For basic questions about limits, derivatives, integrals, and applications, mainly of one-variable functions.
17
votes
2answers
122 views
$\int_0^{\infty}\frac{x^3}{(x^4+1)(e^x-1)}\mathrm dx$
I need to find a closed-form for the following integral. Please give me some ideas how to approach it:
$$\int_0^{\infty}\frac{x^3}{(x^4+1)(e^x-1)}\mathrm dx$$
1
vote
1answer
55 views
Error bounds for $e$
Prove that for all $n\in\mathbb{N}_+$, we have $$(1+\frac{1}{n})^n>\sum_{k=0}^n \frac{1}{k!}-\frac{e}{2n}.$$
2
votes
4answers
69 views
$\frac{d}{dx}\int_{0}^{e^{x^{2}}} \frac{1}{\sqrt{t}}dt$
I'm having trouble understanding how to apply the $\frac{d}{dx}$when taking the anti-derivative.
$$\frac{d}{dx}\int_{0}^{e^{x^{2}}} \frac{1}{\sqrt{t}}dt$$
In class it was mentioned we'll end up taking ...
0
votes
1answer
24 views
Differentiate $y = \sqrt {{{1 + 2x} \over {1 - 2x}}} $ logarithmically
$\eqalign{
& y = \sqrt {{{1 + 2x} \over {1 - 2x}}} \cr
& \ln y = {1 \over 2}\ln (1 + 2x) - {1 \over 2}\ln (1 - 2x) \cr
& {1 \over y}{{dy} \over {dx}} = {1 \over 2} \times {2 ...
2
votes
2answers
80 views
$\int_0^{\pi/4}\!\frac{\mathrm dx}{2+\sin x}$ , $\int_0^{2\pi}\!\frac{\mathrm dx}{2+\sin x}$
Please help me integrate
$$\int_0^{\pi/4}\!\frac{\mathrm dx}{2+\sin x}$$
and
$$\int_0^{2\pi}\!\frac{\mathrm dx}{2+\sin x}$$
I've tried the standard $u = \tan \frac{x}{2}$ substitution but it looks ...
1
vote
1answer
32 views
Understanding the average slope of a curve
This question mainly asks: Is my understanding of the average slope correct? This question is somewhat related to my previous question. However, its different from the previous question to a certain ...
1
vote
1answer
44 views
What about this $\lim_{x \to \infty}\frac{3x+4}{\sqrt[5]{x^9+3x^4+1}}$?
When I saw this limit, I didn't even try to solve it by an algebraic method. I thought about the assyntotic concept.
In the example,
$$\frac{3x+4}{\sqrt[5]{x^9+3x^4+1}}\sim ...
3
votes
5answers
130 views
2
votes
3answers
36 views
Second Derivative of basic fraction using quotient rule
I know this is a very basic question but I need some help.
I have to find the second derivative of:
$$\frac{1}{3x^2 + 4}$$
I start by using the Quotient Rule and get the first derivative to be:
...
-1
votes
2answers
96 views
The euclidean space $\Bbb R^n$ is orientable as a manifold.
I know that
The euclidean space $\Bbb R^n$ is orientable as a manifold.
I think that it is orientable because it has a nowhere vanishing $n$-form.
But I am not sure.
Please can you explain ...
2
votes
4answers
59 views
Approximation of alternating series $\sum_{n=1}^\infty a_n = 0.55 - (0.55)^3/3! + (0.55)^5/5! - (0.55)^7/7! + …$
$\sum_{n=1}^\infty a_n = 0.55 - (0.55)^3/3! + (0.55)^5/5! - (0.55)^7/7! + ...$
I am asked to find the no. of terms needed to approximate the partial sum to be within 0.0000001 from the convergent ...
1
vote
0answers
24 views
Gradient in Cylindrical ccordinates
How can I express grad of $\Phi$ in cylindrical coordinates?. In fact I wanted to check the vector $\nabla\Phi$ is perpendicular to a surface $\Phi=c$ where $c$ is a constant. Where to start? Thank ...
