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

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1
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0answers
28 views

To show a function differentiable

Let $A \in \mathbb{R}^n$ be a fixed vector and $T : \mathbb{R}^n \rightarrow \mathbb{R}^n$ a linear transformation . Define $f : \mathbb{R}^n \rightarrow \mathbb{R}$ by $$f(x) = \langle ...
2
votes
0answers
26 views

Prove continuity of $\frac{x^3y+2xy^3}{x^2+y^2}$ using the definition

$f(0,0)=0$ and $f(x,y)=\dfrac{x^3y+2xy^3}{x^2+y^2}$ when $(x,y) \neq (0,0)$. Is $f$ continuous at $(0,0)$? I went to polar coordinates, $$ f(x,y)=g(r,t)=r^4(\cos^3t \sin t+2\cos t\sin^3t)/r^2=r^2 ...
0
votes
0answers
6 views

How to convert vector field from cartesian to spherical

I have a vector field $A ( r) = \omega \times r$, where $r=(x,y,z)^T$ and now I want to express this field in cylindrical coordinates. How do I do this?
0
votes
1answer
19 views

Separable Differential Equation

The question is: $$t^5\frac{\mathrm{d}y}{\mathrm{d}t} + y^5 = 0$$ The next step says $\frac{1}{y^5}\frac{\mathrm{d}y}{\mathrm{d}t} + \frac{1}{t^5} = 0$ i understand this. However it then says: ...
3
votes
3answers
24 views

Finding a tangent to an ellipse parallel to a given line

Problem: Find the lines that are tangent to the ellipse $x^2 + 4y^2 = 8$ and parallel to $x +2y = 6$. I tried to find the derivative of $x^2 + 4y^2 = 8$ and I got: $$\frac{dx}{dy} = -\frac{x}{2y}.$$ ...
-1
votes
0answers
26 views

Minimise sum of squares [duplicate]

For real numbers $x_{1},..,x_{n}$, minimise $x_{1}^2+..+x_{n}^2$ subsject to the condition $x_{1}+..+x_{n}=2$. This has cropped up in a stats question.
0
votes
1answer
35 views

Given one solution, can a second solution always be found?

Let's consider a second order ODE: $$y''+p(x)y'+q(x)y=f(x)$$ A common procedure is to find linearly independent solutions $y_1,y_2$ to the homogenous ODE, and then apply the technique of variation ...
0
votes
3answers
39 views

How can one determine whether the following series converges or diverges [duplicate]

$$ \sum_{n = 2}^{\infty}\frac{(-1)^{n}}{\sqrt{n} + (-1)^{n}} $$ Wolfram Alpha returns nothing useful, except that the ratio test was inconclusive.
0
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0answers
24 views

How to find $\nabla$ in spherical coordinates

I want to derive(!) just a few components like the $\hat{e}_r$ component of the divergence operator in spherical coordinates and the $\hat{e}_{\phi}$ component of the curl operator in spherical ...
4
votes
1answer
82 views

a doubt with the series $ \sum_{n=0}^{\infty}e^{-nx} $

I know that the series is equal to $$ \sum_{n=0}^{\infty}e^{-nx}= \frac{1}{1-e^{-x}}$$ However, if I expand each exponential term into a Taylor series I get $$ \sum_{n=0}^{\infty}e^{-nx}= ...
0
votes
3answers
21 views

Find all points on the curve $y=2x+x^{-1}$ which have a tangent parallel to the x-axis

Find all the points on the curve $y=2x+x^{-1}$ which have a tangent parallel to the $x$-axis.
0
votes
3answers
35 views

How to find $\int\sqrt{(26x-x^2)}dx $

How do I find $\int \sqrt{(26x-x^2)} dx $ Is this an integration by parts question? Thanks, --Nick
0
votes
2answers
44 views

Is there a difference between limit and “two-sided limit”?

