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

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4
votes
1answer
242 views

Estimating the sum $\sum_{k=2}^{\infty} \frac{1}{k \ln^2(k)}$

By integral test, it is easy to see that $$\sum_{k=2}^{\infty} \frac{1}{k \ln^2(k)}$$ converges. [Here $\ln(x)$ denotes the natural logarithm, and $\ln^2(x)$ stands for $(\ln(x))^2$] I am ...
5
votes
3answers
247 views

Radius of convergence and the endpoints of a power series

Find the radius of convergence and the convergence at the end points of the series: $$\sum_{n=1}^\infty(2+(-1)^n)^nx^n$$ This is what I did: $a_n=(2+(-1)^n)^n\Rightarrow ...
1
vote
3answers
35 views

Difference of Convergence and a Sum

I'm confused on how Convergence does not mean the same thing as the sum of a series. I was asked to find the sum of $\sum_{0}^{\infty}\frac{n+1}{2^n}$. I found that it converged to $\frac{1}{2}$, but ...
1
vote
0answers
75 views

Simplification after chain rule differentiation of: $\frac{1+\frac{x}{\sqrt{x^2+1}}}{x+\sqrt{x^2+1}}$

I'm hitting my head against the wall here trying to figure this out... had to determine the derivative of $\,\,\ln\left(x+\sqrt{x^2+1}\right)$. That part was easy enough, chain rule a bit and ended up ...
8
votes
3answers
238 views

is the following a decreasing function?

I am stuck on figuring out why the following function is a decreasing function when I read a paper. The function is following $$f(x)=-\frac{1}{x}\log[{pe^{-ax}+(1-p)e^{-bx}}]$$ where $a$ and $b$ are ...
0
votes
5answers
130 views

How to find the minimum of the function?

How to find the minimum of the following function $$ {\rm f}\left(w\right) = {1 \over 2}\sum_{i = 1}^{n}\left({1 \over 1 + {\rm e}^{-x_{i}\,w}} -y_{i}\right)^{2} $$ where $x_{i}, y_{i} \in \left(0, ...
2
votes
2answers
502 views

Particle moving along an ellipse; related rates

This is my very first post here, so sorry if I did anything wrong. This is a related rates problem for first semester calculus. I've been trying for some time and still have no idea how to solve it... ...
2
votes
3answers
738 views

Is there a problem in studying analysis before calculus?

Is there a problem in studying analysis before calculus? Most people say that analysis is rigorous calculus, the university I'm studying teaches calculus first because they believe it's better for the ...
2
votes
2answers
246 views

Mean Value Theorem: Real Analysis

I need to show that $\dfrac{2}{\pi}<\dfrac{\sin(x)}{x}<1$ for $0<x<\dfrac{\pi}{2}$. I know I need to use the mean value theorem, would I just say that since $f$ is continuous in the ...
2
votes
0answers
455 views

Compute the first five non-zero terms of the Taylor series about $a=4$ for $f(x)=\sqrt{x}$

This is my first Taylor Series problem and I want to make sure I completed it correctly. Here is the question: Compute the first five non-zero terms of the Taylor series about $a=4$ for ...
1
vote
3answers
342 views

Application of derivative - helicopter problem

A helicopter of enemy is flying along the curve given by $y =x^2+7$. A soldier, placed at (3,7) wants to shoot down the helicopter when it is nearest to him. Find the nearest distance. Please guide ...
1
vote
4answers
80 views

Finding the largest possible value of a non-defined function?

I am confused, I've never had a similar problem. It seems pretty simple but I have no idea of the approach I should take. Hint me please.
0
votes
1answer
50 views

Find the area of these 2 specific regions

Here is the question.... When I put it into wolfram to just get a good visual of it...it looks like this... I believe this is the formula we use The problem is, I don't quite understand what they ...
1
vote
1answer
54 views

Please explain. Really I dont understand and I need to learn. Pde: : example of finding particular integral

When we look at the solution part, there is a statement The PI of the given PDE is obtained as follows After the statement, I dont really understand all of the calculation. Espacially, After the ...
1
vote
1answer
57 views

Normally distributed with probability

Assume the length of waiting at supermarket is approximately normally distributed with mean 6 minutes and standard deviation 1.5 minutes. (1) Fund the probability that waiting time is longer than 8 ...
1
vote
2answers
27 views

Help with power series $f(x)=\frac{2x}{(1-x^2)^2}=\frac{d}{dx}\left(\frac{1}{1-x^2}\right)$

Given that $f(x)=\frac{2x}{(1-x^2)^2}=\frac{d}{dx}\left(\frac{1}{1-x^2}\right)$, find a power series for $f(x)$. What is its radius of convergence? So far I got the following: ...
1
vote
2answers
47 views

Definition of derivative

Well, I know that the derivative of a function $f(x)$is defined this way: $$\frac{df(x)}{dx} = \lim_{\Delta x\to 0}\frac{f(x+\Delta x) - f(x)}{\Delta x}$$ And it's pretty clear that the expression ...
0
votes
1answer
31 views

