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

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2
votes
3answers
43 views

Test if point is in convex hull of $n$ points

I have $n$ points $x_1,\dots,x_n\in\Bbb R^d$, and I would like to check that some other point $y$ lies in their convex hull. How can I do this in some efficient way? I think that there was an ...
2
votes
1answer
23 views

Understanding a necessary step in a solution in variational calculus

I'm reviewing calculus of variations using a pdf that I found online (link) and in the example about the minimal surface of revolution, the writer simplified an equation tagged $(3.16)$ as follows: ...
2
votes
3answers
74 views

how to find the derivate of a function g(x)

It's $g(x)={{x^{2}-1}\over{x^{2}+2}}$ and i have to calculate $g^{13}(0)$. I can't calculate all the derivates so i think to use power series. $g(x)={{x^2\over{x^{2}+2}}-{1\over{x^2+2}}}$ Can i use ...
2
votes
1answer
38 views

Is the Weierstrass function given in Counterexamples in Analysis a typo?

Let $0 < a < 1$; let $b$ be an odd integer; let $ab > 1 + \frac{3\pi}{2}$; let $f: x \mapsto \sum_{n \geq 0}a^{n}\cos (b^{n}\pi x): \Bbb{R} \to \Bbb{R}$. Then $f$ is everywhere continuous ...
2
votes
2answers
53 views

Attempting to draw G(x) from G'(x)

We know $G(0) = 0$ Okay, so I have the above graph but I'm having a difficult time translating it into the graph of $G(x)$. What I know so far is that the slope changes abruptly from 0 to 2 at ...
2
votes
1answer
28 views

Indefinite integral problem: $\int_1^\frac{n+1}{1} \frac{(x - [x])^{[x]}}{[x]} dx$ [duplicate]

$ I =\int_1^\frac{n+1}{1} \frac{(x - [x])^{[x]}}{[x]} dx$ my attempt: $I=\int_1^n \frac{(x - [x])^{[x]}{[x]}}\implies\sum_{i=1}^n\int_r^{r+1} \frac{(x-r)^r}{r}dx $ Now by ...
2
votes
1answer
32 views

Finding integral of $2 \lambda\int_0^{\infty} x^{2n}xe^{-\lambda x^2} \ dx$

Finding integral of $$2 \lambda\int_0^{\infty} x^{2n}xe^{-\lambda x^2} \ dx$$ using integral by parts I get that Finding integral of $$2 \lambda\int_0^{\infty} x^{2n}xe^{-\lambda x^2} \ dx = ...
2
votes
1answer
16 views

Definition of Concavity for Twice Differentiable Functions

Let $f(x)$ be a twice-differentiable function. The definitions for concave upward and concave downward I found in my textbooks are all somewhat wordy, something along the lines of: Definition 1: ...
2
votes
1answer
43 views

How do I find the value of a partial sum without calculator?

Without using a calculator, how do I find: $$\sum_{k=0}^7 \left(\tfrac{2}{3}\right)^k $$ I know about the formula for the sum of a geometric series $\frac{a\cdot(1-r^n)}{(1-r)}$ but im not sure how ...
2
votes
1answer
65 views

When to Use L'Hôpital's Rule

The textbook explanation shows that L'Hôpital's rule can be used on a rational function ${f(x)}\over {g(x)}$ if it is continuous, and $\lim_{x \to c}f(x) = \lim_{x \to c}g(x) = 0$ or $\pm \infty$, and ...
2
votes
2answers
33 views

f differentiable at (0,0) [duplicate]

Let $$f(x,y)=\left \{ \begin{array}{ll} \frac{x^2+y^2}{sin(\sqrt{x^2+y^2})} & \mbox{if } 0<||(x,y)||< \pi \\ 0 & \mbox{if } x=(0,0) \end{array} \right.$$ Determine if $f$ is ...
2
votes
1answer
52 views

Find a maximum product $c_1c_2 \ldots c_n$ such that the sum is 136

I need to find the maximum over all possible products c1*c2*c3*...*cn, where $n$ varies over all positive integers. And all $c_i$'s are any positive real number satisfying $\sum c_i = 136$ First: I ...
2
votes
1answer
29 views

Related Rates Question with Resistor, Finding rate of change of $R$ (Physics)?

