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

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3
votes
3answers
44 views

simplifying $\int{\sqrt{1-4x^2}}\ dx$

i used the substitution $$x=\frac{\sin{u}}{2}$$ and I got to $$\frac{1}{4}(\frac{1}{2}\sin{(2\arcsin(2x))}+\arcsin(2x))+c$$ and $$2x=\sin(u)$$ and drew a triangle now im stuck... the answer is
0
votes
2answers
40 views

Integral of $e^{x^3}$

How do I find the integral of $e^{x^3}$. I have to do find the following integral and when I try to do integration by parts, I cannot find the integral of $e^{x^3}$. $$\int x^2 e^{x^3} ...
0
votes
0answers
17 views

Calculating volume by disc integration

What is the volume $V$ of the object created when the area formed by the lines $$y=x$$ $$y = 2-x^2$$ $$0 \le y \le 2$$ is rotated around the $y$-axis? It says that the answer is $\dfrac{5\pi}{6}$. ...
1
vote
0answers
15 views

Verify the divergence theorem on a ball?

When I verify the divergence theorem on the ball. I got $div(F)=1$, so $\int_{B_R(0)}div(F)dx$ is the volume of the ball, which is $\frac{4 \pi R^3}{3}$. And $n=(x,y,z)/R$, so $\int_{\partial ...
1
vote
2answers
48 views

Antiderivative of $\frac {dy}{dx}$

This is probably a very simple question, but I think its interesting. What I would think, based on my intuition (which I think is correct in this case) is that $$\int \frac {dy}{dx}=y$$ However, ...
-2
votes
1answer
16 views

Let $f(x)=\frac{3x}{x-q}$. Write down the equations of the vertical and horizontal asymptotes of the graph of $f$. [on hold]

This is calculus. Having a lot of trouble with vertical and horizontal asymptotes of lines.
0
votes
1answer
23 views

Expansion for Partial Fractions for $(3-2x)/(x^2+6x+9)$

I'm trying to expand $(3-2x)/(x^2+6x+9)$ into partial fractions to integrate. I'm doing $$(3-2x)/((x+3)^2)=A/(x+3)+B(x+3)^2$$ $$(A(x+3)+B)/((x+3)^2)=3-2x$$ for x=0:$$(3A+B)/9=3$$ for x=1: ...
3
votes
1answer
18 views

from Carathéodory Derivative definition to the derivative of $\sin(x)$

A function $f$ is Carathéodory differentiable at $a$ if there exists a function $\phi$ which is continuous at a such that $$f(x)-f(a)=\phi(x)(x-a).$$ For $f(x) = x^n$, $\phi(x) = x^{n-1} + ...
0
votes
1answer
32 views

Rudimentary calculus question [on hold]

Let $f(x) = g(x)/h(x)$, where $g(2)=18$, $h(2)=6$, $g'(2)=5$, and $h'(2)=2$. Find the equation of the normal of the graph $f$ at $x=2$. I know this is long. This is a calculus question. ...
-2
votes
1answer
24 views

Determine the sum of the following series. [on hold]

Determine the sum of the following series: $$\sum_{n=1}^\infty\frac{(-3)^{n-1}}{n^5}$$
0
votes
2answers
19 views

Let $f(x)=e^{2x}$. The line L is the tangent to the curve of $f$ at $(1,e^2)$. Find the equation of $L$ in the form $y=ax+b$ [on hold]

please help ! calculus ! really need to do this for my final exam. HELP its tomorrow
0
votes
0answers
23 views

Partial derivatives - Chain rule

Let $f(x, y, z)=e^{xz}\tan (yz)$ and $x=g(s, t)$, $y=h(s, t)$, $z=k(s, t)$. We set $m(s, t)=f(g(s, t), h(s, t), k(s, t))$. Find a formula for $m_{st}$ using the chain rule and verify that the result ...
1
vote
2answers
28 views

Find the maximum and minimum of the function $f$

Find the maximum and minimum of $f(x, y)=xy-y+x-1$ at the set $x^2+y^2\leq 2$. I have done the following: Since the region $x^2+y^2\leq 2$ is closed, $f$ has a maximum and a minimum, which is ...
1
vote
4answers
28 views

Use the definition of a limit to prove that the limit is equal to zero?

