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

learn more… | top users | synonyms

3
votes
2answers
59 views

Separating $\frac{1}{1-x^2}$ into multiple terms

I'm working through an example that contains the following steps: $$\int\frac{1}{1-x^2}dx$$ $$=\frac{1}{2}\int\frac{1}{1+x} - \frac{1}{1-x}dx$$ $$\ldots$$ $$=\frac{1}{2}\ln{\frac{1+x}{1-x}}$$ I ...
1
vote
1answer
72 views

Prove $a_n^k$ convergence by induction

Question: $a_n \to L, 0 \le a_n, 0 \le L $ prove that $\forall k \in \Bbb N : a_n^k \to L^k$ What I did: From the limit definition: $\lvert a_n-L \rvert <\varepsilon$. Obviously it has to do with ...
3
votes
4answers
112 views

Expressing a simple sequence in closed form

I just started sequences, and I'm having trouble understanding how to finda pattern. My teacher didn't really explain the technique of finding the closed form. I there any formulas that I should ...
2
votes
3answers
91 views

Solving $\int_0^1\int_{y}^{\sqrt{y}}\frac{\sin x}{x} dxdy$

I am stuck at this question: $$ \int_0^1\int_{y}^{\sqrt{y}}\frac{\sin x}{x} dxdy $$ It seems maybe I can use change of variable theorem to solve this but I don't know how... Can someone show me ...
0
votes
3answers
453 views

Approximating the integral $\int_0^{0.1} \sqrt{1-1/2\sin^2(t)} dt$

How would I go about evaluating $I = \int_0^{0.1} \sqrt{1-1/2\sin^2(t)} dt$ to four decimal accuracy?
2
votes
2answers
96 views

Upper and lower sums - Spivak

I am using the 2nd edition of the book page 218 in Calculus by Michael Spivak in the chapter of the Riemann Integral. $$\sup \{ L(f,P) \} \leq \inf \{ U(f,P) \} $$ It is clear that both of ...
-1
votes
1answer
104 views

Showing $h(x) = \frac{1}{\epsilon}\int_x^{x+\epsilon} f(t)dt$ is differentiable for continuous $f$ and $\epsilon > 0$

Assume $f$ is continuous on $\mathbb{R}$ and $\epsilon>0$. Let $h(x) = \displaystyle\frac{1}{\epsilon}\int_x^{x+\epsilon} f(t)dt$. Show $h$ is differentiable and $h'$ is continuous. Compute ...
0
votes
1answer
73 views

Particular solution of the second order non homogeneous equation

I have a differential equation as follows: $$y''-4y'-4sy=\frac{4}{s} \mathrm{exp}(2x-2x\sqrt{1+s})$$ The general solution of the above equation would be in the form of $$y = C_1 ...
1
vote
1answer
38 views

Question about a part of a proof

I am reading a proof concerning continuity of composite functions. I came across the following in a particular proof. Taking $\epsilon=1$ in the definition of continuity, there exists $\delta_{2} ...
0
votes
1answer
205 views

Prove by Induction using Baseline and splitting into LHS & RHS?

I'm having trouble with this equation mainly because it has a couple of odd things with it, and its these that have thrown me off as i'm not to sure how to tackle them. The equation is: ...
14
votes
2answers
564 views

Evaluating $\int_0^{\infty}\frac{e^{-x}}{1+x^2}dx$

I'm trying to evaluate $$\int_0^{\infty}\dfrac{e^{-x}}{1+x^2}dx$$ By making the substitution $x=\tan\theta$, $$\int_0^{\infty}\dfrac{e^{-x}}{1+x^2}dx=\int_0^{\frac \pi 2}\exp(-\tan\theta)d\theta$$ So ...
1
vote
0answers
84 views

Very basic line integrals (worked out)

my problem is basically (among many others) my textbook sucks and the answers are frequently non nonsensical, sadly i haven't slept in a couple days so im not sure whos out to lunch. gonna post the ...
1
vote
4answers
121 views

Why does $ \frac{1}{1 - z} = 1 + z + z^2 + z^3 + \ldots$ when $|z| < 1$

For $|z| < 1$, $$\frac{1}{1 - z} = 1 + z + z^2 + z^3 + \ldots$$ The fact stated above has appeared in many proofs in my complex analysis textbook, but I have no idea why it is true. Can ...
0
votes
2answers
100 views

Volume by cross-section of a donut

How do I find the volume generated by revolving $(x - H)^2 + y^2 = a^2$ around the y axis specifically using cross sections (I know that shells are easy to use but I can't). I have struggled with ...
0
votes
2answers
71 views

Calculus simple question, who is right?

