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

learn more… | top users | synonyms

4
votes
4answers
145 views

How to calculate the integral $\int_{-1}^{1}\frac{dz}{\sqrt[3]{(1-z)(1+z)^2}}$?

The integral is $I=\displaystyle\int_{-1}^{1}\dfrac{dz}{\sqrt[3]{(1-z)(1+z)^2}}$. I used Mathematica to calculate, the result was $\dfrac{2\pi}{\sqrt{3}}$, I think it may help.
0
votes
2answers
35 views

Find the value of $ \int_0^1\sqrt{\frac{x}{4-x}}$ using $x=4\sin ^2 \theta$

Find the value of $\displaystyle \int_0^1\sqrt{\dfrac{x}{4-x}}$ using $x=4\sin ^2 \theta$ I'm trying to work through this, with the mark scheme, but I don't understand what they do. I understand ...
5
votes
5answers
5k views

Integral $\int{\sqrt{25 - x^2}dx}$

I'm trying to find $\int{\sqrt{25 - x^2} dx}$ Now I know that $\int{\frac{dx}{\sqrt{25 - x^2}}}$ would have been $\arcsin{\frac{x}{5}} + C$, but this integral I'm asking about has the rooted term in ...
0
votes
3answers
28 views

Show a continous function is bounded on a closed interval

For a homework problem, I need to show a function $\pi + 0.5\sin(\frac{x}{2})$ is bounded on the interval $[0,2\pi]$. I'm having trouble conceptualizing a good way to do this though. Can anyone help? ...
0
votes
0answers
28 views

Multiplying double sums formula

Questions: 1)Can I modify the same-endpoint formula somehow to get different-endpoint formula? http://en.wikipedia.org/wiki/Cauchy_product#Finite_summations 2)Or even better do you know any formula ...
-1
votes
0answers
20 views

shell method on both axis find volume [on hold]

the region by y=4x^2 and y=7x is to be rotated about both axes and the volume generated calculated by both the washer and the shell method? the volume is the region bounded by y=4x^2 and y=7x rotated ...
11
votes
5answers
494 views

How to evaluate $\int_{0}^{+\infty}\exp(-ax^2-\frac b{x^2})\,dx$ for $a,b>0$

How can I evaluate $$I=\int_{0}^{+\infty}\!e^{\left(-ax^2-\frac b{x^2}\right)}\,dx$$ for $a,b>0$? My methods: Let $a,b > 0$ and let $$I(b)=\int_{0}^{+\infty}e^{\left(-ax^2-\frac ...
-1
votes
0answers
20 views

Use shell method to find volume of solid generated by revolving the given region … [on hold]

Use the shell method to find the volume of the solid generated by revolving the region bounded by the line $y=6x+7$ and the parabola $y=x^2\ldots$ ...about the following lines A) the line ...
2
votes
3answers
162 views

Area of the field that the cow can graze.

How do we find the area that the cow can graze? The question goes as follows-- There is a circular barn house surrounded by a huge grazing field. A cow is tied to the rope ($AB$) at the end $A$ as ...
0
votes
0answers
25 views

Disk and Washer Method to Figure out Volumes

Well, I'm having a lot of trouble setting up these problems. Here's a problem: We have the area bounded by $y = x^2/25$ and $y =1$. It is revolving about the line $y = 2$. What is the volume ...
0
votes
1answer
57 views

A limit that does not exist

Let $$A_{n}=\{0,1/n,2/n,...\}$$ Let $C_{n}=|S{\cap}A_{n}|$ for a set $S$. Find a set $S$ for which the limit $$\lim_{n\to\infty}\frac{C_{n}}{n}$$ does not exist
19
votes
5answers
465 views

Computing $\lim_{n\to\infty}n\sum_{k=1}^n\left( \frac{1}{(2k-1)^2} - \frac{3}{4k^2}\right)$

What ways would you propose for the limit below? $$\lim_{n\to\infty}n\sum_{k=1}^n\left( \frac{1}{(2k-1)^2} - \frac{3}{4k^2}\right)$$ Thanks in advance for your suggestions, hints! Sis.
0
votes
1answer
22 views

