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

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2
votes
2answers
58 views

Wallis Product for $n = \tfrac{1}{2}$ From $n! = \Pi_{k=1}^\infty (\frac{k+1}{k})^n\frac{k}{k+n} $

How does $$ \prod_{k\ =\ 1}^{\infty}\left(\,\,\sqrt{\, k+1 \over k\,}\,{k \over k + 1/2}\,\right) ={\,\sqrt{\,\pi\,}\, \over 2} ={\sqrt{2\left(\,\pi/2\,\right)} \over 2} ={1 \over 2}\,\,\sqrt{\, ...
1
vote
2answers
53 views

Generating functions for $\log^3(1-x)$ of $\log^3(x)$

I am trying to find generating functions which will give me a power logarithm. I am trying to find generating sums in the form $$\sum_{n=1}^{\infty} a_n\,x^n = -\frac{\log^2(1-x)}{1-x}$$ or ...
2
votes
5answers
90 views

Trigonometric substitution and Integration of $\frac{1}{x^2\sqrt{x^2+1}} $

Regarding the integral $$ \int \frac{dx}{x^2\sqrt{x^2 + 1}} $$ I'm not sure what to do about the extra $x^2$ in the denominator. What can I do about it?
0
votes
1answer
34 views

Proof of limit of a piecewise function, rational, irrational

Prove that: If $f(x) = 0$ for irrational $x$ and $f(x) = 1$ for rational $x$ then $\lim_{x \to a} f(x)$ does not exist for any $a$. So begin by the opposite assumption: Assume $\lim_{x \to a} f(x) ...
3
votes
2answers
55 views

Surface area of a solid of revolution: Why does not $ \int_{b}^{a} 2\pi \,f(x) \,dx $ work?

Why does not $ \int_{b}^{a} 2\pi \,f(x) \,dx $ yield the correct answer when calculating the surface area of a solid of revolution?
1
vote
2answers
61 views

Why does this inequality stand?

I want to ask something about: "Since $i \log_e i$ is concave upwards, it is easy to show that $$\sum_{i=2}^{n-1} i \log_e i \leq \int_2^n x \log_e x \,dx \leq \frac{n^2 \log_e ...
0
votes
2answers
30 views

Directional derivatives exercise from Courant's introduction to calculus and analysis

Show for $z=f(x,y)=\sqrt[3]{xy}$ that $f$ is continuous and that the partial derivatives $\partial z/\partial x$ and $\partial z/\partial y$ exist at the origin but that the directional derivatives in ...
1
vote
2answers
26 views

Raising and Lowering Through Differentiation

I'm calculating the Christoffel symbols of the second kind which is of course defined as multiplying the symbol of the first kind multiplied by the contravariant metric. I was thinking of how to make ...
0
votes
1answer
23 views

Find the volume $z \geq 3x^2+2y^2, \ \ 3x^2+2y^2+5z^2 \le 1$

Find the volume of solid defined by the following inequalities : $$z \geq 3x^2+2y^2, \ \ 3x^2+2y^2+5z^2 \le 1$$ We have an ellipse, which the semi-axis are $\sqrt{\frac{z}{2}}$ and ...
2
votes
1answer
38 views

Determining a radius convergence of a power series

Let $$ \sum_{n=0}^\infty \frac{(-1)^n}{3n+1} x^{3n+1} $$ Is there an immediate way to determine $R=1$?
0
votes
1answer
555 views

proof of the second generalized mean value theorem for integrals

Let $f,g,g´$ be continous on $[a,b]$ and $g$ monotone on $[a,b]$; then there exist $c\in (a,b)$ so that $$\int_{a}^{b}f(x)g(x)dx=g(a)\int_{a}^{c}f(x)dx+g(b)\int_{c}^{b}f(x)dx$$ Ineed to apply the ...
1
vote
3answers
47 views

Fundamental Theorem of Calculus 1 - definite integral

I have two problems, they're not from a book so I can't check the answer for one of them and the other I'm not sure on what to do. $$ {d\over dx}{\int^{1}_{x^{2}}} {\sqrt{t^{2}+1}} {dt} $$ $$=-{d\over ...
0
votes
0answers
21 views

Show existence of a sub-sequence $(f_{n_k})$ which is uniformly convergent to a function in $C[0,1]$

