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

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0
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1answer
66 views

Given $f_X$. Integrate $\int_0^\infty \log_2 (x+1) f_X \, dx$.

Say $Y=Log_2[1+x]=g(X)$ and $f_X = \frac{e^{-\frac{(\mu -\log (x))^2}{2 \sigma ^2}}}{\sqrt{2 \pi } x \sigma }$ is Log-normal density function: [Wiki] Find E[Y]? Since $E[Y] = \int_0^\infty y f_Y \ ...
1
vote
1answer
85 views

Find limit $\lim\limits_{x \to \infty} \int_0^{x} \cos\left(\dfrac{\pi t^2}{2}\right)$

I looked at the graph and found that limit is $\dfrac{1}{2}$ And limit to $-\infty$ is $-\dfrac{1}{2}$ By the way, the function for which we are finding the limit is called Fresnel function
2
votes
3answers
80 views

value of an integral depending on a parameter in complex plane

For each $z\in\mathbb{C}$, evaluate the integral $$ \int_0^1\int_0^{2\pi}\frac{1}{re^{i\theta}+z}d\theta dr. $$ How to evaluate it? Thanks.
1
vote
2answers
54 views

On integration when solving differential equations (specifically separable equations)

So here is the differential equation and inititial conditions: $$x \frac{\mathrm{d}y}{\mathrm{d}x}=y(3−y) $$ and $$y(2) = 2$$ We have to find the equation $y$ in terms of $x ~~[y(x)]$ that is a ...
2
votes
2answers
80 views

Integrating $ \int_0^2 \int_0^ \sqrt{1-(x-1)^2} \frac{x+y}{x^2+y^2} dy\,dx$ in polar coordinates

I'm having a problem integrating $ \displaystyle\int_0^2 \int_0^ \sqrt{1-(x-1)^2} \frac{x+y}{x^2+y^2} \,dy\,dx$. I drew the graph, and it looks like half a circle on top of the $x$ axis. I tried ...
2
votes
2answers
48 views

This function is a multiple of the identity function

Let $f:\mathbb R\to \mathbb R$ be a derivable function such that every tangent lines of its graph intersect the origin. I would like to know how to prove that $f$ is a multiple of the identity ...
1
vote
2answers
880 views

How to draw the graph of this function? (very difficult)

I'm trying to draw the graph of this function: $$f(x) = \begin{cases} |x|^{|x|}, & \text{if $x\neq 0$} \\ 1, & \text{if $x=0$} \\ \end{cases}$$ I divided the question in two parts, if ...
2
votes
1answer
104 views

Alternative definition of complex number, showing it is equivalent to the tradidional one.

The author of a book makes an alternative definition of the complex numbers, later he shows that this definition is equivalent to the ordinary definition where we define $i^2=-1$. Here is his ...
0
votes
3answers
329 views

Surface area of a cone that has the top cut off?

The picture above has all the information about the cone that is given. L is the length between R and 2R (not the side of the cone). I don't have the overall length, so how can I find the angle in ...
0
votes
1answer
20 views

Commpossed percentage over same base

Maybe this sounds a Little trivial. I have a value and Over this value apply percentage (a), then over this result apply other percentage (b) Could say : Value = 145760 a= 8% b= 40% My actual ...
1
vote
1answer
78 views

Extrema and inflection points of the function $y = \cos^2(x) - \cos(x)$

Please help me find the extrema and inflection points of the function $y = \cos^2(x) - \cos(x)$. So far: $$y'=-2 \cos(x)\sin(x)+\sin(x)$$ $$y'' = 2\sin^2(x) - 2\cos^2(x) + \cos(x)$$ $y' = 0$ when $x ...
2
votes
1answer
28 views

Uniform continuity problem

Let $f(x)$ be continuous on $[0, \infty)$, $f'(x)$ and $f''(x)$ be continuous on $(0, \infty)$. Which of the following statements are true: I. If $f'(x) > 0$ and $f''(x) < 0$, then f(x) is ...
4
votes
2answers
114 views