Sketching a curve and finding where the parameter increases

(a) Eliminate the parameter to find a Cartesian equation of the curve. (b) Sketch the curve and indicate with an arrow the direction in which the curve is traced as the parameter increases. $$x = ...
4
votes
2answers
97 views

Analysis of the function $y=x^{\frac{1}{x}}$

The graph of the function $y=x^{\frac{1}{x}}$ for positive $x$ is as shown below: When I calculated $y$ for negative values of $x$ only some of the values between $0$ and $-1$ and only those for ...
0
votes
2answers
71 views

Find the largest $x$ interval containing $0$ on which $y$ is well-defined.

I'm currently taking an intro course on ordinary differential equations and was given this homework problem: Find the solution of the following differential equation:$$\frac{dy}{dx} = y^2(1-2x)$$ ...
1
vote
1answer
75 views

proving a tricky limit is zero

Let $$S=\{x+{\sqrt{2}}:0\leq x\leq 1\}$$ and $$A_{n}=\{0,1/n,2/n,...\}$$ Let $C_{n}=|S{\cap}A_{n}|$. Prove that the $$\lim_{n\to\infty}\frac{C_{n}}{n}=0$$. My attempt: $x+{\sqrt{2}}=\frac{k}{n}$ ...
0
votes
4answers
43 views

number of integers limit proof

find the limit of the number of integers $k$ satisfying $n{\sqrt{2}}{\leq}k{\leq}n(1+{\sqrt{2}})$ as $n$ tends to infinity. How can I obtrain a formula for the no. of integers
1
vote
3answers
28 views

Need help in understanding definition of limit of a function

I am using Thomas/Finney's Calculus and Analytical Geometry and it says "Given any positive radius ε about L, there exists some positive radius δ about c such that for all t within δ units of c ...
1
vote
2answers
34 views

Calculus add formula to derive new formula

I was asked to re-write a formula forward and backward and derive a new formula from it. Here's the problem: Here us formula 5.1.4: I'm not too sure where to start. Thanks!
5
votes
2answers
53 views

Integral with rational functions of powers and exponentials

Any ideas how to solve: \begin{equation} \int_0^\infty x^{n+\frac{1}{2}} (e^{a x }-1)^{-\frac{1}{2}} e^{i x t} dx \end{equation} where $a$ and $t$ are real, positive constants; $n$ is a positive ...
1
vote
1answer
60 views

Integration involving Gamma function

$\int_0^t e^{-\lambda x} x^{n-1} \ dx$ I know that $\int_0^{\infty} e^{-\lambda x} x^{n-1} = \frac{\Gamma(n)}{\lambda^n}$ but am not sure how to do it with the $t$ upper limit.
4
votes
2answers
77 views

What is wrong with the following u-substitution?

We will calculate $\displaystyle\int^{2 \pi}_0 x \, dx$. Let $u=\sin (x)$, and observe that $\sin(2 \pi)=0$ and $\sin(0)=0$. We also have that $\frac{du}{dx}=\cos(x)=\sqrt{1-u^2}$. Hence, $$ \int^{2 ...
0
votes
0answers
11 views

taylor series and zeta transform

let be 2 functins $ f(t) $ and $ g(t)$ related by an integral operator $$ g(s)=s\int_{0}^{\infty}dtK(st)f(t) $$ for some nice function K(t) if we assume that the function $ f(t) $ has a nice power ...
14
votes
2answers
338 views

How can I find this limit involving thrice-iterated logarithm?

$$\lim_{x \to 0}\dfrac{\ln \ln \ln \left[x+(1+x)^{(1+x)^{1/x}/x}\right]+x\left[1-\dfrac{1}{e^{e+1}}\right]}{x^2}$$ How can I find the limit of this question? Any hint. Thank you so much.
0
votes
0answers
26 views

Giant leap of the space goat [migrated]

This problem was inspired by this one and similar problems involving tethered animals. A space goat is tethered to the north pole of a spherical asteroid of uniform density and radius $a$ by a rope ...
-4
votes
1answer
64 views