Let $R_1$,$R_2$,$R_3$ be connected in parallel! See circuit bellow: If $R_1$ increasing at $4\:\frac{\Omega }{s}$, If $R_2$ increasing at $2\:\frac{\Omega }{s}$, If $R_3$ decreases at ...
2
votes
2answers
47 views

$-3\cdot7^x+2\cdot6^x+2\cdot5^x-2\cdot4^x+3^x>0$, for $-1\le x<0$.

Show that $-3\cdot7^x+2\cdot6^x+2\cdot5^x-2\cdot4^x+3^x>0$, for $-1\le x<0$. Here its plotting. By first derivative it involves natural log and expression become more difficult.
2
votes
1answer
39 views

How to maximize the profit for the given equation

It is a maximize profit question. Basically: There is a play which costs $180$ Each attendee costs $0.4$ Ticket price affects the overall attendance. When the ticket price is $5$, then there are ...
2
votes
1answer
29 views

What is wrong with this derivative?

I have the simple function: $$ f=\ln(ka^k) $$ Here are two methods of taking the derivative with respect to $k$: Method 1: $$ \frac{\partial f}{\partial k}=\frac{\partial}{\partial ...
2
votes
2answers
58 views

What is $x^2$-1 applied n times

For the function $F(x)=x^2-1$. How do I write $F^n(x)$ ($F$ applied $n$ times) in terms of $x$?
2
votes
2answers
89 views

Find the solution of the differential equation that satisfies ${{dP} \over {dt}} = 8\sqrt {Pt} ,\,P(1) = 5$

Please help. My homework is grading my answer as incorrect, but I can't tell what I did wrong. The second photo is the work of the problem done correctly but with dp/dt=2sqrt(Pt). I based my work off ...
2
votes
2answers
22 views

Evaluate a certain integral over all space

Evaluate the integral $\iiint e^{-2r} \cos^2\theta \, dV $ over all space. What I have done: I wrote the limit of integration as this: $\int_0^\pi \int_0^{2\pi} \int_0^\infty r^2e^{-2r} ...
2
votes
2answers
57 views

Proving limit of functions involving neighborhoods

Let $A$ and $x_0$ be real numbers, and let $f(x)$ be a real-valued function defined in a deleted neighborhood of $x_0$. Use the definition of limit to prove that $\lim \limits_{x \to x_0} f(x) = A $ ...
2
votes
1answer
39 views

Finding the range for an Extreme Value!

How does one find the range of $xy+\frac{1}{x}+\frac{1}{y}$ I know the domain is $\mathbf |R^2$ except $(0,0)$.
2
votes
2answers
80 views

The general solution for inhomogeneous differential equation

I am working with the following inhomogeneous differential equation, $$x''+x=3\cos (\omega t)$$ The general solution for this is $x(t)=x_h(t)+x_p(t)$ First step is to find $x_h(t):$ So the ...
2
votes
3answers
40 views

if $(x_0,y_0)$ is local extrema in $ax^2 + by^2 + cxy + dx + ey + l$ then its global too.

An exercise on a book asks me to prove that if a point $(x_0,y_0)$ is a local extrema for the function $$f(x,y) = ax^2 + by^2 + cxy + dx + ey + l$$ then it's also a global extrema. The exercise asks ...
2
votes
5answers
55 views

Find and prove the limit of $X_n=$ $\frac {n^{100}}{1.01^n}$

I have to find and prove the limit of the sequence $X_n=$ $\frac {n^{100}}{1.01^n}$ What is the easier way? I tried to use Bernoulli's inequality to say lim$\frac {n^{100}}{1.01^n}$ = lim$\frac ...
2
votes
1answer
30 views

Confusion of proving limits

Ok say we were to prove this simple limit: $\lim \limits_{x \to 2}$ $x^2$ For all epsilon > 0, there exist a delta > 0 such that IF $ 0 < |x-2|< \delta$ THEN $|x^2-4| < \epsilon$ We know ...
2
votes
3answers
72 views