If we have a function of real-variable, then we can talk about one-sided limits $\lim\limits_{x\to a^+} f(x)$ and $\lim\limits_{x\to a^-} f(x)$ and, of course, also about the limit $\lim\limits_{x\to ...
1
vote
2answers
51 views

How to Simplify Sin/tan problem.

I am trying to simplify $\displaystyle\frac{\sin^2}{\tan^2}$ but I don't know how to go about it. Any help is appreciated.
1
vote
1answer
56 views

Does the series $\sum\limits_{n=1}^\infty\frac{\sin(n)n!}{n^n}$ converge?

$\sum\limits_{n=1}^\infty\frac{\sin(n)n!}{n^n}$ Please let me know how you did it. Thank you.
0
votes
2answers
73 views

Maximum among $1, 2^{1/2}, 3^{1/3}, 4^{1/4},…$

What is maximum value among $1, 2^{1/2}, 3^{1/3}, 4^{1/4},....$ ? My approach: let $f(x)=x^{1/x}$ then I found out the derivative of $f$. Since $f(x)$ is maximum where $f'(x)=0$ and $f''(x)<0$ ...
1
vote
2answers
18 views

maximized profit w/ a cost & demand function

I'm having trouble with this problem: If $C(x) = 14000 + 500x − 4.8x^2 + 0.004x^3$ is the cost function and $p(x) = 4100 − 9x$ is the demand function, find the production level that will ...
0
votes
1answer
29 views

Easy calculus question

Let $D = \{ (x,y) : 0 \leq x \leq 1, \; \; x^2 \leq y \leq x, \; \; 0 \leq z \leq x \} $ and suppose $f(x,y,z) = x + y $. Want: $\int_D f $ IS this the correct integral? $$ \int_{0}^1 ...
1
vote
1answer
27 views

Minimize total area of a square and triangle made of 13m long wire

I'm a little bit confused about this problem. I've gotten the first part, but I can't get the second! ...
2
votes
1answer
37 views

Another integral calculus question from Apostol

Am stuck on another question from Apostol "Calculus" Volume 1 (Section 5.11, Question 22). The question reads: Determine a pair of numbers $a$ and $b$ such that $$ \int_0^1 (ax+b)(x^2+3x+2)^{-2} dx ...
1
vote
2answers
80 views

Equality of integrals: $ \int_{0}^{\infty} \frac {1}{1+x^2} \, \mathrm{d}x = 2 \cdot \int_{0}^{1} \frac {1}{1+x^2} \, \mathrm{d}x $

In Street-Fighting Mathematics (page 16), Prof. Sanjoy Mahajan states that $$ \displaystyle\int_{0}^{\infty} \frac {1}{1+x^2} \, \mathrm{d}x = 2 \cdot \displaystyle\int_{0}^{1} \frac {1}{1+x^2} \, ...
1
vote
2answers
24 views

Showing that ${d \over dz}\log\left[ z - a \over z - b \right] = {1 \over (z - a)} - {1 \over (z - b)}$

I'm trying to show that $$ {d \over dz}\log\left[ z - a \over z - b \right] = {1 \over (z - a)} - {1 \over (z - b)} $$ However my attempt yields that $$ {d \over dz}\log\left[ z - a \over z - b ...
0
votes
0answers
12 views

Finding the distance from a parabola (ballistic trajectory) to a point (for use in collision detection)

I need to have some form of collision detection / prevention for an object moving along a ballistic trajectory and a second stationary object on the same plane plane. The ballistic trajectory is ...
0
votes
0answers
12 views

Domain of function of form $f(x)=\frac{g(x)}{k(x)}$

I just want to know did we have a rule to find the domain of function in form of $f(x)=\frac{g(x)}{k(x)}$ .I know $k(x)\ne 0$ . but in general do we have any rule to compute domain of function like ...
1
vote
3answers
34 views

For solid volumes, why does the Integral behave as a summation?