Prove $\displaystyle\sum_{k=0}^\infty \frac{2^k}{\binom{2k+1}{k}}=\frac{\pi}{2}$ [on hold]

How to show that $$\sum_{k=0}^\infty \frac{2^k}{\binom{2k+1}{k}}=\frac{\pi}{2}$$
2
votes
1answer
65 views

Fitzpatrick's proof of Darboux sum comparison lemma

I am just reading Fitzpatrick's advanced calculus. He wants to prove $\lim (\max(x_{i-1} - x_i)) =0$ and $\lim(U(f,P)-L(f,P))$ is equivalent to $f$ is integrable. He used darboux sum comparison ...
2
votes
3answers
57 views

If $f$ is continuous on $[a,b)$ and differentiable on $(a,b)$ such that $\lim_{x\to b^{-}}f(x)=\infty$, Then $f'$ is not bounded above in $(a,b)$.

I got this problem: Let $f$ be a continuous function on the interval $[a,b)$ and differentiable on the interval $(a,b)$, Prove that if $\lim_{x\to b^{-}}f(x)=\infty$, Then $f'$ is not bounded above ...
0
votes
0answers
24 views

Integration with matrices

I have written two equations in matrix format as follows $m(t)={\begin{pmatrix} 200\\ 300\\ 400\\ 500 \end{pmatrix}}^T \begin{pmatrix} ...
2
votes
1answer
263 views

Product rule when differentiate matrix products

I want to differentiate the following expression with respect to $b$ $(Y-Xb)'(Y-Xb)$ Where $Y$ is $n\times1$ and $X$ is $n\times k$ and $b$ is $k\times1$, ' denotes transpose. If i do it term by ...
1
vote
1answer
30 views

Modulus of continuity of $x^2$ on $(0,1)$.

I am working with the modulus of continuity of a function $f:I\rightarrow\mathbb{R}$ defined as $\omega(f,r) = \sup \{ |f(x) - f(y)| : x,y\in I \ \& \ |x-y|\leq r\}$. I am asked to prove that if ...
2
votes
2answers
91 views

Puzzling Proof Involving IVT

If a and b are positive numbers, prove that the equation $$ \frac{a}{x^3+2x-1} + \frac{b}{x^3+x-2}=0$$ has at least one solution in the interval $(-1,1)$. You can't plug and chug ...
-2
votes
0answers
26 views

Set up and evaluate the definite integral for the area of the surface generated by revolving the curve about the x-axis. [on hold]

Set up and evaluate the definite integral for the area of the surface generated by revolving the curve about the $x$-axis. $y = \frac16 x^3 + \frac{1}{2x}$, $1 \leq x \leq 2$
1
vote
1answer
29 views

A Definite Integral Simplification

There was this question on integration that I am correct with the answers section of my textbook upto a point. So I will only ask this last step of evaluation. This is what the book says - And ...
0
votes
2answers
67 views

Find $\int \sin^{-1}\sqrt{x}\hspace{1mm}dx$ [on hold]

Find $\int \sin^{-1}\sqrt{x}\hspace{1mm}dx$ Can someone explain how to approach this problem
7
votes
2answers
324 views

Limit: How to Conclude

I have difficulty to conclude this limit ....; place of my attempts and results, can anyone help? tanks in advance $$\lim_{x\to +a}\, \left(1+6\left(\frac{\sin ...
0
votes
2answers
51 views

Simplification & Differentiation of $\frac{2x}{x^{1/3}}$

Above is the image I had taken a snap shot of. I was working on the problem # 24. I got to rewrite the function as: $y = 2x(x^{-1/3})$ I differentiated it and got the $y'$ as: $y' = ...
52
votes
11answers
2k views

Proving an alternating Euler sum: $\sum_{k=1}^{\infty} \frac{(-1)^{k+1} H_k}{k} = \frac{1}{2} \zeta(2) - \frac{1}{2} \log^2 2$

Let $$A(p,q) = \sum_{k=1}^{\infty} \frac{(-1)^{k+1}H^{(p)}_k}{k^q},$$ where $H^{(p)}_n = \sum_{i=1}^n i^{-p}$, the $n$th $p$-harmonic number. The $A(p,q)$'s are known as alternating Euler sums. ...
3
votes
3answers
76 views

How to compute: $\lim_{x\to 0}x^2 e^{\sin(1/x)}$?

