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

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2
votes
2answers
74 views

Prove that these integrals are equal. How to complete the proof?

$$\int_0^xf(u)(x-u)^2du=2\int_0^x\left(\int_0^{u_2}\left(\int_0^{u_1}f(t)dt\right)du_1\right)du_2$$ Ok, I derived both parts 2 times wrt $x$ and got equal integrals. But I'm suspicious whether it is ...
1
vote
2answers
315 views

Doing Michael Spivak's Exercises

I am doing Spivak's Calculus, and I find it EXTREMELY difficult. I usually ask questions here because I cannot do the problems on my own. How long should it take to do a Spivak problem? Is it ...
0
votes
1answer
22 views

Volume of Revolution $f(x) = x^2$

Suppose you are given $y = f(x)$ I want to use double integrals, instead of the traditional washers. Suppose even better, $f(x) = x^2$ Find the volume of $f(x) = x^2$, $x = 0$, $x = 4$, $y = 0$ ...
1
vote
1answer
43 views

Proof of an inequality

If I am given that $$0<\lambda_2<d_0<\lambda_1$$ and would like to prove that $$c_1e^{-\lambda_1(T-t)}+c_2e^{-\lambda_2(T-t)}>e^{-d_0(T-t)},$$ where ...
0
votes
1answer
34 views

Let $f$ be a continuous function on $[a,b]$ with $f(a) \lt 0 \lt f(b)$. Then there is a largest $x$ in $[a,b]$ with $f(x)=0$.

The following is from Spivak's Calculus. Let $f$ be a continuous function on $[a,b]$ with $f(a) \lt 0 \lt f(b)$. Then there is a smallest $x$ in $[a,b]$ with $f(x)=0$. We show that there is a ...
0
votes
2answers
34 views

Evaluate $\int_{-\pi}^{\pi} te^{-int} dt $

Evaluate $\int_{-\pi}^{\pi} te^{-int} dt $ Using integration by parts: $$\int_{-\pi}^{\pi} te^{-int} dt = t\frac{e^{-int}}{-in}|_{-\pi}^\pi - \int_{-\pi}^\pi \frac{e^{-int}}{-in}dt $$ Let's ...
4
votes
6answers
122 views

Prove that there is a function $w > 0$ such that $\int_{0}^{1}w(x) dx \neq 0$ and $\frac{\int_{0}^{1}w(x)x^{2}dx}{\int_{0}^{1}w(x)dx} = \frac{1}{2}$?

I can see that $w(x) := x$ on $]0, \infty[$ suffices, but I am after a systematic analysis to see this, which I am incapable to do.
2
votes
2answers
48 views

If $\lim_{n\to \infty}a_n\cdot b_n=0$ then either $\lim_{n\to \infty}a_n=0$ or $\lim_{n\to \infty}b_n=0$ or both.

Prove or disprove: If $\lim_{n\to \infty}a_n\cdot b_n=0$ then either $\lim_{n\to \infty}a_n=0$ or $\lim_{n\to \infty}b_n=0$ or both. The answers say it is not necessarily true. I can't find a ...
2
votes
1answer
67 views

If $\{{1\over n}\sum_{k=1}^{n}x_k\}_{n=1}^{\infty}$ converges then $\{{1\over n}\sum_{k=1}^{n}{x_k}^2\}_{n=1}^{\infty}$ converges.

Prove or disprove: a. If $\bigg\{{1\over n}\sum_{k=1}^{n}x_k\bigg\}_{n=1}^{\infty}$ converges then $\bigg\{{1\over n}\sum_{k=1}^{n}{x_k}^2\bigg\}_{n=1}^{\infty}$ converges. b. If $\bigg\{{1\over ...
0
votes
1answer
9 views

Prove the equivalence between the existence of a limit and a corollary about the limit's vicinity.

Prove these are equivalent: a. $\lim_{n\to \infty}{a_n}=L$ b. Every vicinity of $L$ contains all the sequence's elements except for a finite number of elements. I started proving $a. \Rightarrow ...
0
votes
2answers
96 views

Evaluate $\lim_{x \to 0} \dfrac{\sin(2x)}{8x}$

I need to evaluate $$\lim_{x \to 0} \dfrac{\sin(2x)}{8x}$$ I'm doing Calculus One course on coursera.org and here is their explanation that doesn't explain anything for me: Why $\lim_{x \to 0} ...
2
votes
1answer
82 views

Are those Locally Lipschitz definitions equivalent?

