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

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4
votes
1answer
33 views

Why $\lim$ of $\cos(f)$ equals to $\cos$ of $\lim(f)$?

Let $$\lim_{n\rightarrow \infty}\left(\cos\left(\frac{n\pi}{n+1}\right) \right) = \cos\left(\lim_{n\rightarrow \infty}\left(\frac{n\pi}{n+1} \right)\right)$$ Why the $\cos(x)$ function can be ...
2
votes
3answers
37 views

Is the argument I used to evaluate the convergence of the series $\sum_{n=1}^{\infty} (-1)^{n-1}\frac{n+a}{(n+b)(n+c)}$ right?

If $a,b,c$ be real constants, analyze the convergence of $$\sum_{n=1}^{\infty} (-1)^{n-1}\frac{n+a}{(n+b)(n+c)}$$ What I tried to: I compared the general term of my series to $\frac{1}{n}$: ...
1
vote
1answer
17 views

General theorem about all inflection points

Let $f$ be a function. I know that, if $c\in dom(f)$ and either $f''(c)=0$ or $f''(c)$ is undefined, then $c$ may be an inflection point. Can there be inflection points such that $c\in dom(f)$ and ...
0
votes
1answer
64 views

How long will it take me to learn calculus?

First and foremost I want to point out that I am 14 years old currently and love mathematics. Summer is coming and that means I'll have 2 months to try to learn calculus to as far as I can get ...
0
votes
0answers
31 views

Suggestion for modern reference on calculus

I need a book for reference in conference paper. Actually, i use Green's theorem: If functions $P(x,y)$ and $Q(x,y)$ satisfy $$\frac{\partial P}{\partial x} = \frac{\partial Q}{\partial y}$$ in ...
0
votes
1answer
18 views

Taylor Polynomial- Choosing A Point

How does the point we choose to develop the Taylor Polynomial has effect on the approximation? I came across Runge's phenomenon, so roughly speaking we can say we should not develop near the ends of ...
1
vote
0answers
29 views

integral inequalities and continuous functions

Let $f$ be a positive, continuous function on $\mathbb{R}$. Let $c\in (0,1/2)$ be a constant and $\lambda>1$. I want to prove that: (1). for any $a\in\mathbb{R}$, there exists $\delta(a)>0$ ...
0
votes
1answer
13 views

maximum of function in bounded area

How can i calculate maximum of $ \frac{-1}{(x+y+3)^{2}} $ in [-1 1]x[-1 1] with non numeric method. I know that -0.2 is maximum of this function with numeric method and The Hesian matrix is zero . ...
1
vote
1answer
29 views

Wrong derivation of limit of Cesàro mean

It's known that $$\lim_{n\rightarrow\infty}x_{n}=a\Rightarrow\lim_{n\rightarrow\infty}\frac{\sum_{i=1}^{n}x_{i}}{n}=a$$ Consider the following derivation: ...
3
votes
1answer
34 views

Asymptotic ratio of two series

Assume $\{a_n\}$ and $\{b_n\}$ are two positive series such that $$\sum_{n}a_n=\sum_n b_n=1.$$ Assume also for all $n$, $\sum_{k\geq n}a_k\leq \sum_{k\geq n}b_k$ and $$\lim_{n\rightarrow ...
0
votes
1answer
27 views

Non-STEM applications of Calculus? [on hold]

I was talking with a friend, who is a liberal arts major, about the everyday applications of math. We agreed about how Algebra directly applied to the average person's life, but the only examples of ...
0
votes
1answer
38 views

What situations should $\oint$ be used? [on hold]

Can someone clarify when we should use the contour integration symbol?
0
votes
2answers
21 views

Yes or No: The following $\infty\triangleq\sup{\mathbb{R}}$ and $-\infty\triangleq \inf\mathbb{R}$ holds

Can someone verify whether $\infty\triangleq\sup{\mathbb{R}}$ and $-\infty\triangleq \inf\mathbb{R}$ is mathematically rigorous?
0
votes
1answer
23 views

Visualizing Integration

Imagine you have the equation for the circumference of a circle which is $2*\pi*r$ and you want to integrate the perimeter of a circle to get its area. We could do this using an indefinite integral ...
3
votes
3answers
65 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
55 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
1answer
27 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
51 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
24 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
25 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
26 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
26 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
29 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
35 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
22 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
32 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
65 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
66 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 $2|x|$ ?
3
votes
0answers
37 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
26 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
72 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
89 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
20 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
40 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
531 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
68 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.