0
votes
0answers
23 views
Matrix calculus: derivative of product
I'm attempting to find the derivative with respect to $\beta$ of:
$\mbox{RSS}(\beta) = (\mathbf{Y} - \mathbf{X}\beta)^T(\mathbf{Y}-\mathbf{X}\beta),$
where $\mathbf{Y}, \mathbf{X}$ and $\beta$ are ...
1
vote
3answers
43 views
Integration of a rational function from +/- infinity
I am trying to calculate the integral
$$\int_{-\infty}^{\infty}{\frac{a+x}{b^2 + (a+x)^2}\frac{1}{1+c(a-x)^2}}dx$$
where $\{a, b, c\}\in \mathbb{R}$. I have looked in a table of integrals for ...
0
votes
1answer
42 views
$ \int_{0}^{\infty}{\dfrac{\cos(ax)}{(x^2 + 1)^2}dx} $
I have a contour integral problem I need to solve, but I don't know the answer, so I wanted to verify that my work is correct.
$$ \int_{0}^{\infty}{\frac{\cos(ax)}{(x^2 + 1)^2}dx} $$
For this one, ...
0
votes
2answers
33 views
How to solve this integral $\int \frac{(1+2x^2)}{x^2(1+x^2)}dx$
Problem : How to solve this integral
$\int \frac{(1+2x^2)}{x^2(1+x^2)}dx$
I thought it should be $ x + 3x^2$ in the numerator so that I will take $x+x^3$ = u then taking derivative both sides and ...
2
votes
1answer
34 views
Derivative of Linear Map
I'm reading Allan Pollack's Differential Topology and got stuck on this argument: In the second paragraph of page 9, section 1.2 he said
"Note that if $f:U\to \mathbf{R^m}$ is itself a linear map ...
1
vote
4answers
74 views
What is the derivative of $\ln(4^x)$?
What is the derivative of $\ln(4^x)$ (which I believe is also equal to $x\ln4$)?
Is it $\dfrac{1}{x\ln4}$?
1
vote
2answers
37 views
Values of a parameter $x$ in an infinite series that makes it converge
I am required to find the values of $x$ in the following infinite series, which cause the series to converge.
$$\sum_{n=1}^\infty \frac{x^n}{\ln(n+1)}$$
I tried to use the ratio test, and found that ...
2
votes
1answer
38 views
Helpful to review certain calculus topics before first real analysis course?
This is my first time posting, so I apologize in advance if my question is inappropriate here. I wanted to know if it would be beneficial for me to review certain calculus topics before I take my ...
0
votes
1answer
28 views
Using complex logarithms to solve equations
Could someone please just explain the formula/method for solving the complex equation $$e^{iω}=k$$ where $k∈C$.
As an example, I know that when $ω=x+iy$, $e^{2iω}=1$ has solutions $ω = n\pi$ for ...
1
vote
2answers
24 views
Why does $7^{2\ln x}\cdot \ln(7) \cdot (2/x)$ equal to $7^{2\ln x}\cdot \ln(49) /x$?
While reviewing, I came upon this problem which has the derivative
$7^{2\ln x}\cdot \ln(7) \cdot (2/x)$
simplified to
$7^{2\ln x}\cdot \ln(49) /x$
How/why is it simplified like that?
0
votes
2answers
54 views
Integrating $\int{\frac{1}{1+e^{x}}}dx$, Partial Fractions(?)
I need help with this integral:
$$H(x) = \int{\frac{1}{1+e^{x}}}dx$$
It should be easy, but I'm stuck. I thought about using a u-substitution but I didn't get any further. Am I meant to use partial ...
-1
votes
1answer
44 views
Find the amount of nuts and bolts produced when $P=500$
A factory simultaneously manufactures nuts and bolts. Suppose that x and y denote the amount (measured by weight in kilograms) of nuts and bolts respectively produced by the factory in a day, with ...
2
votes
2answers
61 views
How do I solve for $dy/dx$ if $y=\ln (\sin x+\ln x)$?
Solve for $\frac{dy}{dx}$ if $y=\ln(\sin x+\ln x)$.
