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

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2
votes
2answers
98 views

Derivative of $(-2)^{x+1}$ [on hold]

Can we compute the derivative of $(-2)^{x+1}$? This may sound silly, but think about it. We cannot apply any of our formulae on it. I think we may have to go old school with this one
0
votes
4answers
78 views

Proving $\log(b^a) = a \log(b)$ using calculus

Sorry, this is a really simple question, but I'm trying to teach myself calculus and can't figure it out. If we define $\log(b) = \frac{db^x}{dx}(0)$ how does one prove $\log(b^a) = a\log(b)$? I ...
0
votes
1answer
28 views

Double checking my progress on finding critical numbers

still cramming for the test tomorrow. (This cram has lasted most of my weekend) I'm having trouble with determining the critical values for this example problem. The problem is as follows: $$F(x) = ...
2
votes
2answers
79 views

A transcendental number from the diophantine equation $x+2y+3z=n$

Let $\displaystyle n=1,2,3,\cdots.$ We denote by $D_n$ the number of non-negative integer solutions of the diophantine equation $$x+2y+3z=n$$ Prove that $$ \sum_{n=0}^{\infty} ...
-1
votes
1answer
42 views

Math question: Calculus [on hold]

"A rancher would like to enclose two adjacent rectangular corrals that cover a total area of $12,000 ft^2$. If material for the fence costs $3.5$ usd per foot, find the dimentions (length and width) ...
2
votes
1answer
27 views

convergence of infinite integral

Given f(x) continuous in $[0,\infty)$, and $$\int_0^\infty \left|f(x)\right|dx ~, ~ \int_0^\infty f^4(x)dx$$ converges, prove that $$\int_0^\infty f^2(x)dx$$ converges. My motivation is to show that: ...
8
votes
2answers
168 views

A Binet-like integral $\int_{0}^{1} \left(\frac{1}{\ln x} + \frac{1}{1-x} -\frac{1}{2} \right) \frac{x^s }{1-x}\mathrm{d}x$

I met this integral $$ \int_{0}^{1} \left(\frac{1}{\ln x} + \frac{1}{1-x} -\frac{1}{2} \right) \frac{ \mathrm{d}x}{1-x} \qquad (*) $$ while evaluating this log-cosine integral. I made several ...
0
votes
0answers
44 views

How prove $S_{k}(x)=\sum_{i=1}^{n}x^k_{i}$ this System of equations The only solution?

when I read a china book,I see this follow interesting problem (the author says it is clear have follow) if give the number $S_{k}(x),k=1,2,3,\cdots,n$ ,and such $$\begin{cases} ...
0
votes
0answers
19 views

Interesting properties of functions and sets that depends on dimension of space.

For $n=1$ (or $m=1$), we have some basic properties of functions and sets that are not valid (or not necessarily valid) for $n\neq 1$ (or $m\neq 1$). For exemple: Calculus. Let $[a,b]$ be a closed ...
2
votes
0answers
48 views

Ordinary differential equation­

$$\dfrac{dy}{dx}-\dfrac{\tan y}{1+x}=(1+x)e^x\sin y$$ I tried $\sin y=t$ but failed. It seems to immune to methods I know of or I am just unable to make the right substitution... Wolfram alpha ...
3
votes
1answer
33 views

Calculating limit in parts. Why possible?

Let $f$, continuous function, differentiable at $x=1$ and $f(1)>0$. Consider the following equation: $$\lim \limits_{x\to 1} ...
1
vote
0answers
11 views

Derivative of a generalized hypergeometric function

Let $$f(a)={_2F_3}\left(\begin{array}c1,\ 1\\\tfrac32,\ 1-a,\ 2+a\end{array}\middle|-\pi^2\right).$$ How to find $f'(0)$ in a closed form?
5
votes
1answer
116 views