When you take a definite integral, you can think about calculating the area under the curve (via Riemann rectangle slices approximation) Now, when you take the volume of a 3D object, you sum the ...
0
votes
1answer
30 views

Convergence when integrating a not-quite power series

This is a question where I don't have serious doubts about the truth of the statements; it's more about how to prove things rigorously. Consider $\displaystyle\frac{1-x^t}{1-x}$ where $t$ may be a ...
2
votes
3answers
60 views

$\int_0^\frac{\pi}{2}\cos ^2x\log(\tan x)dx.$

Evaluate $\int_0^\frac{\pi}{2}\cos ^2x\log(\tan x)dx.$ Sidenote:Via mathalpha I know that answer is $-\pi/4$ but do not know how to derive that.
-5
votes
1answer
74 views

How to solve this crazy problem $\int^{\int_0^9 (2x^2+x+3)dx}_{\int_{4}^6 (x-1)dx}\frac{1}{x}dx$ [on hold]

How to solve this crazy problem $$\int^{\int_0^9 (2x^2+x+3)dx}_{\int_{4}^6 (x-1)dx}\frac{1}{x}dx$$ that's IMPOSSIBLE! -_-
2
votes
0answers
34 views

Understanding underlying algebra for calculus convergence problem

I'm working on series convergence/divergence problems in my Calc 2 class, and (as has happened often), I'm hung up on some underlying algebra. The first step in the solution manual for a problem I'm ...
0
votes
1answer
31 views

Partial derivative of a Piecewise function

If I have the following equation: $$ f(x,y) = \begin{cases} x; & y \ge 0 \\ y; & y < 0 \\ \end{cases} $$ What are the partial derivatives (both x and y) of the function? I have trouble ...
1
vote
2answers
41 views

school exersise/ Differentiation

If $f$ is differentiable in ${\bf R}$ and for every $x \in {\bf R}$, $$ f(x+\cos x)-f(1-x) \leq x\cos x , $$ then prove that $f'(1)=1/2$. How is a school kid supposed to solve this exercise ? ...
0
votes
0answers
27 views

Help on proving surjectiveness

How do you prove the function $\gamma :S \rightarrow \dot{\mathbb{R}} $, (where S denotes the complex unit circle), defined by $$\gamma(t)=\frac{i(1+t)}{1-t}$$ is surjective.? Where ...
2
votes
2answers
53 views

Integral $I=\int_0^\infty \frac{x^4}{(\alpha+x^2)^4}dx$

Hi I am trying to show $$ \int_0^\infty \frac{x^4}{(\alpha+x^2)^4}dx=\frac{\pi}{32\alpha^{3/2}},\quad \Re(\sqrt \alpha)> 0. $$ I am looking for a solution to this NOT using contour integration, but ...
1
vote
4answers
55 views

Show that in ℝ[x], no polynomial of odd degree > 1 is irreducible. [duplicate]

I think that logically, I understand the concept because no matter what polynomial you have you can always factor it into something with a x to a power plus or minus some real number, and that real ...
-8
votes
1answer
69 views

Solve this if you can. [on hold]

Solve this if you can: $$x^{x^x}=e^{x^{e^{x^2}}}$$ I think this problem cannot be solved by anyone. Thanks.
0
votes
1answer
37 views

how does this converges? Sequence and series convergence

Consider the following problem- Converges or Diverges? $$(1-2)-(1-2^{1/2})+(1-2^{1/3})-(1-2^{1/4})+....$$ I said it converges but then my work i showed in paper got wrong How would you prove that ...
11
votes
2answers
81 views

Second derivative of $f(f(\cdots f(x)\cdots )?$

For convenience, let's write $f_n(x)=f(f(\cdots f(x)\cdots )$ where $f$ is iterated $n$ times. Suppose: $$f(0)=0,\quad f'(0)=\alpha,\quad f''(0)=\beta$$ What is $f''_n(0)?$ I've found ...
0
votes
0answers
31 views

sequence and series of convergence problem

We have a sequence/series problem. and i need help Consider the following: How can we show that $2-2^{1/2}+2^{1/3}-2^{1/4}+2^{1/5}-2^{1/6}+\ldots$ diverges? I am so lost as to solving this problem. ...
2
votes
1answer
88 views