All I can think of to start is to state that: $$|n-∞| < \delta \Rightarrow |(c/n^2)-0| < \epsilon$$ But I don't know where to go from there
0
votes
2answers
21 views

In complex variables, why is |z-1| < 5 an open disk centered at +1, where the boundary is a circle of radius 5?

How can I justify this basic concept? Use the definition of the modulus? Write z = $e^{i\theta}$? ...and why is |z+1| < 5 ...centered at -1 and not +1? Thanks, Edit: it is always the basic ...
0
votes
1answer
31 views

Solve the system of differential equations

I plan on adding more into later just a bit stuck, researching it at the moment. Solve the system of differential equations $$\begin{bmatrix} x'\\y' \end{bmatrix} - \begin{bmatrix} -11&15\\ ...
1
vote
3answers
59 views

prove that $\lim_{x \rightarrow 0^+}\frac{1}{x} \int_0^x\sin(\frac{\pi}{t})dt =0$ [on hold]

I want to show that \begin{equation*} \lim_{x \rightarrow 0^+}\frac{1}{x} \int_0^x\sin(\frac{\pi}{t})dt =0. \end{equation*} Any idea?
0
votes
4answers
63 views

Does there exist a bijection [on hold]

Does there exist a bijection from (0,1) to $\Bbb{R}$? How to prove there is or not?
-2
votes
1answer
43 views

How to integrate a function with a nested absolute value: $|x^2 - 2|x||$? [on hold]

I need help with this problem, $$\int_0^4|x^2 - 2|x||dx$$ what should I do with $2|x|$ ?
2
votes
0answers
31 views

How to evalute: $\int_0^1 \frac{e^{-ax}}{ax} -\frac{e^{-abx}}{1- e^{-ax}}((1-x)\cos (\pi x) + \frac{3}{\pi} \sin(\pi x)) dx$ and $a, b >0$

How to evalute: $$\int_0^1 \left[ \frac{e^{-ax}}{ax} -\frac{e^{-abx}}{1- e^{-ax}}\left((1-x)\cos (\pi x) + \frac{3}{\pi} \sin(\pi x)\right) \right] dx$$ and $a, b >0$
0
votes
0answers
27 views

How to check whether the following function is concave or convex or neither.?

Let $\pi$ be a vector such that all its elements sum to 1. i.e, $\sum_1^n \pi(i) = 1$ where $\pi(i)$ denotes the i$^{th}$ component and $n$ is the length of the vector. Let $D$ be a diagonal matrix ...
1
vote
2answers
25 views

Second order differential equations where rhs $= 6e^2\cos(3x)$

Solve the differrential equation $$y'' - 4y' + 13y' = 6e^{2x}\cos(3x)$$ where $y(0)=3$ and $y'(0)=-8$ I think we start like... For the homogenous case $$\lambda^2 -4\lambda + 13 = 0 $$ ...
-1
votes
2answers
60 views

Limits and Trigonometry

Consider an function $f$ , defined as : $$f^k (\theta) =\sum_{r=1}^n \left( \frac{\tan \left( \frac {\theta}{2^r} \right) }{2^r} \right)^k +\frac 1 3 \sum _{r=1}^n \left( \frac { \tan \left( ...
3
votes
3answers
83 views

evaluate the sum $\sum_{n=1}^{\infty}\sum_{k=n}^{\infty}\frac{1}{(n^2+n-1)(k^2+k-1)}$

I'm trying to evaluate this sum $$\sum_{n=1}^{\infty}\sum_{k=n}^{\infty}\frac{1}{(n^2+n-1)(k^2+k-1)}$$ I have no idea how to deal with it. With one sum I can, with partial-fraction decomposition, ...
0
votes
1answer
16 views