Let $f_n:[0,1]\rightarrow R$ be a sequence of continuously differentiable function, Let $M>0$ be such that for any $0\le x \le 1$ and natural $n$, $|f_n(x)|$, $|f'_n(x)|<M$ Show existence of a ...
5
votes
6answers
113 views

Show that $h(x)=x^5+3x+6$ is one to one

How do I show that $$h(x)=x^5+3x+6$$ is one to one? I set $$f(a)=f(b)$$ and try to isolate for $a$ and $b$ but I get stuck because I have a term of "$a$" that is degree $5$ and a term of "$a$" that is ...
1
vote
4answers
69 views

Maximum of subtended angle $\theta$

Following Problem, from Jim Fowler's Mooculus class: A painting is mounted on a wall. The bottom of the painting is 5 feet above eye level, and the top of the painting is 14 feet above eye level. If ...
1
vote
2answers
28 views

Spline approximation for $g(t) = \frac{t e^{-t}}{(x+t^2)^2}$

Is there any nice way to do a spline approximation for $$ g(t) = \frac{t e^{-t}}{(x+t^2)^2}\,, $$ where $x$ is some constant? I tried finding nice interpolation points, however this proved very ...
2
votes
4answers
114 views

Finding the function of these numbers $1, 2, 5, 13, 34, 89, 233, 610$

Firstly I used the differences between them but I found the numbers return again. How can I find the function of these numbers
1
vote
0answers
65 views

Prove the Swartz inequality using $ 2xy \leq x^2 + y^2 $

Im really bad at analysis and this problem was recommend to me to help me grasp some basics of $\epsilon $ $\delta $ So im doing a problem ( though its like 12 pieces ) this is i guess the fourth ...
0
votes
1answer
25 views

Computing the value of a function whose derivative is another function

Apologies if this is something relatively trivial, my calculus is a bit rusty. Let say I have function $f(t)$ which is increasing at a non-constant rate. This rate is also a function of $t$, lets say ...
16
votes
6answers
159 views

What is the importance of $\sinh(x)$?

I stumbled across $\sinh(x)$. I am only a calculus uno student, but was wondering when this function comes into play, and what is its purpose? Last, does it have world applications, or is it a ...
2
votes
1answer
50 views

Surface of revolution of an ellipse

I have been working on this question, but I end up getting the wrong answer overtime: The ellipse $$\frac{x^2}{a^2}+ \frac{y^2}{b^2} = 1$$ where $a>b$ is rotated about the $x$-axis to form a ...
0
votes
1answer
28 views

Is this enough to demonstrate divergence of an improper integral?

The integral in question is $$\int_0^\infty (f(x)-a)^2dx$$ Where f(x) is some continuous function and a is some constant. When we expand the integrand,we end up with an $a^2$ term. We can then ...
0
votes
0answers
29 views

Minimize total cost of one kilometer

The cost of the fuel consumption of a locomotive is proportional to the square of its speed plus 100 pounds per hour without regard to its speed. The cost of the fuel consumption is 25 pounds per hour ...
0
votes
2answers
20 views

Calculus minimum cost for an open box

An open box with a squared base of volume $128 \ m^3$. The cost of the material used for the base of the box is $2$ pounds per $m^2$, and that of the material used for the lateral faces is $0.5$ ...
4
votes
2answers
103 views
+100

Find Minimum value of $P=\frac{1}{1+2x}+\frac{1}{1+2y}+\frac{3-2xy}{5-x^2-y^2}$

Given: $x,y\in (-\sqrt2;\sqrt2)$ and $x^4+y^4+4=\dfrac{6}{xy}$ Find Minimum value Of $$P=\frac{1}{1+2x}+\frac{1}{1+2y}+\frac{3-2xy}{5-x^2-y^2}$$ Could someone help me ?
1
vote
1answer
41 views

Proving the existence of a sequence of polynomials convergent to a continuous function $f$.

