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

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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 ...
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
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 ...
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 ...
1
vote
2answers
27 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 ...
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
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 ...
1
vote
0answers
66 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
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 $0.5$ ...
-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)' ...
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, $y=x^2+3$
5
votes
2answers
55 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 ...
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 ...
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) ...
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 ...
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 ...
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 $$(x^3)\:\text{dt}+(1-t)[(x^2+1)t-1]\:\text{dx}=0$$ ...
-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 ...
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.
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 ...
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
1answer
17 views

Optimization, minimizing volume of an open top box given the volume

The question is: An Anacleto box is a square open box: the bottom is a square, the four sides are equal rectangles, and there isn’t anything on the top. The box should have a volume of 1000 ...
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 ...
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 ...
0
votes
2answers
35 views

How do I find this distance?

Find the minimum and maximum distances between the ellipse $x^2+xy+2y^2 = 1$ and the origin. This is what I've attempted so far: Maximize $x^2+y^2+z^2$ with respect to $x^2+xy+2y^2 = 1$. Using ...
5
votes
7answers
321 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 ...
2
votes
1answer
29 views

Using Lagrange multipliers to find the extrema of $f(x,y) = e^{2xy}$ subject to $x^2+y^2 = 16$

Find the maximum and minimum values of $f = e^{2xy}$ with respect to $x^2+y^2 = 16$. Using Lagrange multipliers, $\nabla f = \lambda\nabla g$. Therefore, the constraints are the following: ...
1
vote
3answers
47 views

Evaluate $\int {x+3\over x^2+6x+10}dx$ [on hold]

$$\int {x+3\over x^2+6x+10}dx$$ Could anyone help me with this substitution problem?
1
vote
2answers
18 views

Intervals and Signs

In the first and second derivative tests, I find whether the derivative is positive or negative by picking a random number within that open interval. The number I pick is arbitrary; however, what ...
0
votes
1answer
35 views

Minimizing a function in Mathematica

Edit: I simplified the function using $\textbf{Simplify[...]}$ How can I minimize this function of $x$, where $l$ is a positive constant? $$\frac{1}{2} \sqrt{\frac{x}{l}+\frac{l}{x}+4 x^2-2}$$ ...
2
votes
2answers
57 views

Proof of Max (x,y)

The problem states that $ \max(x,y) = \dfrac { x+y+|y-x|} {2} $ where $x,y \in \mathbb{R}$ Part 1) Prove that this is true. Part 2) Derive a formula for $\max (x,y,z)$. 1) Intuitively i see this as ...
1
vote
2answers
27 views

Given a curve, such as $1/x$, how to find which tangent is closest to its OWN interception with the y-axis

As title mention, if I have a function such as $\dfrac{1}{x}, x>0$, how can I find which tangent of the curve is closest to its y-axis interception. Using pythagorean theorem, one sees that the ...
2
votes
0answers
26 views

Partial Integral of an ellipse

this is my first question on stack exchange so please bear with me. I am trying to generate a synthetic image of an ellipse in Matlab where each pixel is shaded according to how much of that pixel ...
0
votes
3answers
97 views

Evaluation of the integral $\int 3x \cos x^2 \, dx$

I want to solve this: $$\int 3x \cos x^2 \, dx$$ I get this answer: $$ \frac{\sin 2x}{2}+\frac{\cos 2x}{4}+C $$ but the answer should be: $$ \frac{3 \sin x^2}{2}+C $$ Am I doing anything wrong ...
1
vote
3answers
64 views

Want to ensure my proof is rigourous enough.

Question. Prove: $ 0 \leq x < y $ then $ x^n < y^n$ $ \forall n \in \mathbb{N} $ I'm particularly bad at proving obvious things but here it goes. ( please be super strict on analyzing my proof ...
0
votes
1answer
66 views

How to approach, substitution - definite integral

So I have this problem $${\int^{\pi/2}_0} {{\cos\theta \sin\theta}\over \sqrt{\cos^{2}\theta +8}}d\theta $$ and I'm not sure if this is the right direction to begin. If I have $u = \cos\theta$ ...
0
votes
2answers
27 views

Solution of given differential equation using Laplace Transforms.

I need solution of DE $$y'' + 2y' + 5y = 0$$with initial conditions $$y(0)= 1 \text{ and } y'(0)=0$$ I tried this but problem came when i started taking laplace inverse of F(s), so i need a complete ...
0
votes
2answers
40 views

How to PROVE there are only finite number of sub limit in this sequence that not converge

for example, let $A,B,C\:\in \mathbb{R}\:$ be some constants, and $$ a_n=\begin{cases} A, & n=3k-2,\ k\in \mathbb{N} \\ B, & n=3k-1,\ k\in \mathbb{N} \\ C, & n=3k,\ k\in \mathbb{N} ...
1
vote
4answers
76 views

Differentiating $ \left( 1 - \frac {1}{x} \right)^x $

I have a calculus question. How does one differentiate $\left(1-\frac{1}{x}\right)^x$, for x>1? It should be positive right?
2
votes
1answer
76 views

Evaluating $\int_0^{2} \frac{dx}{\sqrt[3]{2x^2-x^3}}$

How to calculate this integral? $$\int_0^{2} \frac{dx}{\sqrt[3]{2x^2-x^3}}$$ I suppose that it should be parted like this: $$\int_0^{1} \frac{dx}{\sqrt[3]{2x^2-x^3}} + \int_1^{2} ...
1
vote
2answers
54 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 ...
-1
votes
1answer
31 views

Finding a differentiable inverse of $f(x)=x/\cos x$

Let $$ f:\left(-\frac{\pi}{2},\frac{\pi}{2}\right) \rightarrow \mathbb{R} $$ be defined by $$ f(x) = \frac{x}{\cos x}. $$ We're supposed to show that $f$ has a differentiable inverse $$f^{(-1)}$$ ...
0
votes
2answers
43 views

Taylor expansion of the Error function

The error function $\operatorname{erf}(z)$ is defined by the integral $$ \operatorname{erf}(z)=\frac{2}{\sqrt{\pi}} \int_0^z e^{-t^2}\,dt,\quad t\in\mathbb R$$ Find the Taylor expansion of ...
3
votes
1answer
54 views

Writing a proof of the convergence of a series defined recursively

Define the sequence $a_n$ recursively by $a_1=1$ and $$a_{n+1}=\frac13\left(a_n^2+\frac1n\right)$$ (a) Prove, by induction or otherwise, that $(a_n)$ is decreasing. (b) Prove that the series ...
0
votes
1answer
31 views

Finding the surface area of a sphere

Today I was going over some calculus that I had long forgotten, and I made the following mistake when trying to find the area of a sphere: I though it would be this: ...