Trigonometric Substitution on $\frac{1}{x\sqrt{(x^2 +25)}}$

How can I find $$\int\frac{1}{x\sqrt{(x^2 +25)}} \space dx$$ using trigonometric substitution?
2
votes
1answer
35 views

Improper Integral $\int _ {0}^{1} \frac{1}{\sqrt{(1-x) \sin{x}}} dx $

The question is : Does the following improper integral converges? $$\int _ {0}^{1} \frac{1}{\sqrt{(1-x) \sin{x}}} dx $$ I have tried some approaches but I'm not sure whether it was correct or not. ...
2
votes
1answer
31 views

Is f(x,y) differentiable at the origin?

My book defines $f(x,y)$ is differentiable at $(a,b)$ when $\lim_{(x,y)\to(a,b)} \frac{R_{1,(a,b)} (x,y)}{\|(x,y)-(a,b)\|}=0$ where $R_{1,(a,b)}(x,y)=f(x,y)-L_{(a,b)} (x,y)$ and ...
2
votes
1answer
40 views

Laplace transform of the zeroth-order Bessel function [duplicate]

Consider $J_0$ the zeroth order Bessel function. I'm trying to compute the Laplace transform $$\mathcal{L}[J_0](s) = \int_0^\infty J_0(t) e^{-st}dt,$$ but until now I couldn't find a good way to do ...
2
votes
1answer
20 views

How to evaluate the line integral?

I've solved part (a) and (b), but I'm unsure about part (c). My attempt is: $\int_C f(x,y,z)ds$ = $\int_0^4 ((2t*\frac{4}{3}t^{3/2}*\frac{1}{2}t^2$)(t+2) = $\frac{2}{13}4^{3/2} + \frac{8}{33} ...
2
votes
1answer
64 views

Verification of Green's Theorem homework help

QUESTION: Verify Green's Theorem for $$\oint (x^2-2xy)dx+(x^2y+3)dy$$ around the curve $y^2=8x$ and $x=2$. My attempt: L.H.S. comes out to be $\frac{128}{5}$ which is correct acc. to the book. For ...
2
votes
1answer
26 views

Divergence of $ \sum_n\sqrt{2\pi}^{-1}{n^{-r^2/2}}\left(\frac{1}{r\sqrt{\log n}} - \frac{1}{r^3(\log n)^{3/2}}\right)$.

From this MathOverflow post, we have the following. $$ \sum_n\sqrt{2\pi}^{-1}{n^{-r^2/2}}\left(\frac{1}{r\sqrt{\log n}} - \frac{1}{r^3(\log n)^{3/2}}\right)$$ This diverges if $r^2/2\le 1$, ...
2
votes
1answer
46 views

Ellipse Perimeter

I've seen lots of methods of getting an approximation of the perimeter of an ellipse, however, I was wondering if there is an exact method that exists, no matter how complex. Thanks.
2
votes
4answers
66 views

Show that $E_r(a)=\{x\in\mathbb{R}^n:\|x-a\|>r\}$ is path-connected

For $a\in\mathbb{R}^n, r>0,n\ge2,$ show that $E_r(a)=\{x\in\mathbb{R}^n:\|x-a\|>r\}$ is path-connected and hence connected. So I'm trying to use the idea that for x, y $\in E_r(a)$ if ...
2
votes
2answers
37 views

Instantaneous rates of change

I am having problems solving the following question. The volume, $V$, of a sphere of radius r is given by $V=f(r)=\frac{4}{3}\pi r^3$. Calculate the instantaneous rate of change of the volume, $V$, ...
2
votes
1answer
70 views

Sequence diverging to infinity is not bounded above?