I need to show that if $f$ is continuous function ($f:\mathbb{R}\rightarrow \mathbb{R}$), then there exists a sequence of polynomials which converges to $f$ on any compact subset of $\mathbb{R}$. I ...
-1
votes
2answers
28 views

Calculus: Maximum profit for a factory [on hold]

A factory earns 20 pounds for each unit if it produces 800 units per week. If the production will increase, the profit for each unit will decrease 0.02 pounds. Find the number of units to be produced ...
2
votes
1answer
20 views

Finding the PDF from the CDF where the CDF is not differentiable at some point

I got the following problem: Let $X$ be a continuous random variable with $CDF$ denoted $F_X$ defined as follows: $F_X(x)= \begin{cases} 1-x^{-4/3}, & x\in[1,\infty) \\ 0, & x\in ...
2
votes
3answers
22 views

Determine monotone intervals of a function

Let $$ f(x) = \int_1^{x^2} (x^2 - t) e^{-t^2}dt. $$ We need to determine monotone intervals of $f(x)$. I tried to differentiate $f(x)$ as follows. $$ f'(x) = \left(x^2 \int_1^{x^2} e^{-t^2}dt \right)' ...
5
votes
5answers
112 views

Why is the antiderivative of $1/(1+x^2)=\tan^{-1}(x)$?

My textbook says the antiderivative of $1/(1+x^2)$ is $\tan^{-1}(x)$. To confirm this to myself I took the derivative of $\tan^{-1}(x)$ expecting to get $1/(1+x^2)$, but instead I ended up with ...
5
votes
2answers
54 views

Evaluating sums using residues $(-1)^n/n^2$

I am an alien towards compelx analysis, with very little know I am posing a question, who someone may want to help with. Evaluate: $$\frac{1}{4}\cdot \sum_{n=1}^{\infty} \frac{(-1)^n}{n^2}$$ In ...
0
votes
1answer
28 views

Find the area between the two functions--integrals [on hold]

Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Then find the area of the region $y=5x^2$ and, $y=x^2+3$
2
votes
8answers
112 views

How to show that $f(x) = 0$ if $\int_a^bf(x)\,\text{d}x=0$ for all $a,b\in\mathbb{R}$?

I found this problem on the web: Let $f(x)$ be a real-valued, continuous function with the property that $$\int_a^bf(x)\,\text{d}x=0$$for all real numbers $a,b$. Prove that $f$ is identically $0$. ...
7
votes
1answer
44 views

Differentiating a constant and switching order

Why does this work? $$\int x^2e^{ax}dx = \int \frac{d^2}{da^2}e^{ax}dx = \frac{d^2}{da^2}\int e^{ax}dx = \frac {d^2}{da^2} \frac {e^{ax}}a = \frac{e^{ax}(a^2x^2-2ax+2)}{a^3}$$ $a$ is a constant, so ...
7
votes
1answer
136 views

Alternative ways to evaluate $\displaystyle \int^1_0 \frac{\text{Li}_2(x)^3}{x}\,dx$

In the following link here I found the integral & the evaluation of $$\displaystyle \int^1_0 \frac{\text{Li}_2(x)^3}{x}\,dx$$ I'll also include a simpler version together with the question: is ...
0
votes
3answers
626 views

Optimize volume of an open cardboard box made from flat square of cardboard…

Consider the following problem: A box with an open top is to be constructed from a square piece of cardboard, 3 ft wide, by cutting out a square from each of the four corners and bending up the sides. ...
4
votes
2answers
352 views

If $\,x>1$, then $\displaystyle\lim_{n\rightarrow\infty}\frac{\left\lfloor x^{n+1} \right\rfloor}{\left\lfloor x^n \right\rfloor}=x$.

How can I prove that $$ \lim_{n\rightarrow\infty}\frac{\left\lfloor x^{n+1} \right\rfloor}{\left\lfloor x^n \right\rfloor}=x, $$ whenever $x>1$. Here $\left\lfloor \cdot\right\rfloor$ denotes the ...
0
votes
2answers
70 views

Does $\displaystyle\lim_{x \to 1}x\ln(x - 1)$ exist? WolframAlpha says yes

The solution to one exercise says that $$\lim_{x \to 1}x\ln(x - 1) = -\infty$$ How can this be, if $\operatorname{dom} \ln(x - 1) = (1, +\infty)$? Only the limit from the right exists, but the other ...
4
votes
2answers
93 views

How to integrate a fraction of the type $\frac{1}{(ax+b)^c(dx+e)^f}$?