If a function is continuous at 0, find the values of two constants a and b

What approach should I take? I am practising for my Calc Final and I've some problems trying to understand these kind of examples.
2
votes
1answer
257 views

Is there an analytical solution to the integral of Weibull cdf $1-e^{-(x/a)^b}$

As part of a bigger exercise I need to try and find the integral of $$1-\exp\{-(x/a)^b\} dx.$$ Note: I can't quite get LaTex to format the equation properly but the exponential should be raised to ...
0
votes
2answers
30 views

Related Rate of Change

After t years the population of a certain town is $P(t) = 50+ 5t$ thousand people. A population $P$ has an associated CO$_2$ level, $C(P) = \frac{\sqrt{P^2 + 1}}{2}$. After $2$ years, the rate at ...
2
votes
2answers
105 views

Finding the derivative of $y^x = e^y$

I have a problem that I've been working on and I don't understand how to start: If $y = y(x)$ satisfies $y^x = e^y$, then when $(x, y) = (2\sqrt{e}, \sqrt{e})$, the derivative $\frac{dy}{dx} = \ ...
0
votes
1answer
42 views

Help setting up integral

Let $A$ be the region in $\mathbb{R}^3$ bounded by the planes $x=0$, $y=0$, $z=2$, and the surfance $z=x^2+y^2$. Evaluate $$\int_A x\, \mathrm{d}x\, \mathrm{d}y\, \mathrm{d}z$$ Here's what I have but ...
0
votes
1answer
61 views

Find outlet of $x^3y-x^2-9xy+2x+8=0$

Find outlet of: $$x^3y-x^2-9xy+2x+8=0$$ I keep getting $y'= \frac{-3x^2y+2x+9y{\color{red}{-2}}}{x^3-9x}$ as a solution, but it's not correct, the book says the solution is: $y'= ...
1
vote
1answer
63 views

Find an integral for the area of the surface generated by revolving the curve $y=sin(x)$ between $0 \le x \le \pi$, about the x-axis

So here is my problem: Find an integral for the area of the surface generated by revolving the curve $y=sin(x)$ between $0 \le x \le \pi$, about the x-axis Just thinking about the problem I feel ...
2
votes
4answers
126 views

Solve$(log_{2}(x+1))^2=4$ [duplicate]

$$(log_{2}(x+1))^2=4$$ $$log_{2}(x+1)*log_{2}(x+1)=log_{2}16$$ $$x^{2}+2x-15=0$$ $$(x+1)*(x+1)=16$$ $$x^{2}+2x+1=16$$ $$x^{2}+2x-15=0$$ $$(x+5)(x-3)=0$$ $$x_1=-5; x_2=3$$ The solution is only $x_1=3$. ...
0
votes
3answers
642 views

Calculus problem interest compounded continuously

Suppose that \$1000 is invested at 5% interest compounded continuously. At 20 years, what is the rate of increase? After how many years will the account reach \$2000? Could you explain how solve ...
0
votes
1answer
28 views
0
votes
3answers
99 views

Find the limit: $\displaystyle\lim_{x\to \infty} \sqrt{x^2+x}-\sqrt{x^2-x}$.

Find the following limit: $$ \lim_{x\to \infty} \sqrt{x^2+x}-\sqrt{x^2-x} $$ I tried to simplify using conjugation. This gave me the following: $$ \lim_{x\to \infty} ...
1
vote
2answers
88 views

Solve $\frac{\log(2x+1)-\log 4}{1-\log(3x+2)}=1$

My attempt:$$\frac{\log(2x+1)-\log4}{1-\log(3x+2)}=1$$ $$\frac{\log(2x+1)-\log4}{\log10-\log(3x+2)}=\log10$$ $$\frac{\log\frac{(2x+1)}{4}}{\log\frac{10}{(3x+2)}}=\log10$$ ...
0
votes
0answers
37 views

Rate of change in the gas bubble in someones eye.

They put a gas bubble in someone's eye. The volume of a gas bubble changes from $0.4$ cc to $1.6$ cc in $74$ hours. Assuming that the rate of change of the radius is constant, find ...
0
votes
1answer
32 views

Is the intederminate form: $-\infty/\infty$ valid for L'Hôpital's Rule?

in my calculus course, we learned that in order to use L'Hôpital's Rule, the function must have the form of either $0/0$ or $\infty/\infty$. However, I wanted to ask if the forms $-\infty/\infty$ or ...
1
vote
4answers
79 views

for what $a$ values does this series converge?

for what values of a does this series converge\diverge, absolutely converge diverge $\sum_{n=1}^{\infty}{(\frac{an}{n +1})^n} , a\in \mathbb{R}$ at first i wanted to do the root test, but ...
1
vote
1answer
34 views

definite integral to evaluation (what am I doing wrong here 3)

I was given the below definite integral to evaluate. below is the equation and my answer after I plug in the values but it is wrong. correct answer should be 29/6 what am I doing wrong? thank you ...
2
votes
2answers
82 views

does this series converge / diverge

does this series converge/diverge conditionally or absolutly $\sum_{n=2}^{\infty} (-1)^n \cdot \frac{\sqrt{n}}{(-1)^n + \sqrt{n}} \cdot \sin(\frac{1}{\sqrt{n}})$ i can use the facts that: ...
1
vote
1answer
33 views

Proving the monotonicity of a recurrence.