Let $f:\mathbb{R}^{+}\times \mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ be locally Lipschitz in the sence that there exists a positive $C^{0}$ function $\ell :\mathbb{R}^{+}\times \mathbb{R}^{+} ...
1
vote
2answers
61 views

Stationary points of general function

When studying the stationary point(s) of the following $$ Q=\frac{K(x)}{x} $$ I find $$ \frac{dQ}{dx}=\frac{x\frac{dK(x)}{dx}-K(x)}{x^2}=0 $$ and hence $$ \frac{K(x)}{x}=\frac{dK(x)}{dx} $$ I'm ...
0
votes
0answers
26 views

How to estimate parameters of a parametric function if its values for a set of arguments are known?

Suppose we have a parametric function $F(\alpha_1, ..., \alpha_{N_p}, \mathbf{x})$. For a set of arguments $\mathbf{x_1}$ ... $\mathbf{x_N}$ it's values $F(\mathbf{x_1})$ .... $F(\mathbf{x_N})$ are ...
3
votes
0answers
47 views

Thomae function is continuous at $x\notin\mathbb Q$, discontinuous at $x\in\mathbb Q$ [duplicate]

Let f be the Thomae function aka the popcorn function. Use the epsilon-delta to prove that f is continuous for all irrational numbers and discontinuous for all rational numbers.
0
votes
1answer
37 views

Max of $f^2$ in terms of max of $f$?

Is it true that $\max(|f(x)|^2) = (\max |f(x)|)^2$? Where $f: [a,b] \to \mathbb{C}$ and $f$ is continuous.
1
vote
0answers
94 views

How to differentiate $(x!)\uparrow\uparrow(!x)$?

I need help in differentiating the following expression with respect to x, which I recently came up with when trying to differentiate expressions involving subfactorials... ...
2
votes
3answers
98 views

Trapezoidal rule to find $\int_{0}^{1} \frac {\cos(2x)}{x^{1/3}} dx$

I am supposed to find the integration of the given function in the interval $[0,1]$ using trapezoidal rule (as an assignment problem). $$ \frac {\cos(2x)}{x^\frac{1}{3}} $$ As it can be seen, the ...
3
votes
2answers
496 views

Equation for Tangent Line that passes through $(0,1)$ on the curve $y = \ln x$

I'm totally lost. I've been trying to figure this out. This is what I've figured out: $dy/dx = 1/x$ $y$-intercept $= 1$ So I try to do $y-y_1 = m(x-x_1)+b,$ which I get as $y-1 = 1/x(x-0)+1,$ ...
1
vote
3answers
82 views

Compute $\lim_{x\rightarrow \infty}(\frac{ax+b}{ax+c})^{rx}$

Compute $\lim_{x\rightarrow \infty} (\frac{ax+b}{ax+c})^{rx}$ I try taking the natural log, which is $\ln(y)=rx\ln(\lim_{x\rightarrow \infty} \frac{ax+b}{ax+c})$ but that turns into $0x\infty$ no ...
0
votes
0answers
78 views

Is there a difference between inequality and equality sign when using Lagrange multiplier?

For example, find the extreme values of z=xy subject to the condition x+y=1 This is quite simple example of finding extreme using Lagrange multiplier When the constrain is changed from x+y=1 to ...
-1
votes
1answer
1k views

How do I differentiate sin(1/x)

How can I differentiate $\sin(1/x)$. Do I take $\sin^{-1}(x)$ or what? If I let $u = 1/x$, then $\sin(u)'$ equals $\cos(u)$, then replace $u = 1/x$, $\cos(1/x)$.
1
vote
1answer
65 views

$f_n(x)=n\sin^{2n+1}x\cos x$. Find $\lim_{n\to\infty}f_n(x)$

Let $f_n(x)=n\sin^{2n+1}x\cos x$. Then find the value of $\lim\limits_{n\to\infty}\displaystyle\int_0^{\pi/2}f_n(x)\;dx-\displaystyle\int_0^{\pi/2}(\lim\limits_{n\to\infty}f_n(x))\;dx$ My Thoughts: ...
0
votes
0answers
34 views

How do I find the volume of a solid rotated on the the line $y = -1.5$?

