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

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2
votes
2answers
70 views

integrate $\int \frac{(16-9x^2)^{\frac{3}{2}}}{x^6}dx$

$$\int \frac{(16-9x^2)^{\frac{3}{2}}}{x^6}dx$$ $$\int \frac{(16-9x^2)^{\frac{3}{2}}}{x^6}dx=\int \frac{3\left(\frac{16}{9}-x^2\right)^{\frac{3}{2}}}{x^6}dx$$ $x=\frac{4}{3}\sin\theta$ ...
1
vote
2answers
50 views

Problem with Indefinite Integral $\int\frac {\cos^4x}{\sin^3x} dx$

I'm stuck with this integral $\int\frac {\cos^4x}{\sin^3x} dx$ which I rewrote as $\int \csc^3x \cos^4xdx$ then after using the half angle formula twice for $\cos^4x$ I got this $\frac 14\int ...
1
vote
1answer
51 views

Evaluate $\int_{-\pi}^{\pi}\frac {\sin nx}{(1+2^x) \sin x} $

Evaluate $$\int_{-\pi}^{\pi}\frac {\sin nx}{(1+2^x)\sin x}dx \:\:\: n \in \mathbb{N}$$ $$\int_{-\pi}^{\pi}\frac {\sin nx}{(1+2^x)\sin x}dx = \int_{0}^{\pi}\frac {\sin nx}{(1+2^x)\sin x}dx + ...
1
vote
2answers
63 views

Why is the function continuous at a point which gives the case 0/0?

I have this function : $f(x) = \frac{6x^2+18x+12}{x^2-4}$, the domain is R. How come its graph is continuous at $x = -2$? I know it can be simplified to $\frac{6(x+1)}{x-2}$ ( firstly $f(x) = ...
-5
votes
0answers
20 views

Calculus: Proving Continuous Function by Intermediate Value Theorem [duplicate]

Prove step by step: Let $f(x)$ be a continuous function from the closed interval $[a, b]$. Use the Intermediate Value Theorem to show that $f(x)$ has a fixed point, that is, there is a point $x \in ...
1
vote
0answers
68 views

Sum involving binomial coefficient and gamma function

I was wondering if anyone has ever seen the following sum: \begin{equation} \sum_{j=0}^{n} \left(-1\right)^{j} \binom{n}{j}\frac{\Gamma\left(\mu+j\right)}{\Gamma\left(\mu+j+n+1\right)} ...
1
vote
2answers
98 views

How to differentiate this integral?

Given $$g(x)=\int_{0}^{x} (x-t)e^{t}dt$$ find out $g''(x)$ I thought of using Lebnitz theorem to differentiate it but using Lebnitz I get this $g'(x)=1\cdot (x-x)e^{x}=0$ I don't know how to find ...
0
votes
0answers
38 views

On summation of series [on hold]

Consider the equality of summations $\sum_{a} f(a) = \sum_{a}f(1-a)$ where both sums are convergent. What conditions need to be satisfied such that $f(a) = f(1-a)$ for all $a$, where $a$ is a ...
0
votes
1answer
30 views

Use Rolles Theorem to show that the function $x^{n}+kx+l=0$ has at most 2 roots if $n$ is even?

"Use Rolles Theorem to show that the function $x^{n}+kx+l=0$ has at most 2 roots if $n$ is even, and at most 3 roots if $n$ is odd?" To do this, I assume I must show that there are certain values of ...
0
votes
3answers
57 views

Proving equation has only one solution

So i want to prove that $$x^2e^x=1$$ has at least one solution for $$x\in\mathbb{R}$$ I am kinda lost and would appreciate any help. This is suppose to be solved using basic calculus but i am not ...
0
votes
2answers
28 views

What is the instantaneous rate of change in the real world?