Closed form of $\displaystyle\sum_{n=1}^\infty x^n\ln(n)$

Is there a closed form of this : $$\sum_{n=1}^\infty x^n\ln(n),$$ where $|x|<1$. Thanks in advance.
0
votes
1answer
152 views

Calculus 3 Riemann Sum

I do not really know where to start for this problem. I have solved ones with rectangular shape where the points were aligned but never a problem like this. If anyone could help me start this ...
0
votes
3answers
177 views

Why $2x$? Can't it be $x$? [duplicate]

So today in my school our neighbor class monitors were complaining to that few of our students were yelling and making noise. Actually the case was that we were having very aggressive debate over a ...
0
votes
2answers
46 views

Double Integration: $\iint_D\ e^{30x}\ dA$

I am having trouble with this double integral. I know I must set it up to have the $y$ values go from $x$ to $x+1$ and the $x$ values from $0$ to $1$. When I solved the integral I got the answer ...
11
votes
3answers
438 views

Evaluating a sum involving binomial coefficient in denominator

I came across the following sum: $$\sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+1)^2}\frac{4^k}{{2k \choose k}}$$ I thought that this can be evaluated using the expansion of ...
0
votes
2answers
87 views

Integrating $\iint_R \sin(9x^2+4y^2)\ dA$

The question I'm trying to solve is: $\displaystyle \iint_R \sin(9x^2+4y^2)dA$, where $R$ is the region in the first quadrant bounded by $9x^2+4y^2=1$. I'm a little confused in solving this. Does ...
3
votes
4answers
92 views

Convergent $\{a_n\}$ $\nRightarrow$ convergent $\{\frac{a_{n+1}}{a_n}\}$

Suppose $\{a_n\}$ converges and $\forall n \; a_n \neq 0$. Then $\{\frac{a_{n+1}}{a_n}\}$ converges. How to disprove this?
5
votes
2answers
160 views

Can the limit $\lim_{h \to 0}\frac{f(x + h) - 2f(x) + f(x - h)}{h^2}$ exist if $f'(x)$ does not exist at $x$?

The second derivative of $f$ can be written as $$f''(x) = \lim_{h \to 0}\frac{f(x + h) - 2f(x) + f(x - h)}{h^2}$$ while it can also be written as (in fact, I believe this is the definition of ...
1
vote
3answers
394 views

Integrating $\int_1^2 \int_0^ \sqrt{2x-x^2} \frac{1}{((x^2+y^2)^2} dydx $ in polar coordinates

I'm having a problem converting $\int\limits_1^2 \int\limits_0^ \sqrt{2x-x^2} \frac{1}{(x^2+y^2)^2} dy dx $ to polar coordinates. I drew the graph using my calculator, which looked like half a ...
1
vote
1answer
53 views

How do I prove the following inequality $\sum_{k=n+1}^{\infty}\frac{1}{k^{2}\log k}\leq\frac{1}{n\log n}$?

I would appreciate some help proving the inequality $$\sum_{k=n+1}^{\infty}\frac{1}{k^{2}\log k}\leq\frac{1}{n\log n}.$$ Thanks in advance!
2
votes
2answers
60 views

$\lim x^{2/3} -(x^2+1)^{1/3}$

How to find the limit $$\lim_{x \rightarrow \infty} x^{2/3} -(x^2+1)^{1/3}$$ I've tried: $$\lim x^{2/3} -(x^2+1)^{1/3} = \lim x^{2/3} \cdot \lim \left[1 - \left(1+\frac{1}{x^2}\right)^{1/3} ...
1
vote
1answer
41 views

Bound the derivative norm of a convolution by the function norm

Is there a bound of the form $$ \|(f*\phi_\epsilon)'\|_{L^2}\leq C(\epsilon) \|f\|_{L^2}, $$ where $\{\phi_\epsilon\}$ are standard mollifiers, and $C(\epsilon)$ does not depend on $f$?
1
vote
1answer
75 views