I am going crazy right now and I can't seem to figure out what I am doing wrong here, can someone explain to me why I am wrong? We are given $$y = 9\sin^2x $$ and $$y = 9$$ and we are to rotate it ...
1
vote
2answers
85 views

convergence and divergence of series

I have been working a bit on series and came across two problems I couldn't solve: Determine if the series diverge or converge conditionally/absolutely: 1) $\displaystyle \sum_{n=1}^ \infty ...
11
votes
4answers
217 views

show $\lim_{x\to 0}\frac{e^x-1}{x}=1$ without L'Hopital

how would you show that $$\lim_{x\to 0}\frac{e^x-1}{x}=1$$ without using derivatives or l'hopital but using basic ideas that are generally introduced just before derivatives in a typical introductory ...
3
votes
3answers
99 views

Find all continuous functions $f(x)^2=x^2$

Find all functions $f$ which are continuous on $\mathbb R$ and which satisfy the equation $f(x)^2=x^2$ for all $x \in \mathbb R$. Clearly $f(x)=x, -x, |x|, -|x|$ all satisfy the condition. However, ...
1
vote
3answers
79 views

Integral of an inverse

Let $f(x)=x^3−2x^2+5$. Then find the integral $$\int_{37}^{149} \! f^{-1}(x) \, \mathrm{d}x$$ I know the inverse theorem for differentiation.( I don't think we can apply it here). Is there other ...
4
votes
4answers
4k views

Find a closed form of the series $\sum_{n=0}^{\infty} n^2x^n$

The question I've been given is this: Using both sides of this equation: $$\frac{1}{1-x} = \sum_{n=0}^{\infty}x^n$$ Find an expression for $$\sum_{n=0}^{\infty} n^2x^n$$ Then use that to find an ...
0
votes
2answers
58 views

Stuck on proving $\int_{-\infty}^\infty \cos(\frac{\pi}{a}x)\cos(\frac{3\pi}{a} x) \, \mathrm{d}x$ = 0

Can someone please help me to show how $$\int_{-\infty}^\infty \cos(\frac{\pi}{a}x)\cos(\frac{3\pi}{a} x) \, \mathrm{d}x = 0$$ Attempt: Trig Identity yields $$= \frac{1}{2} \int_{-\infty}^\infty ...
1
vote
2answers
61 views

How to factor this expression

I am having a little problem factoring this problem. $$ % Commenting out image ![enter image description here][1] \newcommand{\red} [1]{\color{red}{#1}} v(\red 2) = \lim_{t \to \red 2} ...
1
vote
2answers
61 views

Finding the volume of the solid generated by revolving about the y-axis

The two functions are: $f(x)= 4x$ and $f(x) = 2x^2$ I found the POI at x= 0 and x=2. From the graph I drew I can tell its a washer so I did, $V=\pi \int ...
0
votes
0answers
25 views

Rabbit population in regards to wolves eating them once they reach a certain population.

A rabbit population increases exponentially with growth rate $k=0.12 months^{-1}$ When population reaches $R=300$ at, for example, time $t=0$, the wolves begun eating rabbits at a rate of $r$ rabbits ...
2
votes
0answers
40 views

Limits algebra. Prove $a_n$ converges and find its limit.

Let $0<a_0<1$, $a_{n+1}=a_n-{a_n}^3$. Prove $\{a_n\}_{n=1}^{\infty}$ converges and find its limit. What I did so far is: Let us show that $a_n>0$ by induction on $n$. The case $n=0$ is ...
1
vote
1answer
75 views

Rabbit population - how do I know which equation??

I have attached an image of the question, #40. I'm struggling with part A because I'm not sure if I have the following equation right. $\frac{dR}{dt} = Ce^{0.12 t} + 300$ Or is it $\frac{dR}{dt} ...
0
votes
1answer
619 views

Find the area of the surface generated when the given curve is revolved about the x-axis

Find the area of the surface generated when the given curve is revolved about the x-axis The part of the curve $y=12x-2$ between the points $(\frac{5}{12},3)$ and $(\frac{13}{12},11)$ I understand ...
1
vote
2answers
132 views

Trig substitution of $\sqrt{x^2-9}/x$

$$ \int {\sqrt{(x^2-9)} \over x} dx $$ I used $ x=3\sec u $ for the substitution. I simplified it down to $ {\sqrt{(x^2-9)}-3\operatorname{arcsec}{x \over 3}}+c $ However, wolfram alpha simplifies ...
1
vote
1answer
29 views

Lake with capacity of 1000 fish. How long will it take to reach 900?