Differential of the greatest integer function

So I know that the derivative of the greatest integer function is zero. That is if $f(x) = [x]$ then $df/dx = 0$. Then, a friend asked me for the differential , $df$ of $f(x)$. My answer was zero. He ...
-1
votes
1answer
17 views

continuous functions and limit existance

Let, $C\in \mathbb R$ and let $f(x)= Cx^2+1$ if $x \geq 2$ , $f(x)= 10-x$ if $x<2$ for what value of $C$ is $f(x)$ a continuous function.
0
votes
0answers
17 views

Find the extremas of the fuction $f$

I have to find the extremas of $f(x, y)=3x+2y$ subject to $2x^2+3y^2 \leq 3$. Since the region $2x^2+3y^2 \leq 3$ is closed, $f$ has a maximum and a minimum, which is either at the boundary or at ...
-1
votes
1answer
38 views

Theorem of Lagrange multipliers - Extremas of $f$

I have to find the extremas of $f(x, y, z)=x+y+z$ subject to $x^2-y^2=1$, $2x+z=1$. I have done the following: We will use the theorem of Lagrange multipliers. The constraints are ...
0
votes
1answer
12 views

Is the following property suffictient for second order differentialbility?

Let $U\subset R^n$ be an open set, and $f:U\to\mathbb R$ a $C^1$ function. Suppose that for any $x_0\in U$, there exists a $n$-variable-polynomial $T_{x_0}$ of degree at most $2$ such that, ...
0
votes
4answers
37 views

First order differential equation: did i solve this equation right

So i'm trying to solve: $$x^2\frac{dy}{dx} + 2xy = y^3$$ I'm given this differential equation, that Bernoulli equation: $$\frac{dy}{dx} + p(x)y = q(x)y^{n} $$ I think i've solved it and ...
0
votes
0answers
8 views

Optimal Space-Travel Departure Time (Issues deriving and solving complex expressions).

Problem This problem aims to determine the optimal time to depart for an intergalactic destination, taking into account the fact that in a number of years technology back on the planet you left may ...
3
votes
4answers
520 views

Prove that limit doesn’t exist anywhere? [on hold]

I'm doing some practice problems and am having trouble answering these problems: Consider the following function $$f(x)=\begin{cases}1, & \text{if } x\in \Bbb Q\\ -1, & \text{if } x\in \Bbb ...
6
votes
2answers
61 views

A simple way to find $\lim_{n\rightarrow\infty}{\frac{1}{n^2}\sum_{k=1}^n{\sqrt{n^2-k^2}}}$

I was reading an exam paper used to identify gifted high-school students, and I encountered the following problem: $$\lim_{n\rightarrow\infty}{\frac{1}{n^2}\sum_{k=1}^n{\sqrt{n^2-k^2}}}$$ Using ...
0
votes
0answers
17 views

continue on some strange summation formulas ..by william Gosper

could you show if is it true the following expressions? $$\sum _{z=1}^{\infty } \frac{(-1)^z \cos \left(\sqrt{\pi ^2 a^2+b z^2+c}\right)}{z^2}=\frac{b \sin \left(\sqrt{\pi ^2 a^2+c}\right)}{4 ...
1
vote
1answer
57 views

How to find $\frac{0}{0}$ limit without L'Hôpital's rule

I am having trouble solving this limit. I tried applying L'Hôpital's rule but I got $\frac{0}{0}$. $$\lim_{x\to0} {\frac{\frac{1}{1+x^3} + ...
1
vote
1answer
39 views

How to find bounds of this integral $\int_0^{10} \frac{x}{\sinh \frac{x}{2}}dx$

How to find bounds of this integral: $$\int_0^{10} \frac{x}{\sinh \frac{x}{2}}dx$$ I try but I get that integral not converges. Thank you.
1
vote
1answer
36 views

First order differential equation: how do I prove that $u$ satisfies the differential equation