Problem: Prove that $(a_n)$ -> $\infty$ implies that $(a_n)$ is not bounded above. My attempt: Let $C>0$ be arbitrary. Let $(a_n)\to\infty$ as $n\to\infty$. By definition, $\forall ...
2
votes
2answers
57 views

$\sum_{k=n}^{\infty}\left(n-k\right)e^{-\lambda}\frac{\lambda^{k}}{k!}= ?$

Could you please help me. How do I sum the following: $$\sum_{k=n}^{\infty}\left(n-k\right)e^{-\lambda}\frac{\lambda^{k}}{k!}$$ If the summation had started at 0, then it would be simply an ...
2
votes
1answer
77 views

Can anyone solve this odd integral? [closed]

Can anyone solve this odd integral? $$\int\frac{e^{-50(\frac 1x-1)^2}}{x}\,dx$$ for $x>0$. I couldn't . . .
2
votes
1answer
26 views

Solving differential equation and obtain expressions for unknowns?

I have the following differential equation $my'' + \beta y' + mg = 0$ , with condition $y(0)=0$. I need to solve the equation and obtain expressions for the unknowns. I have attempted to use the ...
2
votes
2answers
76 views

what is wrong with this derivation?

We have simple function : $$Y = X^2$$ Writing $X^2$ as : $X^2 = X+X+X+...........+X$ $(X times)$ We can write above equation as : $$ Y = X+X+X+X+...........+X$$ Differentiating with respect to X, ...
2
votes
4answers
82 views

Prove Newton's iteration will diverge for these functions, no matter what real starting point is selected. $f(x)=x^2+1$ and $g(x)=7x^4+3x^2+\pi$.

Prove Newton's iteration will diverge for these functions, no matter what real starting point is selected. $f(x)=x^2+1$ and $g(x)=7x^4+3x^2+\pi$. We know that $f(x)>0$ and $g(x)>0$ for all ...
2
votes
2answers
64 views

Convert Power Series to function

I tried to solve the attached Power Series, however I can't get to the right answer. I wrote down the correct answer at the top-right of the page. Appreciate your help!
2
votes
1answer
14 views

Question about $o$, $O$ and $\sim$

Let $f$ and $g$ be functions defined on real line and let $g(x)\ne 0$ from some sufficiently large $x$. Please let me know if this is correct: $f\sim g \iff ...
2
votes
1answer
37 views

Isometric problem using compact spaces

Let $A\subset \mathbb{R}^{n}$ a compact subset and $f:A\rightarrow A$ an isometry (i.e , a function such that $\| f(x)-f(y)\| = \| x-y\|$ for all $ x,y \in A$). Show that $f(A)=A$ Now, I've already ...
2
votes
1answer
31 views

Integral of odd angle of sin/cos

Couldn't manage evaluating the integral: $\int_{i=1}^\infty cos(\theta ^{5})d\theta$ What I could figure out is that this integral doesn't converges absolutely because the function isn't bounded, ...
2
votes
1answer
30 views

Proof for determining Fourier coefficients

While determining Fourier coefficients we have this equation $$\int^{T}_{0} x(t) e^{-jn\omega_0t} dt = \sum^{+\infty}_{k\ =\ -\infty} a_k [\int^{T}_{0} e^{j(k-n)\omega_0t}dt]$$ I want to ask that how ...
2
votes
2answers
61 views

Find the points on the graph of $f(x) = 12(x + 9) − (x + 9)^3$ where the tangent line is horizontal.

I cannot figure out how to get started on this question. Would I First simplify, and then take the derivative? Please help!
2
votes
1answer
45 views

How to find the inflection point of $f(x)=\frac{e^x}{1+e^{2x}}$

With the second derivative, the solution gives logarithm of negative number, but I know the graph and the function changes the concavity. $$f(x)=\frac{e^x}{1+e^{2x}}$$ What are the inflection ...
2
votes
2answers
59 views

a very basic question on uniform continuity of function

A very basic and very strange question came to me. Let $D\subseteq\mathbb {R}$, $f:D\to\mathbb{R}$ a continuous function. Then $f$ is uniformly continuous on $D$ iff. $$\forall\epsilon>0\ ...
2
votes
1answer
21 views

Given $y=\frac{3+sin(2x)}{2+cos(2x)}$ find the equation of the tangent

Given the function $y=\frac{3+\sin(2x)}{2+\cos(2x)}$ find an equation of the tangent C at the point C such that $x=\frac{\pi}{2}$. I calculated the derivative ...