I know how to solve for integrals involving $du$ and $u$, but how do I do this type of problem (I think it's the opposite of the integral problem)?
...
2
votes
1answer
49 views
How to place a limit that it's inside the integral, outside.
I did this:
$$\int_{1}^t x^{-1}dx=\int_{1}^t\lim_{n\rightarrow -1}{x^n}dx =\lim_{n\rightarrow -1}\int_{1}^t{x^n}dx $$ just to have a way to approximate $\ln t$. $$\ln{t}=\lim_{h\rightarrow ...
1
vote
2answers
35 views
Spivak problem on Schwarz inequality
I have a question regarding problem 19 in the 3rd Ed. of Spivak's Calculus. Specifically, part (a). The question concerns the Schwarz inequality:
$$
x_1y_1 + x_2y_2 \leq ...
0
votes
1answer
25 views
Suppose you invest \$10 at 10.2% per annum compounded annually. How many years would it take for your investment to grow to \$15 000?
I'am solving a simlar equation to this and just trying to figure out how they did it?
the only part I don't understand is how they got the number.... 1.102
15000 = 10(1.102)n ¬1 mark
1500 = 1.102n
4
votes
4answers
119 views
Differentiate $\log_{10}x$
My attempt:
$\eqalign{
& \log_{10}x = {{\ln x} \over {\ln 10}} \cr
& u = \ln x \cr
& v = \ln 10 \cr
& {{du} \over {dx}} = {1 \over x} \cr
& {{dv} \over {dx}} ...
1
vote
0answers
22 views
Reduction of order to find general solution of linear nonhomogenous differential equations. [closed]
Find the general solution of y''+p(t)y'+q(t)y=g(t) assuming y=v(t)y(t) is a solution. Use reduction of order to find the general solution.
1
vote
4answers
112 views
How to integrate $\int_0^\infty \frac{1}{1+y^4} dy$ [duplicate]
I tried the trigonometric substitution $y^2 = \tan \theta, sec^2\theta = 1 + y^4$
But now I'm stuck with $\frac12 \int \frac{\sqrt{\sin \theta}}{(\cos\theta)^{\frac92} } d \theta$
I ran out of ...
2
votes
1answer
52 views
Solving for a matrix from its quadratic form
I have a set of vectors that I am trying to predict from another set of vectors using a matrix $W$. To find this matrix, I decide I want to minimize the $\ell^2$ norm of the error, e.g.:
$$
...
0
votes
1answer
22 views
Integral, set and parametric representation
I am to compute the following: $\displaystyle\iiint\limits_V 1\, dx\, dy\, dz$,
where $V= \{{(x,y,z) \in \mathbb R^3 : (x-z)^2 +4y^2 < (1-z)^2} \text{ and } 0<z<1\}.$
Does anyone have idea ...
0
votes
5answers
36 views
Parametric equations for given line
How would you find the parametric equations for:
1) a line through $(3,1)$ and $(-5,4)$.
2) a segment joining $(1,1)$ and $(2,3)$.
Can anyone show me the steps of doing it cause the way my textbook ...
2
votes
5answers
115 views
Can someone please explain $e$ in layman's term? [duplicate]
I never really understood what $e$ means and I'm always terrified when I see it in equations. What is it? Can somebody dumb it down for me? I know it's a constant. Is it as simple as that?
1
vote
1answer
20 views
Finding the value of t where tangent line is perpendicular to x axis
For the curve x = t$^2 - 1, y = t^2 - t$, the tangent line is perpendicular to x-axis, where
Options are :
a ) t = 0
b) $t \to \infty$
c) $t = \frac{1}{\sqrt{3}}$
d) $t = \frac{-1}{\sqrt{3}}$
...
0
votes
4answers
89 views
How to integrate $\int_0^\infty e^{-ty^2} \sin t dt$
My book suggests that I do some sort of limiting
$\lim_{A \to \infty} \int_0^A e^{-ty^2} \sin t d t$
But I'm not getting anywhere.