Evaluation of $\int_{0}^{\frac{\pi}{2}}\frac{\sin (2015x)}{\sin x+\cos x}dx$

Evaluation of $\displaystyle \int_{0}^{\frac{\pi}{2}}\frac{\sin (2015x)}{\sin x+\cos x}dx$ $\bf{My\; Try:}$ Let $\displaystyle I = \int_{0}^{\frac{\pi}{2}}\frac{\sin (2015x)}{\sin x+\cos x}dx\tag{1}$ ...
3
votes
3answers
94 views

$\int_{0}^{\pi/2}\ln\left(1+4\sin^4 x\right)\mathrm{d}x$ and the golden ratio

We already know that, for any real number $t$ such that $t\geq-1$, $$ \int_{0}^{\pi/2} \ln \left(1+t \sin^2 x\right) \mathrm{d}x = \pi \ln \left( \frac{1+\sqrt{1+t}}{2} \right). $$ Prove that ...
2
votes
2answers
30 views

Establish the absolute maximum of a function

We have this function:$$f(x)=\begin{cases} \sin(x) \cdot\ln(\sin2x), & \mbox{if }0<x<\pi/2 \\ 0, & \mbox{if }x=0,\mbox{or }x=\pi/2 \end{cases}$$ So, how to prove that it decreases and ...
2
votes
2answers
74 views

Pick a smart function

Our teacher wants us to find a function $f$ on $(0,\pi)$ such that $$\sqrt{\sin(x)} f(x)^{\frac{1}{4}} =k_1 + \cos(x)$$ and $$\sqrt{\sin(x)} f(x)^{-\frac{1}{4}} = k_2 + \cos(x).$$ The two constants ...
0
votes
0answers
12 views

Numerical solution of first order ODE

I have an in-homogeneous ODE. $R'(x)-(C_1 +C_2 x) R(x) = R_1-C_1 R_0\, x \tag 1$. What I know is the constant matrix $ R(0)$ as initial condition. Question:- how to find out R(1) by numerical ...
1
vote
1answer
45 views

Finding Cauchy principal value for: $ \int_1^\infty \frac{ a x^2 + c }{x^4 - b x^2 - c} \mathrm{d}x $

I need to solve the integral $ \displaystyle \mathcal{P} \int_1^\infty \frac{ a x^2 + c }{x^4 - b x^2 - c} \mathrm{d}x $, where $\mathcal{P}$ is the Cauchy principal value, $ - 1 \leq c \leq 1$ and ...
1
vote
1answer
67 views

Alternate series [duplicate]

The alternate series $S=\displaystyle \sum_{k=2}^{\infty} \frac{(-1)^n}{\sqrt{n}+(-1)^n} $ converges? $S$ is absolutely convergent?
2
votes
2answers
133 views

integrate $ \int \frac {x dx}{\sqrt {1+x^{10}} } $

This is a tough one. Thanks. $$\int \frac {x dx}{\sqrt {1+x^{10}} } $$ This is not a homework problem. I spend 10 hours over the course of 3 days on this. I tried: 1) substituting u for x^5 to get ...
3
votes
2answers
33 views

The quadratic case in nonlinear programming

I'm reading about nonlinear programming and I stumbled into the following statement where I started to wonder a bit: Consider the function $$f(\textbf{x}) = ...
0
votes
1answer
13 views

Determine the maximum cross‐sectional area.

The client wants to maximise the volume of a materials store to be constructed next to a 3 metre high stone wall (shown as OA in the cross section in the diagram). The roof (AB) and front (BC) are ...
2
votes
4answers
123 views

Using integral definition to solve this integral

I'm trying to solve this question using the definition of integral: $$\int^5_2 (4-2x)dx$$ Definition of integral: We define first the inferior and superior sum: Let $f:[a,b]\to \mathbb R$ be a ...
0
votes
1answer
23 views

Find if a rectangle passes through another in cartesian plane

I want to know how to prove or find out if the red big rectangle passes through one of these small rectangles i have the coordinates of the big rectangle (the top left) and i have it's width and ...
0
votes
0answers
27 views

Weierstrass substitution for solving trigonometric eqations

I'm trying to solve a set of equations of a parallel robot . The equations can be writen as $x(\cos(\theta),\sin(\theta))$ $y(\cos(\theta),\sin(\theta))$ so to solve the equation I used Weierstrass ...
3
votes
2answers
87 views

