a,b)$ and $g^{'}(x)\neq 0$ for every $...
-1
votes
2answers
54 views

Integrable functions (domain)

"Verify if both of this functions are integrable in their domains: $g(x) = ( \|x\|^2 + 1)^{ -\frac{a}{2} }, \forall a>n$, domain in $ \mathbb{R} ^n$ $h(x) = \|x\|^{-\|x\|}$, defined in $\mathbb{R}...
0
votes
2answers
283 views

What does it mean if the derivative of a function is a constant?

I was doing a homework problem to find the derivative of an equation and got "7" as the answer. I was trying to think about what it means if a derivative is a constant like that, is it just that the ...
2
votes
2answers
73 views

Why can't $x$ be negative in $x^{\ln{y}}$

According to wolfram alpha, the domain of $x^{\ln{y}}$ is $x>0$ and $y>0$ but putting $(-1,1)$ for $(x,y)$ I get a perfectly fine answer of 1? $y>0$ makes sense, since $\ln{y}$ is only ...
1
vote
1answer
126 views

Find $(f^{-1})'(a) = f(x) = 2x^3 + 3x^2+7x+4, a=4 $ [duplicate]

Find $(f^{-1})'(a) = f(x) = 2x^3 + 3x^2+7x+4, a=4 $ How do I go about solving a problem like this? What are the steps?
0
votes
1answer
20 views

Schwarz inequality and uniform converges

Let $$f_n(x) = \frac{x}{1+nx^2}$$. Show that for $x\ne 0$ $f_n(x)$ converges uniformly to some $f$. So the solution suggests the Schwarz inequality, yielding: $$\left|f_n(x)\right| \le \frac{\left|...
2
votes
2answers
49 views

When is the series converges?

Let the series $$\sum_{n=1}^\infty \frac{2^n \sin^n x}{n^2}$$. For $x\in (-\pi/2, \pi/2)$, when is the series converges? By the root-test: $$\sqrt[n]{a_n} = \sqrt[n]{\frac{2^n\sin^n x}{n^2}} = \...
1
vote
1answer
85 views

Explicitly demonstrating Stokes' theorem over a tetrahedron.

Consider the vector field: $$\vec{F} = \left(2x-y,-yz^2,-y^2z\right)$$ We are to explicitly show stokes theorem: $$\oint_\Gamma \vec{F}\cdot d\vec{x} = \int\int_{\partial V}\nabla \times \vec{F}\cdot ...
0
votes
1answer
51 views

The height of right isosceles triangle decreases with the speed proportional to the area of this triangle

The height of right isosceles triangle decreases with the speed proportional to the area of this triangle. At time $t=0$ the area of triangle is $2$, and at time $t=1$ the area of triangle is $\frac12$...
2
votes
2answers
43 views

Prove integral inequality of $C^1$ function

Let $f$ be $C^1$ class over $[a,b]$. Asuume that $f(a)=f(b)=0$ prove that: $\displaystyle \sup_{t\in [a,b]} |f(t)| \le \frac{1}{2}\int_a^b |f'(t)| dt$ I tried to show it from C-S ineq. for integrals ...
0
votes
2answers
65 views

Find a limit containing an definite integral

Let $f\in C[0,1]$ and $f\ge 0$. Find the limit: $$\lim_{n\to\infty} \left( \int_0^1 (f(x))^n \ dx \right)^{1/n}$$ My thought: We denote $f_n(x) = f(x)^n$. If we could show that $f_n(x)\to f$ ...
1
vote
1answer
51 views

Examine the continuity of function $f(x)=\frac{2x^2-4x}{|x+1|+|x-3|-2}$

Using the definition of absolute value for $$|x+1|=\begin{cases} x+1, & x\ge -1\\ -x-1, & x>-1 \end{cases}$$ and $$|x-3|=\begin{cases} x-3, & x\ge 3\\ -x+3, & x>3 \end{cases}$$ ...
0
votes
1answer
42 views