I can't grasp this concept of an instantaneous change of rate. How could a point on a function graph have a rate of change in the first place? In this moment I just know that it is named the ...
2
votes
2answers
19 views

Property of real functions when derivative approaches zero

This is a question from my exam in Calculus 1. Problem 6 Let $f: [0,\infty[ \to \mathbb{R}$ be continuously differentiable and $\lim_{x \to \infty} f'(x) = 0$. a) Show that for $n \in ...
2
votes
4answers
61 views

compute the value of an indefinite integral

Help me please with this indefinite trigonometric integral. How can I solve this kind of integrals? $$\int\limits \frac{1}{\left(\cos^4(x) \cdot \sin^2(x)\right)}dx$$
1
vote
1answer
18 views

Trigonometric Function Simplification: $T_2 (x) = \cos (2 \arccos x)$

Let $T_n (x) = \cos (n \arccos x)$ where $x$ is a real number, $x \in [–1, 1]$ and $n$ is a positive integer. Show that $$T_2 (x) = 2x^2 – 1.$$ My attempt: $T_2 (x) = \cos (2 \arccos x)$ ...
8
votes
2answers
64 views

Solving an exponential equation with different bases

Solve the equation $2^x + 5^x = 3^x + 4^x$. I can figure out two special solutions $x=0$ and $x=1$, and I try to prove that they are the only two solutions. However, I find it hard to do so because I ...
0
votes
0answers
22 views

meaning of a dense subset

I'm trying to understand something - say I want to prove a certain property of functions in a space X. Is it enough to prove this property over functions which belong to a dense subset of X?
0
votes
0answers
29 views

Let ${a_n}{x^n} + … + {a_1}{x^1} - f(t) = 0$ and $f(t) $ is decreasing function of $t$. Can we say that $y(t)$ is decreasing function of $t$?

Let $t\in (0,1)$ and ${a_n}{x^n} + .... + {a_1}{x^1} - f(t) = 0$ $f(t) $ is continuous decreasing function of $t$. $a_i\ge0$ for all $i$. $y(t)$ is zero of equition. Can we say that $y(t)$ is ...
0
votes
1answer
22 views

Find the volume of the solid generated by the region [on hold]

Find the volume of the solid that is generated when the region enclosed by $ y = \cosh 2x, y = \sinh 2x, x = 0, $ and $ x = 5 $ is revolved around the x-axis.
0
votes
0answers
47 views

Prove that the following function is convex?

I am trying to prove that the function $$g(\alpha)=\ln\Big(\sum_{i=1}^{n}t_i^\alpha+A(\alpha)\Big) ~~t_i, \alpha>0,$$ where $A(\alpha)=\int_{a}^{b}x^{\alpha+\beta-1}e^{-\lambda x^\beta}\,dx$,is ...
1
vote
1answer
30 views

Finding the absolute maximum and minimum within an interval?

For some arbitrary function: $f(x)$ within the interval $a<x<b$, should I just calculate the roots for $f'(x)$, and the points $f(a)$ & $f(b)$, and make deductions based around what is going ...
2
votes
1answer
70 views

Find $\int\limits^{\infty}_{0}\int\limits^{\infty}_{0}{\frac{1}{(x+y)^{3/2}}\exp\left\{-\frac{a^2}{2(x+y)}\right\}}\,dy\,dx$.

In my posterior probability computation, I got the following integration and I could not figure it out. ...
6
votes
0answers
65 views

Confusion with Courant: Which of his two calculus books is THE one?

Since I've worked my way through Spivak's Calculus book, I thought I'd give Courant's allegedly fantastic exposition of the subject a go as well. However, I've run into a problem. People in ...
0
votes
1answer
17 views

Are difference and differential operators commutative?

I was searching information, theorems, etc... about commutativity between difference and differential operators but I dont found explicit statements so Im unsure about my assumptions (so I ask here in ...
1
vote
1answer
26 views

Creating an integral for finding the volume of this revolution

I need to find the volume of a solid that is created by rotating the area within the following boundaries: $y=x^3$ $y=8$ $x=0$ which is rotated over $x = 3$. I thought I had the correct integral ...
0
votes
1answer
29 views

Bifurcation Diagram question for Population harvesting model $P' = rP (1-\frac{P}{K}) - hP$