Solving $\frac{d f(x)}{dx} + f(x-1) = x^2$

Given following differential equation: $$\frac{d f(x)}{dx} + f(x-1) = x^2$$ where $ f(x)=0 $ for $x \leq 0 $. How do I find the solution for $ x \geq 0 $ ? I understand that for $ 0 \leq x \leq 1 ...
0
votes
1answer
39 views

Use mean value theorem on $f(x) = x^{1/5}$, to show that $2< \sqrt[5]{33}<2.0125$

The problem specifically aks us to use mean value theorem on the interval $[32, 33]$ It has always puzzled me that mean value theorem can be used to prove Inequalities. Can anyone show how mean ...
2
votes
1answer
29 views

Does the limit $\lim_{k\to\infty}\int|\cos kx |f(x) d\lambda(x)$ always exist?

Let $f$ be a Lebesgue integable function. Does the limit $$\lim_{k\to\infty}\int|\cos kx |f(x) d\lambda(x)$$ always exist?
1
vote
0answers
14 views

existence of solution of volterra integral equation of the first kind

$$ \int_0^t k(s,t)f(s)ds=g(t) $$ To know the existence and uniquness of solution of volterra integral equation(VIE) of the first kind, we differentiate it and convert to the VIE of the second kind. ...
1
vote
4answers
34 views

equation of line tangent to integral

Given a function, $$F(x)= \int_{-2}^x 3-4t \;\mathrm{dt}$$ find the equation of the line tangent to $F(x)$ at $x=1$ I'm having difficulty understanding why evaluating $F(1)$ (equal to $15$) is ...
1
vote
1answer
60 views

Evaluation of $\int e^x\tan x\left(1-2\sec^2 x\right)dx$

Evaluation of Following Integrals. $(a)\;\;\displaystyle \int e^x\tan x\left(1-2\sec^2 x\right)dx\;\;\;\; (b)\;\; \int \left(1+x-\frac{1}{x}\right)e^{x+\frac{1}{x}}dx$ $\bf{My\; Try::}$ For $(a)$ ...
1
vote
1answer
40 views

Solution to diff eq

Check whether the function $y=\sin(3x)/3$ is a solution of $xy'+y+3\cos3x$ with the initial condition $y(\pi)=0$ Find $xy'$ for the function $y=\sin(3x)/3$ I am a ex-math minor who is just trying ...
2
votes
1answer
91 views

What does “calculus” mean?

"calculus" and "formal system" From http://en.wikipedia.org/wiki/Propositional_calculus#Terminology a calculus is a formal system that consists of a set of syntactic expressions ...
0
votes
3answers
53 views

Showing $\lim_{n \to \infty}\left(1 + \frac{x}{n}\right)^n = e^x$ using implicit and log differentiation

Hey guys I'm looking over my review sheet for an upcoming test and I'm having trouble with this problem. Apparently I'm supposed to use implicit differentiation and log differentiation, and I'm ...
3
votes
2answers
26 views

radius of convergence when root test fails

I'm stuck on this problem: Find the radius of convergence of $\sum \limits_{n=1}^\infty \frac{x^n}{((2+(-1)^n)^n} $ An attempt: From the root test, it seems $L$ does not exist: $$L= \lim_{n ...
-1
votes
1answer
28 views

determine the maximum cross‐sectional area.

The client wants to maximise the volume of a materials store to be constructed next to a 3  metre high stone wall (shown as OA in the cross section in the diagram). The roof (AB) and  front (BC) are ...
31
votes
6answers
2k views

Fun Geometric Series Puzzle

I recently was reminded of a puzzle I solved in college and thought I'd give it a shot again. However, being distanced from college math, I am having a harder time remembering how I arrived at the ...
0
votes
0answers
30 views

In general, let $f \in L^2(-1,1)$ and let $g:R \to R$, $g(x)=f( \{ x \} )$. How are the Fourier transforms of f and g related?