Here is my work as well. I always get stuck when I have to do a u-sub and deal with a negative LN. Some help or a strategy for the future please? Thank you!
6
votes
1answer
206 views

Moment method for weak convergence

Say, we have a sequence of probability distributions $\mu_n$ on $\mathbb R$, that are uniformly subgaussian in the sense that $$\mu_n(\mathbb R\setminus[-R,R])\leq Ce^{-CR^2}$$ for some positive ...
4
votes
2answers
177 views

Let $f$ be a function such that every point of discontinuity is a removable discontinuity. Prove that $g(x)= \lim_{y\to x}f(y)$ is continuous.

Let $f$ be a function with the property that every point of discontinuity is a removable discontinuity, i.e., $\lim_{y \to x}f(y)$ exists for all $x$, but $f$ may be discontinuous at some (even ...
0
votes
1answer
23 views

Area of the surface when the curve is revolved about the x-axis

Find the area of the surface generated when the given curve is revolved about the x-axis. $y=2x+7$ on [0,4] This is what I have : $$S=\int_{a}^{b} 2π \left(f(x)\right)\sqrt{1+f'(x)^2} $$ ...
0
votes
2answers
62 views

How to sketch trigonometric functions?

I was given this as an assignment in Calculus for Life Sciences and I really would like to figure it out: sketch: y=sinx, y=cosx, y=tanx over -2x≤x≤2x
0
votes
0answers
39 views

Differential Equation result

So im doing a differential equation and I turned it into the integration form but now im having trouble at solving it, and cant find anything yet. The integral: $$\int \frac{ds}{\ln(s)+5}$$
0
votes
0answers
27 views

Can this problem be solved using induction

Given: $y=(x^2-1)^n$ Can this be proven using induction taking base case $n=0$: $y^{(n+2)}-2xy^{(n+1)}+n(n+1)y^{(n)}=0$ I don't want a complete solution, just a hint in the right ...
1
vote
1answer
191 views

Find the surface area generated when the curve is revolved around the x-axis

Find the surface area generated when the curve is revolved around the x-axis $y=\frac{x^3}{10}$ on $[0,\sqrt{10}]$ This is what I have so far: $$f'(x)=\frac{3x^2}{10}$$ $$f'(x)^2=\frac{9x^4}{100}$$ ...
0
votes
2answers
59 views

What is $\lim_{(x,y) \rightarrow (0,0)} \frac{2x^6y}{x^6+y^2}$?

Wolfram Alpha says: $$\lim_{(x,y) \rightarrow (0,0)} \frac{2x^6y}{x^6+y^2} = 0,$$ but provides no explanation. Would the function even have a limit?
0
votes
3answers
103 views

Solve this integral $ \frac{1}{2 \pi} \int_{0}^{2\pi} e^{\frac{i}{2} \left( n - m \right)x} \cos(x) dx$

I am supposed to calculate the integrals $$ \frac{1}{2 \pi} \int_{0}^{2\pi} e^{\frac{i}{2} \left( n - m \right)x} \cos(x) dx$$ and $$ \frac{1}{2 \pi} \int_{0}^{2\pi} e^{\frac{i}{2} \left( n - m ...
0
votes
2answers
61 views

Show that $\sum_{n=1}^\infty nx^{n-1}$ converges uniformly on $[0,\frac{9}{10}]$

Show that $\sum_{n=1}^\infty nx^{n-1}$ converges uniformly on $[0,\frac{9}{10}]$ First I think we need to show pointwise convergence: $$\sum_{n=1}^\infty nx^{n-1} = \lim_{N\to\infty} ...
5
votes
2answers
86 views

Proving $y=\lfloor x\rfloor$ doesn't have a primitive function

Prove that $y=\lfloor x\rfloor$ doesn't have a primitive function. I have the proof from a book here but I don't understand it: Suppose that there is $F(x)$ such that $F'(x)=\lfloor ...
0
votes
0answers
58 views

Differentiability question (function of two or more variables)

Imagine I have a function $f:\mathbb{R}^m\times \mathbb{R}^n \rightarrow \mathbb{R}^d$, denote by $|\cdot|$ the Euclidean norm in all spaces and I assume the following hypotheses: there is a finite ...