Find the Fourier coefficient of $f(x)$

Let $$f(x) = \begin{cases} \sum_{k=0}^\infty \frac{e^{inx}}{1+k^2} &\mbox{if } x \ne 2\pi k \\ 0 & \mbox{if } x = 2\pi k \end{cases}$$ Find the Fourier coefficients of $f(x)$ What I ...
0
votes
2answers
40 views

Proving the inequality $2^{-1+\frac{1}{n}}\left(x+1\right)\leq\left(x^{n}+1\right)^{\frac{1}{n}}$ for $n\in\mathbb{N} $ and $x>0$

Prove that for all $n\in\mathbb{N}$ and $x>0$, $$2^{-1+\frac{1}{n}}\left(x+1\right)\leq\left(x^{n}+1\right)^{\frac{1}{n}}$$ The last class was about Taylor polynomial of functions, so I ...
5
votes
3answers
140 views

evaluating $\lim_{n \to \infty }(\frac{n!}{n^n})^\frac{2n^4 + 1}{5n^5 + 1}$

I was trying to work out the following limit $$\lim_{n \to \infty }\Big(\frac{n!}{n^n}\Big)^{\Large\frac{2n^4+1}{5n^5+1}}$$ I have some idea about how to do it. Basically I expanded the $n!$ and ...
9
votes
5answers
813 views

What is the difference between necessary condition & sufficient condition?

My book says : For having extreme point $a$ of function $f$, the necessary condition is that $f'(a) = 0$. However, it isn't a sufficient condition. Now, what is the difference between necessary ...
-2
votes
4answers
127 views

Why isn't $0$ an extreme point of $f(x) = x^3$? [closed]

My book writes $0$ isn't a point of extremum for $f(x) = x^3$ inspite of being $f'(0) = 0$. Why is it so?
0
votes
0answers
68 views

Can someone explain presheaf to a calculus student?

From reading some stuff online, there is the claim that continuous functions are presheafs, so that the toy functions I play with in class such as $f(x) = x^2$ can also be thought of as presheafs. ...
0
votes
1answer
23 views

Vector-valued function to describe a hyberboloid

I need to find a vector-valued function to describe the quadric surface $x^2+y^2-z^2=1$. I could use the identity $\cosh^2 u - \sinh^2 u = 1$, but I'm not sure how. The best I could arrive at is $\vec{...
4
votes
1answer
79 views

Can someone explain to a calculus student what “dual space is the space of linear functions” mean?

I ran across this phrase today in a post and I am slightly confused. From my understanding, the dual space is the space of functions that sends a vector to a real number. There are two confusions: ...
0
votes
1answer
99 views

How do I study the varition of the function $f(x) =\sqrt{-|x|}$ in $\mathbb{R}$?

I want to know how can I study the variation of the function $f(x) =\sqrt{-|x|}$ in $\mathbb{R}$ and how I can draw it's graph? Thank you for your help.
5
votes
5answers
577 views

Advanced / In-Depth Calculus Book for Self-Edification

I am a pre-engineering student currently taking a Single Variable Calculus course at a community college. I recognize that my future success (or not so much) as an engineer will be based, in large ...
2
votes
1answer
880 views

Find the inverse $f(x) = 2x^2-8x, x>2 $

$$ 2x^2-8x, x>2 $$ What is the best way to solve this problem. $$x = 2y^2-8y $$ $$x = y (2y-8) $$ do I divide both sides by $y$ so as to solve for $y$? Help
4
votes
7answers
230 views

Evaluate $\lim\limits_{x\to\ 0}(\frac{1}{\sin(x)\arctan(x)}-\frac{1}{\tan(x)\arcsin(x)})$