So I'm given this differential equation, that Bernoulli equation: $$\frac{dy}{dx} + p(x)y = q(x)y^{n} $$ now it says: Show that if $y$ is the solution of the above Bernoulli differential ...
1
vote
2answers
20 views

Deriving energy equation (Kinetic)

A particle of mass $m$ moves on the $x$-axis under a force $$F(x)=-2x+2\epsilon x^2$$ Use newton's second law, $F=m\ddot x$ to derive the energy equation $$\frac{1}{2}m\dot x^2+V(x)=E_0$$ where ...
1
vote
1answer
35 views

Apply chain rule to $u = y^{1 - n}$ in order to find $\frac{du}{dx}$

Let $u = y^{1 - n}$. I know that, by using the chain rule: $$\frac{du}{dx} = \frac{du}{dy} \cdot \frac{dy}{dx}$$ Also, I know that $\frac{du}{dy} = (1 - n)y^{-n} = \frac{1 - n}{y^{n}}$ Now, for ...
4
votes
3answers
280 views

Decomposition into partial fractions to compute an integral

I'm having problems with: $$\int_{-\infty}^{\infty}\frac{x^4+1}{x^6+1}dx$$ I was thinking: $\frac{x^4+1}{x^6+1}$ is an even function and the interval $(-\infty,\infty)$ is symmetric about 0, we ...
1
vote
0answers
36 views

Is $lim$ an operator? [duplicate]

In my calculus I lecture notes the prof said that $lim$ satisfies the properties of linearity as well as multiplicity. This looks like what an operator might do. Can we characterize $lim$ as an ...
3
votes
2answers
36 views

How to precisely define $C^\infty$ in $f(x) \in C^\infty$

In single variable calculus, a common way to denote a function that is continuous for all derivatives is to write $f(x) \in C^\infty$ i.e. $f(x) = \exp(x)$ Is there a more rigorous way to define ...
0
votes
0answers
31 views

How fast is the distance between two points changing.

I am having a difficulty with the following question from my calculus unit. Bus station A is located 100km west of bus station B. At 12pm a bus leaves station A driving south at 70km/h and a bus ...
0
votes
1answer
23 views

Taylor Polynomial - intuition

How do adding higher derivatives of the function on the same point gives a better approximation?
0
votes
3answers
43 views

What is the volume and surface area of the 1-Sphere?

I am reading a post on here that mentioned something about the 1-sphere. I know that a 2-sphere is a circle, and 3-sphere is a volume, but what is this 1-sphere and how do you calculate the volume and ...
3
votes
2answers
73 views

Why are radians used in calculus. [duplicate]

Ok, please ignore my silliness. So, why do we use radians in calculus and why is it considered more scientific than degrees. And how did mathematicians know or prove that radians would work for all ...
1
vote
1answer
16 views

Shortest Path with Constraint

What is the length of the shortest path that goes from $(0,2)$ to $(12,1)$ that touches the $x$-axis? I tried using calculus to solve this problem (i.e.: distance is: $$ \sqrt{(x-0)^2 + (0-2)^2} + ...
1
vote
0answers
36 views

How to compute the unit outer normal at the point in a curve?

Given a smooth closed curve $f(x,y)=0$, How to compute the unit outer normal at each point $(x_{0},y_{0})$ in the curve?
0
votes
0answers
23 views

what would the equation of a torus be by making the circunference $(y-2)^2+ z^2 = 0$ and $x=0$ turn along the $z$ axis

What I understand of the question is that I have to, somehow, give the equation of the torus that results of spinning the circumference $$(y-2)^2 + z^2 = 0$$ and $$x=0$$ which as far as I know is just ...
0
votes
4answers
42 views

Big-O notation — is it mainly used to classify rate of growth or rate of decay to zero?

For example, $e^{x} = 1 + x + x^2/2 + O(x^3)$, and we interpret $O(x^3)$ as the remainder term that goes to zero like $x^3$. What's the primary usage of Big-O notation? (strictly in math classes, ...