$\lim_{x\rightarrow 0}\frac{\sqrt{1-\cos2x}}{\sqrt{2}x}$

$\displaystyle\lim _{x\to \:0}\left(\dfrac{\sqrt{1-\cos \left(2x\right)}}{\sqrt{2}x}\right)$ = $\displaystyle\lim _{x\to \:0}\left(\dfrac{\sqrt{2\sin\left(x\right)^2}}{\sqrt{2}x}\right)$ = ...
5
votes
5answers
222 views

Interpreting higher order differentials

I'm trying to understand Taylor's Theorem for functions of $n$ variables, but all this higher dimensionality is causing me trouble. One of my problems is understanding the higher order differentials. ...
0
votes
1answer
61 views

Evaluate $\lim_{x\to0^+}(\sin x)^x$

Let $L=\lim _{x\to \:0+}\left(\sin \left(x\right)^x\right)$ Then we have: $\log L = \lim _{x\to \:0+}\left(\sin \left(x\right)^x\right)$ $\log L = \lim \:_{x\to \:\:0+}\left(\dfrac{\ln \:\left(\sin ...
0
votes
3answers
86 views

Complex Roots and calculations

roots of the equation $z^6 =1-\sqrt3 i $ are $$z_1,z_2,z_3,z_4,z_5,z_6 $$ calculate:$$|z_1|^3 +|z_2|^3+|z_3|^3+|z_4|^3+|z_5|^3+|z_6|^3$$ also calculate: $$z_1^6 +z_2^6+z_3^6+z_4^6+z_5^6+z_6^6$$ ...
0
votes
1answer
45 views

Derivative of two-variable exponential function [closed]

How do you get the derivative of this thing? $f(x,y)=20x^{3/2y}$
0
votes
1answer
55 views

How to simplify linear algebra equation

Im a trying to understand the derivation of an linear algebra equation. It is from a paper about 3D mbICP scanmatching. I am not that good at linear algebra but I am trying to learn. The equation ...
2
votes
2answers
42 views

Let $f: \Bbb{R} \to [0; \infty)$ .Prove that $\forall n \in \Bbb{N}$ $\forall y \in \Bbb{R}$ $ \exists t=t(n;y)$ such that $\int_{y}^{t}f(x)dx=n$

The problem goes like this: Let $f: \Bbb{R} \to [0; \infty)$ be a continuous function such that $\lim_{x \to \infty}f(x)=\infty$. Prove that $\forall n \in \Bbb{N^{*}}$ and $\forall y \in \Bbb{R}$ ...
5
votes
5answers
124 views

$\cos x -1+\frac{x^2}{2!} \geq 0$ for every $x\in \mathbb{R} $

I am having a problem in showing that $\cos x -1+\dfrac{x^2}{2!} \geq 0$ for every $x\in \mathbb{R}$. I have tried the following: I understand that $\dfrac{x^2}{2!}$ is always non negative. But ...
3
votes
4answers
167 views

what is a smart way to find $\int \frac{\arctan\left(x\right)}{x^{2}}\,{\rm d}x$

I tried integration by parts, which gets very lengthy due to partial fractions. Is there an alternative
0
votes
2answers
95 views

eliminate trigonometric terms

Consider the following simultaneous equations. $$ \begin{align} & 2\beta^3\cos(3\omega\tau) = ...
0
votes
1answer
36 views

How far will my car roll given a function representing the slope of the landscape I'm driving on?