A deer population grows logistically and is harvested at a rate proportional to its population size. The dynamics of population growth is modeled by $P' = rP (1-\frac{P}{K}) - hP$ where $r$ (the ...
0
votes
0answers
10 views

Partial derivative of polynomial dependant on previous time values

I have not touched calculus for a few years, and I am not sure what is going on here. Any help would be greatly appreciated :) Essentially, let $p_{t} = \log p(y_{t}|h_{t},h_{t+1})$, where ...
7
votes
2answers
130 views

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that $f'(x)$ is continuous and $|f'(x)|\le|f(x)|$ for all $x\in\mathbb{R}$

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that $f'(x)$ is continuous and $|f'(x)|\le|f(x)|$ for all $x\in\mathbb{R}$. If $f(0)=0$, find the maximum value of $f(5)$. $f'(x)=f(x)$ ...
4
votes
2answers
512 views

If a function is discontinuous at one point, then filled in, is it now continuous?

I am looking at the continuity of the following function $f(x) = \sin(1/|x|), f(0) = 0$ So this is $f(x) = \sin(1/|x|)$ filled in at $x = 0$ Clearly, $\lim\limits_{x \to 0} f(x) = 0 $ by squeeze ...
-1
votes
3answers
69 views

Without computing, is the integral of $\int_0^1 t(t-1)(t-2)\,dt$ positive or negative? [on hold]

I have to graph the function, but I don't think I'm doing it right. Here is a picture of it Sorry, this is my first time using this site and I don't know how to use MathJax yet.
1
vote
0answers
37 views

Differentiating a matrix product

In one of the books I found that given that for a linear system $x'=Ax$, there exists a matrix $Q:=\int\limits_0^\infty B(t)dt$, where $B(t)=e^{tA^T}e^{tA}$, and $V(x) = x^T Q x$, ...
0
votes
2answers
27 views

Confusing result obtained taking second derivative of ye^y

I was doing my calculus homework, and one of the questions asked for the first and second derivative of $ye^y=x$, I did the computations and arrived at $-(x+1)^{-2}$, which was a lot neater and ...
1
vote
4answers
65 views

Solving $\lim_{n \to \infty} \sqrt{n} \sin\left({\sqrt{n+3}-\sqrt{n-2}}\right)$

I have trouble finding the value of the following limit: $$\lim_{n \to \infty} \sqrt{n} \sin\left({\sqrt{n+3}-\sqrt{n-2}}\right)$$ For now I have rewritten the term into: $$ \lim_{n \to \infty} ...
1
vote
0answers
19 views

Function with both easy to find Fourier and Hermitian coefficient

I'm writing some notes on Spectral theory and I would like to make a simple example finding the generalized fourier coefficient of a function in respect of two different bases. I was thinking about ...
1
vote
3answers
34 views

Integrate the following equation. (exponential function)

Integrate $$\frac{e^x -2}{e^{x/2}}$$ This is my calculation: but it is wrong....
2
votes
2answers
25 views

Vector-Valued Functions and Continuity

Why is it that when a vector-valued function $r(t)$ is continuous at some time $t$ then $\|r(t)\|$ is also continuous at that time $t$, but the converse is not true? That if $\|r(t)\|$ is continuous ...
0
votes
0answers
39 views

Shortest distance between two functions/curves

I'm completely stumped with this, I've tried looking at other questions asking the same kind of thing without success. I am given two functions, $f(x)=x^2+4x+6.2 $ and $h(x)=-3x^2-5$ and I am asked ...
0
votes
2answers
42 views

Why is $\cos\left(\frac{3\pi}{2}-t+2k\pi\right) = -\sin(t)$ [on hold]

Why is this true? $$\cos\left(\frac{3\pi}{2}-t+2k\pi\right) = -\sin(t)$$
0
votes
2answers
49 views

Evaluating $\lim_{x\to 0}{\frac{\sin^2x}{2x^2}}$ without L'Hospital

I have been trying to evaluate $$\lim_{x\to 0}{\frac{\sin^2x}{2x^2}}$$ Finally, I used the L'Hospital's Theorem and I got the answer $1/2$, but I wonder if there is a way to solve this without this. ...
0
votes
3answers
48 views