0
votes
2answers
29 views
Perimeter or Calculus Word problem
A rectangular plot of farmland will be bounded on one side by a river
and on the other three sides by a single strand of electric fence.
With 1400m of wire at your disposal, what is the largest ...
1
vote
4answers
57 views
Integral of $\int \frac{x^4+2x+4}{x^4-1}dx$ [duplicate]
I am trying to solve this integral and I need your suggestions.
$$\int \frac{x^4+2x+4}{x^4-1}dx$$
Thanks
7
votes
1answer
134 views
Prove that $f(1)-f(1/e)\le \int_0^1 \sqrt{x} f'(x) dx$
Let $f:[0,1]\rightarrow \mathbb{R}$ be a differentiable function such that
$$f(x^2)+f(y^2)\le2 f(\sqrt{x y}), \space x,y\ge0 $$
Prove that
$$f(1)-f(1/e)\le \int_0^1 \sqrt{x} f'(x) dx$$
Where should ...
0
votes
2answers
41 views
Finding the indefinite integral $\int \frac{3x+2}{(6x^2+8x)^7}\,\mathrm dx$
I'm not too familiar with how to solve this. Could anyone present a step by step guide on how to get the answer?
$$\int \dfrac{3x+2}{(6x^2+8x)^7}\,\mathrm dx$$
7
votes
5answers
159 views
What does $x^\pi$ mean? [duplicate]
I was just wondering, what does $x^\pi$ or for that matter, $x$ raised to any irrational number mean? For example, I want to represent $x^2$ then that would mean $x * x$ or if I want to do ...
3
votes
3answers
100 views
How do you integrate the following trigonometric function involving sin and cos?
How do you integrate the following functions:
$$\int \left( \frac{\cos\theta}{1+\sin^2\theta} \right)^2 \, d\theta$$ and $$\int \left( \frac{\cos\theta}{1+\sin^2\theta} \right)^3 \, d\theta
$$
...
-1
votes
1answer
35 views
prove that $||d^2f(x)||\le M \Rightarrow ||df(x)||\le \sqrt{2Mf(x)}$
let E be a banach space , $f : E \to \mathbb R$ a function of $C^2$ / $f>0$ we suppose that $\exists M $ cte and :
$||d^2f(x)||\le M $
prove that :
$||df(x)||\le \sqrt{2Mf(x)}$
2
votes
0answers
33 views
Stoke's theorem application to curl theorem. I did. Please can you check it?
Now, I need to apply stoke's theorem to curl theorem.
My teacher gave a hint.
Accourding to the hint, I accept $w=Pdy∧dz +Q dz∧dx + R dx∧dy$ $\in Ω^2(M)$
$dim(M)=2$
M is the subset of $\Bbb ...
0
votes
1answer
54 views
Theorem or just a change of varibles?
I have a formula in my text:
$$\int \int_{S} F \cdot n dA= \int \int_{w} F(G(u,v)) \cdot (dG_{u}\times dG_{v}) du dv$$
I am really lazy and hate remembering formulas to me this looks like a ...
1
vote
2answers
31 views
Help making Lipschitz proof rigorous
A function $f:R→R$ is defined to be Lipschitz if there is a constant $K>0$ such that for all $a,b∈R$ $$|f(a)−f(b)|≤K|a−b|$$
Suppose $f:R→R$ is Lipschitz. Prove that $f$ is continuous.
Could ...
0
votes
1answer
41 views
How to express the sum of a set?
Suppose I have a set of numbers. How can I express in set-theory terms the sum of the elements in that set?
1
vote
1answer
25 views
how to calculate $d\Omega(f)$ here
the question was to find $d \Omega(f)$ with :
$$ \Omega : (E,[.]) \to (F,||.||) \\f \to -f'' +f^3$$
$
[f] = |f'(0)| + ||f''|| $
; $ ||f|| = Sup_{[0,1]}|f(x)| $
the answer is given to me like this ...
0
votes
1answer
65 views
How to calculate $\sum_{i=2}^n {\frac 1{\log_2 i}}$
How to calculate
$$\sum_{i=2}^n {\frac 1{\log_2 i}}$$