Define the following recurrence for $n = 1, 2, \cdots$ $T(n) = ( 1 - \operatorname{H}(\frac{1 - P^{\frac{1}{n}}}{2}))^n$ where $0 < P < 1$ is a constant, function $\operatorname{H}(\cdot)$ is ...
1
vote
1answer
29 views

What is the angle of inclination joining (1,4) and (3,7)

I have found the slope of this line to be 3/2 I used the equation $\dfrac{y_2-y_1}{x_2-x_1} = (7-4)/(3-1)=3/2$ I then found the equation of the line to be $y=3/2x+ 2.5$ I used the equation ...
1
vote
2answers
102 views

Are the differential and derivative of a single-variable function exactly the same thing?

I just started taking a calculus class but I got in late and it had already started like weeks ago, so I'm completely lost. I believe the teacher uses this same formula in order to get the ...
1
vote
2answers
103 views

Finding maxima of multivariable functions in bounded regions

Find the maximum and minimum values of $xz+yz$ over the ball $x^{2}+y^{2}+z^{2} \leq 1$ I frequently have trouble solving these kind of exercises that ask you to find extrema on a bound region. How ...
0
votes
3answers
43 views

second derivative of exponential $e^{x^2}+3x-2$

I have to find the first and second derivative of $e^{x^2}+3x-2$, the first one i can do ok but can someone please help me with the second. thanks
1
vote
2answers
42 views

Differential equations doubt

How can I solve this kind of differential equation? $y' + 2xy^2 = 3x$ Note that it is not variables separable, and it can't be expressed in the first order linear differential equation form, in ...
3
votes
1answer
110 views

Why is the derivative the tangent vector?

I'm trying to understand, at least intuitively why the derivative of a function at a point is the tangent vector at this point. If we see the functions of this form $f:\mathbb R\to \mathbb R$ we see ...
7
votes
3answers
564 views

Limit $\lim_{n \to \infty }\sqrt[n]{ \frac{\left | \sin1 \right |}{1}\cdot\cdot\cdot\frac{\left | \sin n \right |}{n}} $

Help me please with the limit: $\lim_{n \to \infty }\sqrt[n]{ \frac{\left | \sin1 \right |}{1}\cdot\frac{\left | \sin2 \right |}{2}\cdot\cdot\cdot\frac{\left | \sin n \right |}{n}} $ Thanks!
2
votes
2answers
42 views

integrate the following equation

here is the equation: here is my answer: the correct answer: $-\sqrt {1 - 2x} +c$
2
votes
2answers
75 views

Series prove or disprove statement

Let $a_n$ be a sequence. $\lim_{n\to\infty} a_n =0 $ then the series: $\sum_{n=1}^\infty a_n, \ \sum_{n=1}^\infty (a_n+a_{n+1}) $ converge and diverge together. It doesn't seem right, ...
8
votes
1answer
146 views

A continuous function integral inequality

Let $m$ be a positive integer. $f\colon[0,\infty)\to[0,\infty)$ is a continuous function such that $f(f(x))=x^m,\forall x\in[0,\infty)$. Show that $$\int_0^1f^2(x)\,dx\ge\frac{2m-1}{m^2+6m-3}$$
2
votes
1answer
37 views

How would I write this statement in mathematical terms

I have a function $y(t)$ and an estimate of time at which the function is maximum $t_i$, I want the final estimate $t_f$ to be the value that makes $y(t)$ maximum in the interval $t_i - n < t < ...
2
votes
1answer
512 views

Use polar coordinates to find the volume of the given solid

Use polar coordinates to find the volume of the given solid bounded by the paraboloid $z=1+2x^2+2y^2$ and the plane $z=7$ in the first octant. I did it. Is that right ? $$\int_0^{\pi \over 2} ...
3
votes
1answer
58 views

Does this Manifold exist?

The excercise is the following: Give an example or disprove: There is at least one m-dimensional manifold that is compact in some $\mathbb{R}^n$ such that one chart is sufficient to get the whole ...
0
votes
2answers
108 views

Does the series converge/converge absolutely/diverge

$$\sum^{\infty}_{n=1}\frac{(-1)^n}{n^a\ln n}$$ $$a>0$$ Does the series converge/converge absolutely/diverge ? I tried to divide to cases and factor the series: ...
0
votes
2answers
53 views

Prove that $\limsup\left(b_{n}a_{n}\right)=\limsup\left(a_{n}\right) $ when $\lim_{n\rightarrow\infty}\left(b_{n}\right)=1$

I'm having trouble with this homework question: "Let there be a sequence $\left(b_{n}\right)_{n=1}^{\infty} $ such that $\lim_{n\rightarrow\infty}\left(b_{n}\right)=1$. Let there also be some ...