Area of a rectangle within a curve

The cargo space of a bulk carrier is 60m long. The shaded part of the diagram represents the uniform cross-section of this space. It is shaped like a parabola with equation ${{1\over 4}x^2, - 6 \le ...
0
votes
1answer
18 views

An “extra” solution to an initial value problem

So I came up with this example when I was teaching: consider the IVP $$ y'(x) = xy-x-5y+5, y(0)=1. $$ The standard approach is to separate variables: $y'(x) = (x-5)(y-1)$, which allows me to ...
0
votes
2answers
50 views

Find the inverse $\dfrac{x}{\|x\|}$ in $\mathbb{R^2}$

I wish to find the inverse of $\dfrac{x}{\|x\|}$, where $x \in \mathbb{R}^2$ Let's do this. Let $$y_1 = \dfrac{x_1}{\sqrt{x_1^2+x_2^2}}$$ $$y_2 = \dfrac{x_2}{\sqrt{x_1^2+x_2^2}}$$ Then $$y_1 = ...
1
vote
2answers
33 views

Can I prove a function is continuous by looking at the domain?

I came across the following question in a calculus book: For the function $$f(x)=1-\sqrt{1-x^2}$$ show that it is continuous on the interval $$-1≤x≤1$$ The solution in the book showed that the one ...
1
vote
1answer
32 views

Combining two results from partial integration

I have a set of two PDEs: $$\partial_{\tau}\theta+\partial_{\eta}\psi=0$$ $$\partial_{\tau}\psi=-\partial_{\eta}\theta+\alpha\partial_{\eta}^{2}\psi$$ These can be combined into a wave equation of ...
1
vote
1answer
20 views

Computing a line integral where the curve is in polar coordinates

Compute $\int \limits_{C} F.dr$ for $F(x,y)=(y,x)$ and $C$ is the curve given by $r=1+\theta$ for $\theta \in [0,2\pi]$ My Attempt Am I correct in saying that $F$ is a conservative vector field ...
0
votes
2answers
26 views

How to simplify inverse trigonometric function

How to simplify the following equation: $$\sin(2\arccos(x))$$ I am thinking about: $$\arccos(x) = t$$ Then we have: $$\sin(2t) = 2\sin(t)\cos(t)$$ But then how to proceed?
1
vote
0answers
30 views

Fourier coefficients of the Gaussian.

I would need to find the fourier coefficient of this gaussian for a problem. I'm now stuck with this integral, \begin{equation} c_{n}=\int_{-1}^{1}e^{\frac{x^{2}}{2}}\left(\cos\left(\pi ...
0
votes
3answers
26 views

How do we find more appropriate constants for expansions of functions?

We all knonw that the expansion of $e^x$ is $$1+x+x^2/2+...$$. But what if I want to find more approximate expansion of $e^x$. I try that $$e^x-1-c_0(x)+(c_0+c_1)(x^2/2)-(c_0+c_1+c_2)(x^3/3)=0$$ and ...
3
votes
2answers
67 views

How to derive: $\left(1 + \frac{1}{n}\right)^n < 1 + 1 + \frac{1}{2}+…+\frac{1}{2^{k-1}}$

In my textbook they write the following inequality: $\left(1 + \frac{1}{n}\right)^n < 1 + 1 + \frac{1}{2}+...+\frac{1}{2^{k-1}}+...+\frac{1}{2^{n-1}}$ They say that they derive this inequality by ...
4
votes
2answers
45 views

Prove that $\lim\limits_{x\to\infty} f'(x)=0$

Let $f$ be a function in $(0,\infty)$ such that $f'(x)$ exists. In addition, $\lim\limits_{x\to \infty} f'(x)=L$ (finite) and $f(n)=0$ for every $n \in \Bbb N$. Prove that ...
0
votes
1answer
17 views

How to represent y as a function of w?

Assume : $F(y)=G(w)$ where $F,G$ are two real-valued functions from $R \to R$. We want to find the function $C(w)$ such that : $F'(y)=C(w)$ and C should be built based on F and G. Thanks so much.