Evaluate $$\lim\limits_{x\to\ 0}\left(\frac{1}{\sin(x)\arctan(x)}-\frac{1}{\tan(x)\arcsin(x)}\right)$$ It is easy with L'Hospital's rule, but takes too much time to calculate derivatives. What are ...
3
votes
1answer
53 views

Doubt about conservative fields in 2D and 3D

Regarding a conservative field $\vec{F}$ in a region $D \subseteq R^2$, I know that the requirements are: Curl of $\vec{F}$ is $0$. $\vec{F}$ is defined in D (doesn't have singularities in D). But ...
0
votes
2answers
111 views

Curve of intersection of two surfaces

Show that the curve of intersection of the surfaces $x^2+2y^2-z^2+3x=1$ and $2x^2+4y^2-2z^2-5y=0$ lies in a plane. I was able to do this by doubling both sides of the first equation and subtracting ...
0
votes
2answers
28 views

Changing differentiated variable in a mixed partial derivative

What operations do I need to perform the following conversion? $$ \frac{\partial ^2y}{\partial x\partial z} \mapsto \frac{\partial ^2y}{\partial x\partial t} $$
0
votes
1answer
24 views

What are the equilibria?

I have the following equations: $\frac{du(t)}{dt}=-au^2(t)+(a-1)u(t)$ $\frac{dv(t)}{dt}=av^2(t)-(a-1)v(t)$ I have to find equilibrium points for these equations. I figured I should look for the ...
1
vote
2answers
30 views

Rotation Around the Y-Axis

I have an equation: $y = -0.0122625x^2 + 120.38736$ and I want to rotate this around the y-axis and find the volume from the range 0 to 99. I have no idea how to do this and would greatly appreciate ...
3
votes
5answers
156 views

Evaluate $\lim\limits_{n\to\infty}(1+x)(1+x^2)\cdots(1+x^{2n}),|x|<1$

I get that the limit is $$\lim\limits_{n\to\infty}f(x)=1$$ because $$\lim\limits_{n\to\infty}(1+x^{2n})=1,\,\,\lvert x\rvert<1.$$ Is this right?
1
vote
1answer
121 views

Assume that f is a one to one function: If $f(x) = x^5 + x^3 +x$ , find $f^{-1}(3)$ and $f(f^{-1}(2))$

If $f(x) = x^5 + x^3 +x$ , find $f^{-1}(3)$ and $f(f^{-1}(2))$ How do I go about solving this? For example, since I am giving f inverse should $I = x^5 +x^3 + x = 3$ ?
7
votes
4answers
424 views

Can you show me a good approach for taking the limit of this function?

I tried to use binomial expansion, but I didn't get the same result. I would like to know how to approach this please. I know the answer is $\sqrt{e}$. My problem is : $$\lim\limits_{x\to 0} \left(1+...
1
vote
6answers
95 views

Determine whether function is 1 to 1: $f(x) = x^2-2x$

I am following these steps: 1) Write $y= f(x)$ 2) Solve this equation for $x$ in terms of $y$ (if possible). 3) To express $f^{-1}$ as a function of $x$, interchange $x$ and $y$. The resulting ...
3
votes
3answers
72 views

Examine the convergence of a sequence $\{a_{n}\}$ which is given by $a_{1}=a>0,a_{2}=b>0, a_{n+2}=\sqrt{a_{n+1}a_{n}},n\ge 1$

I used inequality between arithmetic and geometric means to show that a sequence $\{a_{n}\}$ is bounded: $$a_{n+2}=\sqrt{a_{n+1}a_{n}}\le \frac{a_{n+1}+a_{n}}{2}$$ Solving this, I get quadratic ...
2
votes
2answers
41 views

confusion in using Lebiniz integral rule

I was trying this question - Let $$f: (0,\infty )\rightarrow \mathbb{R}$$ and $$F(x) = \int_{0}^{x}tf(t)dt$$ If $F(x^2)= x^{4} + x^{5} $, then the value of $\sum_{r=1}^{12}f(r^{2})$ is I applied ...