I'm working on obtaining chemical reactions' speed, and this is one of the problems I met with. $$ \int \frac{1}{(ax+b)^c(dx+e)^f}dx $$ Can this equation could be solved? If possible, please show ...
5
votes
7answers
320 views

If $x$ and $y$ are not both $0$ then $ x^2 +xy +y^2> 0$

Can't quite finish this proof: Prove that if $x$ and $y$ are not both $0$ then $ x^2 +xy +y^2> 0$ $ x^2 +xy +y^2 +xy -xy> 0$ $ (x +y)^2 -xy> 0$ Without loss of generality define $x\geq ...
1
vote
3answers
48 views

Greatest value of $f(x)= (x+1)^{1/3}-(x-1)^{1/3}$ on $(0,1)$

Greatest value of $f(x)= (x+1)^{1/3}-(x-1)^{1/3}$ on $(0,1)$ Please guide me to solve this problem. I have differentiated it with respect to $x$ and make equal to zero, but couldn't get any point.
8
votes
3answers
135 views

How to solve the differential equation $(2x^3y)\:\text{dy}+(1-y^2)(x^2y^2+y^2-1)\:\text{dx}=0$?

Solve $$(2x^3y)\:\text{dy}+(1-y^2)(x^2y^2+y^2-1)\:\text{dx}=0$$ I tried the substitution $y^2=t$ ; $2y\:\text{dy}=\text{dt}$ to get $$(x^3)\:\text{dt}+(1-t)[(x^2+1)t-1]\:\text{dx}=0$$ ...
4
votes
1answer
65 views

The equality case of the Schwartz inequality

Question: The fact that $a^2 \geq 0$ $ \forall a \in \mathbb{R}$; elementary as it may seem, is nevertheless the fundamental idea upon which most important inequalities are ultimately based. The ...
36
votes
8answers
1k views
10
votes
6answers
666 views

Evaluating $\int_0^\infty \frac{\log (1+x)}{1+x^2}dx$

Can this integral be solved with contour integral or by some application of residue theorem? $$\int_0^\infty \frac{\log (1+x)}{1+x^2}dx = \frac{\pi}{4}\log 2 + \text{Catalan constant}$$ It has two ...
-2
votes
0answers
39 views

perfect competition [on hold]

A (perfectly) competitive firm has total cost given by $$TC(Q) = 5,000,000 + 5Q +\frac{Q^2}{10,000}$$ Regarding its fixed cost of \$5 million, \$4 million can be avoided if the firm produces $0$, but ...
2
votes
2answers
52 views

Calculus of Variations. Lagrangian Hamiltonian Mechanics Mathpages.

Over at http://www.mathpages.com/home/kmath523/kmath523.htm is an article about Lagrangian and Hamiltonian Mechanics with a derivation of the Euler-Lagrange equations of motion. Mid-way through is ...
2
votes
1answer
87 views

How prove $\frac{2}{3}<\frac{3x^6+15x^2+2}{2x^6+15x^4+3}\le\frac{3}{2}$

Let $x\in(0,1]$, show that $$\dfrac{2}{3}<\dfrac{3x^6+15x^2+2}{2x^6+15x^4+3}\le\dfrac{3}{2}$$ My try: since $$\begin{align}\dfrac{3x^6+15x^2+2}{2x^6+15x^4+3} ...
2
votes
2answers
46 views

How find this limit $\lim_{x\to 0}\frac{1}{x^4}\left(\frac{1}{x}\left(\frac{1}{\tanh{x}}-\frac{1}{\tan{x}}\right)-\frac{2}{3}\right)=?$

Find this following limit $$\displaystyle \lim_{x\to 0}\dfrac{1}{x^4}\left(\dfrac{1}{x}\left(\dfrac{1}{\tanh{x}}-\dfrac{1}{\tan{x}}\right)-\dfrac{2}{3}\right)=?$$ My try: since ...
1
vote
2answers
118 views

How solve equation:$y^4+4y^2x-11y^2+4xy-8y+8x^2-40x+52=0$

Let $x,y\in \mathbb{R}$, solve this follow equation:$$y^4+4y^2x-11y^2+4xy-8y+8x^2-40x+52=0$$ My try: Since $$8x^2+4x(y^2+y-10)+y^4-11y^2-8y+52=0$$ then I can't. Maybe this problem have other nice ...