How to prove that $e^\pi>\pi^e$ [duplicate]

How can one prove that $e^\pi>\pi^e$ ? Should I use logarithms or something like that? Or is there a gentle smart way of proving so?
0
votes
1answer
18 views

consequence of Mean Value Theorem

Let $f$ a continuous function on $[a, b]$ $a < b$ ,derivable on $(a, b)$ then there exist $c_1, c_2 \in (a, b)$ ,$c_1 \ne c_2$ such that $\frac{f (b) - f (a)}{b - a} = \frac{f '(c1) + f' ...
0
votes
1answer
29 views

Question regarding trigonometry

I've got this thing on my mind : we know that $cos(x)$ is a periodic function , hence integral from $2(k-1) \pi$ to $2k \pi$ will yield the same value for any $k \geq1$. My question is , why is ...
0
votes
3answers
42 views

what are the equilibrium points of the following: [on hold]

where $x$ represents susceptible individuals, $y$ represents infected individuals. Find the two biologically meaningful equilibria. $$ \frac{\mathrm{d}x}{\mathrm{d}t} =12−3xy−3x $$ $$ ...
-1
votes
1answer
38 views

Advanced Calc proof help

Assume that for $a,b>0$ and any $0 < t< 1$ $$ a^tb^{1-t} ≤ ta+(1-t)b $$ Prove given $a_1,a_2,...,a_n ≥ 0$, $b_1,b_2,...,b_n \geq 0$ and $b_1+b_2+...+b_n=1$ We have $$ \left(\sum_{i = ...
1
vote
0answers
15 views

difference of the values of a function is an integral

This is a very simple quesiton but something I don't understand. From Taylor expansion: $$f(y)-f(x)=f'(x)(y-x)+O((y-x)^2)$$ so, if I just picture that, on the left is the difference between two values ...
2
votes
2answers
39 views

Geometric intuition: Seeing the regions in double integrals

Context: solving double integrals. I had the formula $$x^2+y^2=1-x-y$$ yet I could not see what shape it had. This is even more true with 3D pictures like $$2x^2+2y^2 \le 1+z^2.$$ Is there a summary ...
1
vote
1answer
28 views

Ho To Perform U-Substitution On Given Intergal

$\int{x^2\sqrt{2+x}}{dx}$ I haven't been able to find what u should be in this intergal, where should I start? I've gotten as far as: let $u = 2 + x$; $du=\frac{1}{x}dx$
3
votes
1answer
27 views

chain rule question help me please

Let $f:\mathbb{R}^2\to\mathbb{R}$ be a $C^2$ function and let $c(t)$ be a $C^2$ curve in $\mathbb{R}^2$. Write a formula for the second derivative $\frac{\mathrm{d}^2}{\mathrm{d}t^2}(f\circ ...
1
vote
1answer
29 views

A question about a continuous function that satisfy certain limits at $\pm\infty$

I got this question: Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function such that $\lim_{x\to\infty}\frac{f(x)}{x^2}$ and $\lim_{x\to -\infty}\frac{f(x)}{x^2}$ exist and are real numbers. ...
-2
votes
0answers
28 views

Equilibrium question [on hold]

Consider the differential equation $$x' = x^3 − x^2 − 6x.$$ (a) Find all equilibria. (b) Determine the stability of each equilibrium analytically (not from the phase line diagram). (c) Sketch ...
1
vote
1answer
25 views

Continuous function $f:\mathbb{R}\to\mathbb{R}$ that got no extrema must be one to one

I got this question: Prove that if $f:\mathbb{R}\to\mathbb{R}$ is a continuous function that got no extrema then $f$ is one to one. I tried to prove it but I don't know how to proceed. I started by ...