Recall that $\{ x \}$ is a decimal part of a real number $x$. For example, if $x=3.41$, then $\{ x \}=-0.41$. Part A: Let $f(x)=2x$, with $x \in (-1,1)$ and let $g:R \to R$, $g(x)=2 \{ x \}$. Sketch ...
1
vote
2answers
35 views

Use the Fourier inversion formula to compute h(x) when $\hat{h}(x)=\frac{1}{(1+y^2)^2}$

Use the Fourier inversion formula to compute $h(x)$ when $\hat{h}(x)=\frac{1}{(1+y^2)^2}$. $$\hat{h}(x)=\frac{1}{(1+y^2)^2}=\frac{1}{1+y^2} \times \frac{1}{1+y^2}=\widehat{\frac{1}{2}e^{-|x|}} \times ...
2
votes
4answers
111 views

Proving exponential is growing faster than polynomial

Let $P(x)$, a polynomial which isn't the zero-polynomial. I want to prove the following limits $$\lim \limits_{x\to\infty} \left|P(x)\right|e^{-x} = 0$$ $$\lim \limits_{x\to-\infty} ...
0
votes
1answer
38 views

Calculus question that is driving me nuts. Rotation volume. [on hold]

Find the volume formed by rotating the region enclosed by: $y=\sqrt[5]{x}$ and $y=x$ about the line $y=25$.
0
votes
2answers
63 views

Proof that a function is continuous in R

I have some trouble with this problem, I'll write what I did Problem: $ f(x) = x^2-2x $ Prove f continuous in $\Bbb R$. My solution: I need to prove that: $$\lim_{x \to x_0} f(x) = f(x_0)$$ Or ...
4
votes
1answer
106 views

find the minimum value of $x^2-6x+9+ \dfrac{64}{x^2}$

Looking for an elegant solution. I can do by brute force, that is finding derivative and double derivative. All Ideas will be appreciated and tried by me.
1
vote
2answers
114 views

Evaluate $\int_0^1 \sqrt{2x-1} - \sqrt{x}$ $dx$

I'm trying to calculate the area between the curves $y = \sqrt{x}$ and $y= \sqrt{2x-1}$ Here's the graph: I've already tried calculating the area with respect to $y$, i.e. $\int_0^1 ...
1
vote
0answers
16 views

Integrating with indicator functions

I want to evaluate $$\int_{-\infty}^{\infty}(A_1e^{-\beta_1(b-x-y)}+B_1e^{-\beta_2(b-x-y)})(pn_1e^{-n_1y}1_{\{y\geq0\}}+qn_2e^{n_2y}1_{\{y<0\}})dy,$$ $b>x, \beta_1<n<\beta_2$. I am trying ...
0
votes
0answers
31 views

Convergence of a series using limit test

As per my understanding: When we take $\lim_{n \rightarrow \infty}$ of a function, it should approach a finite number, it converge. And if the opposite is true, it diverges. Now, the test for ...
2
votes
2answers
43 views

Partial Fractions with an irreducible quadratic factor

$\int \frac{2}{(x-4)(x^2+2x+6)} dx$. this is a partial fraction with irreducible quadratic factors. I know how to set it up and I found A, B, and C. 2 = A((x^2)+2x+6) +(x-4)(Bx+C). then I plugged ...
1
vote
0answers
58 views

To show a power series is a Taylor series

Is it possible to prove if $f(x) = \sum_{n = 0}^\infty a_n(x - a)^n$ then the series is the Taylor series of $f$ without using complex analysis, as done here?
1
vote
0answers
31 views

Summation of Recurrence (Convergent series)

I have solved this issue. Would you please verify whether I am correct or not? Motivation for the post is our previous discussion link.I am restating my problem with additional elaborated explanation ...
2
votes
3answers
86 views

Prove that $\lim \limits_{x\to \infty}e^{\frac{\ln(x)}{x}}=1$

How to prove that $\lim \limits_{x\to \infty}e^{\frac{\ln(x)}{x}}=1$? I know that $x$ grows much faster to infinity then $\ln(x)$, therefore the limit equivalent to $e^0 = 1$ but that's not a ...