So I was driving in my car thinking to myself "I wonder how far I would go (before starting to roll backwards) if I just took my foot off the brakes" I tried to figure it out myself but could not. SO ...
2
votes
0answers
142 views

optimization word problem in calculus

You are asked to build an open cylindrical can (i.e. no top) that will hold $665.5$ cubic inches. To do this, you will cut its bottom from a square of metal and form its curved side by bending a ...
1
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1answer
44 views

Optimization question for calculus

Could anyone tell me where is my mistake? I took the derivative and I solved for r and ended up with this answer
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3answers
36 views

Optimization in Calculus

As you can see I found the equation but I don't know how to find the points. As far as I tried was $(7, 49)$ but it was wrong.
1
vote
1answer
50 views

Derivative in calculus

What is the derivative of $$ \tan^{-1}\left(\frac{88}{d}\right) - \tan^{-1}\left(\frac{44}{d}\right) $$ My answer looks weird, but that is as far as I got. $$ ...
5
votes
1answer
68 views

How many decimal representations are possible for the number 1

I know that there at least two $0.\overline{9}$ and 1 Is there a neat and more concrete way to understand this problem.
1
vote
1answer
69 views

How do I construct such a numerical method for solving ODE?

I am asked to expand $x(t+h)$ and $x(t+2h)$ around $t$ up to the rest term of the third order, find $A, B, C \in \mathbb R$ such that $$x'(t)=\frac{Ax(t)+Bx(t+h)+Cx(t+2h)}{h} + O(h^2)$$ and based on ...
0
votes
1answer
102 views

Advanced calculus integral [closed]

How do you integrate the following? $$\int_0^1 \! \frac{(x^2-y^2)}{(x^2+y^2)^2} \, \mathrm{d}y.$$ and $$\int_0^1 \! \frac{(x^2-y^2)}{(x^2+y^2)^2} \, \mathrm{d}x.$$
3
votes
1answer
87 views

Formula for Alternating Geometric Series

I am aware of the following formula: $$\sum_{n=0}^{\infty}(-1)^nr^n=\frac{1}{1+r}$$ However, I am having difficulty understanding if there is a simple formula for the following equation: ...
2
votes
1answer
195 views

Show that the function $F(x) =(x-a)^2 (x-b)^2 +x$ has the value $(a+b)/2$ at some point $x$

Show that the function $$F(x) =(x-a)^2 (x-b)^2 +x$$ has the value $\frac{a + b}{2}$ at some point $x$. I think it might be something with the the intermediate-value theorem, not sure though. ...
3
votes
3answers
124 views

How can I evaluate $\lim_{x \to \infty} (2^x + 3^x)^{1/x}$?

Someone can explain how can I resolve this limit please? $$ \lim_{x \to \infty} (2^x + 3^x)^{1/x} $$ I tried to convert to exponential $$ \lim_{x \to \infty} \exp \left(\tfrac{1}{x} \ln(2^{x} + ...
5
votes
7answers
231 views

Is There a Difference Between $d^2x$ and $(dx)^2$?

I've just started reading through Calculus Made Easy by Silvanus Thompson and am trying to solidify the concept of differentials in my mind before progressing too far through the text. In Chapter 1 ...
2
votes
3answers
76 views

Integral test for convergence: $\sum _1^\infty \frac{e^{1/n}}{n^2}$

Integral test for convergence: $$\sum _1^\infty \frac{e^{1/n}}{n^2}$$ I tried approaching this as an IBP but I haven't been able to sort the solution. Can this be made into a improper integral? and ...
-1
votes
1answer
58 views

Calculus rate questions!

Question 1: Sand forms a comical pile whose height is always $2$ times the base radius. If the base radius of the pile increases at a rate of $7~ \text{feet/hour}$, find the rate of change of the ...
0
votes
0answers
80 views

Related Rates Conical Problem

Water drips from a conical tank with height 16 feet and diameter 8 feet into a cylindrical tank which has a base of area $900\pi$ sq.ft.. The depth $h$ of the water in the conical tank is changing at ...
5
votes
1answer
136 views

What is the expected value of the number of randomly chosen real numbers between $0$ and $1$ needed to reach a sum of $1$? [duplicate]

My friend told me that the answer to this question was $e$, which intrigued me, but he refused to tell me why. My initial intuition was completely wrong. I thought that since the expected value of ...