Does anyone know how to compute the following limit? $$\lim_{x\to 0}x^2 e^{\sin(1/x)}$$ My attempt is that this limit does not exist because the limit of $\sin(1/x)$ as $x$ tends to $0$ does not ...
4
votes
6answers
80 views

Given $f(x)=\int_5^x \sqrt{1+t^2}\,dt$, find $(f^{-1})'(0)$

If $f(x)=\int_5^x \sqrt{1+t^2}\,dt$, find $(f^{-1})'(0)$. Here is what I have done so far. I have took $f'(x)=(1+x^2)^{1/2}$ and I have found $1/f'(0)$ which should equal $1$. I don't think this ...
2
votes
1answer
21 views

Finding the limit by factoring the denominator and canceling

I have the problem $$\lim_{x\to10} \frac{x-3}{x^2+7x-30}$$ If I factor it to $\dfrac{x-3}{(x+10)(x-3)}$ then $x-3$ cancels and I'm left with $0$. I know the real answer is $1/20$, but why is zero ...
0
votes
1answer
33 views

Sketch the region enclosed by $y=|x|$, $y=x^2-4$. Decide whether to integrate WRT x or y, then find the area of the region

One of those wonderful problems that was never discussed in class - any help would be much appreciated - I've been going at this darn thing all night :/
0
votes
4answers
91 views

Given $I_{n}=\int_{0}^{1}x^ne^{\sqrt{x}}dx$ find $\lim_{n \to \infty}I_{n}$

I need some guidance regarding the following problem, given the integral: $$I_{n}=\int_{0}^{1}x^ne^{\sqrt{x}}dx$$ where $n=0,1,2,...$ I have to find: $$\lim_{n \to \infty}I_{n}$$
10
votes
5answers
746 views

$\int_0^{\infty}\frac{\ln x}{x^2+a^2}\mathrm{d}x$ Evaluate Integral

Anyone remember the method for this? I think this should been done on the site $$\int_0^{\infty}\frac{\ln x}{x^2+a^2}\mathrm{d}x$$
0
votes
3answers
96 views

Prove rigorously: $\displaystyle \lim_{x\rightarrow 2}x^2+5x=14$

Prove rigorously: $\displaystyle \lim_{x\rightarrow 2}x^2+5x=14$ I am using an epsilon-delta proof, and this is as far as I got: $|f(x)-L|<\epsilon$ $|x^2+5x-14|<\epsilon$ ...
1
vote
4answers
67 views

Find $ \lim\limits_{x \to 0^+}x\int_x^1 \dfrac{\cos t}{t^2}\hspace{1mm}dt$

Find $\lim\limits_{x \to 0^+}x\int_x^1 \dfrac{\cos t}{t^2}\hspace{1mm}dt$ This looks like an interesting problem,but i cannot figure out where to start, can anyone explain
-5
votes
0answers
35 views

The limit of $\sin (\theta)/\theta$ as $\theta$ approaches $0$? [duplicate]

How to find the limit of $\sin (\theta)/\theta$ as $\theta$ approaches $0$? Rewrote this to make it clearer. I have thought about this and can't make sense of how to begin.
1
vote
3answers
51 views

$\sin ^{-1}(\cos (40 {}^{\circ}))=50 {}^{\circ}$

I know this is very basic but I am struggling with understanding this? $$\sin ^{-1}(\cos (40 {}^{\circ}))=50 {}^{\circ}$$ The step is this but I do not understand where what laws are used to get 90 ...
2
votes
2answers
45 views

Calculus for enzyme kinetics

I will try my best to make sense. Calculus is a distant memory for me... I would like to see if there is a way to generate an equation to determine the amount of remaining substrate as a function ...
0
votes
0answers
23 views

basic differential equation question

The following statement arises in a proof I am reading, and I do not understand why this is: Suppose $J$ is an open interval containing zero and $x: J \to W$ satisfies $x'(t)=f(x(t))$ and $x(0)=x_0$. ...