I have the following budget constrain: $p_1\cdot P+p_2\cdot C=100$ and I also have this equality that I have to plug into the budget constrain: $2P=C$ So: $p_1\cdot P+p_2\cdot 2P=100$ , so now I want ...
10
votes
1answer
264 views

Equality of integrals

this is q.2 of ahlfors p. 241: Show that $$\int_{-1}^{1}\frac{dt}{\sqrt {(1-t^2)(1-k^2t^2)}}=\int_{1}^{\frac{1}{k}}\frac{dt}{\sqrt {(t^2-1)(1-k^2t^2)}}$$ if and only if $k=(\sqrt{2}-1)^2$ . Thank you. ...
1
vote
1answer
77 views

Finding ring endomorphisms.

I need to find $\varphi \in \operatorname{End}(\mathbb{R}[x])$ such that there's a function $\psi \in \operatorname{End}(\mathbb{R}[x]), \psi \neq 0$ such that $\psi \circ \varphi = 0$ but there's no ...
1
vote
0answers
52 views

Integrating $\dfrac{1}{(1+2x)(2\sqrt{x})}$.

I'm trying to integrate the following: $\int_0^\infty \frac{1}{2\sqrt{x}(1+2x)} $ After a series of $u$ & $s$ substitutions, I end up with the following: $ \frac{1}{\sqrt{2}} \arctan(s) $ with ...
3
votes
3answers
640 views

Evaluating the limit of $\ln(2x)/\ln(x)$ as $x\to\infty$

I'm trying to evaluate the following limit: $\displaystyle\lim_{x \to \infty} \displaystyle\frac{\ln(2x)}{\ln(x)}$ Using L'Hospital's rule, I end up with: $\displaystyle\lim_{x \to \infty} ...
-1
votes
2answers
1k views

How does one calculate the centroid of an equilateral triangle using integration [closed]

Edit: The vertices are $(0,0)$, $(1, 0)$, $(1/2, \sqrt{3}/2)$. Please confirm. Confirmed.
1
vote
2answers
46 views

Lagrange remainder to approximate $3^{2.1}$ less than 0.1

How do I solve this problem: Use the appropriate Taylor polynomial $P_n(x,c)$ to estimate $3^{2.1}$ with error less than $0.1$, given $\ln 3$ is about $1.099$. I understand that the remainder ...
5
votes
2answers
85 views

Integrate rational function $\frac{x^2}{1+x^4}$

Integrate $$\int\frac{x^2dx}{1+x^4}$$ I've factored the denominator to $(x^2-\sqrt{2}x+1)(x^2+\sqrt{2}x+1)$ and got stuck.
4
votes
1answer
136 views

Is this a valid proof of the derivatives of the trigonometric functions?

For the sake of this proof, the trigonometric functions $\cos$ and $\sin$ are defined as the coordinates of a point on the unit circle, rather than any of the modern analytic definitions. Let $\vec ...
2
votes
1answer
226 views

Suppose that $f(x)= xg(x)$ for some function $g$ which is continuous at $0$. Prove that $f$ is differentiable at 0 , and find $f'(0)$ in terms of $g$.

Suppose that $f(x)= xg(x)$ for some function $g$ which is continuous at $0$. Prove that $f$ is differentiable at 0 , and find $f'(0)$ in terms of $g$. Okay, so far I have: $$f(x) = x g(x) \quad ...
4
votes
1answer
131 views

Evaluating the integral $\int_{-1}^1 \frac{1}{\sqrt{1-x^2}}\ln|z-x|dx$

I don't know how to deal with this integral: $$\int_{-1}^1 \frac{1}{\sqrt{1-x^2}}\ln|z-x|dx,$$ where z is a complex number.
2
votes
4answers
1k views

Sketching the graph of a trig function

How would I sketch the following function. $f(x)=\cos(x)+\sin(x)$ on $[0,2\pi]$ for my first derivative I got $f'(x)=-\sin(x)+\cos(x)=0$ but How would I find the critical points I mean I know ...
0
votes
1answer
98 views

Range of values for optimization?

Example 1: A window is being built and the bottom is a rectangle and the top is a semicircle. If there is 12 meters of framing materials what must the dimensions of the window be to let in the ...
3
votes
1answer
92 views

Problem about finding region and function preserving areas

This problem is driving me crazy. "Let $S$ be a parametrized surface by $\varphi:[a,b]\times[c,d]\rightarrow\mathbb{R^3}$, $\varphi$ is of class $C^1$. Show that exists an open, connected set ...
0
votes
0answers
63 views

Limit of an integral - $\lim_{t \to \infty} \frac{\int_{a}^{b}e^{\frac{-x^2}{1 + t^2}} dx}{\sqrt\pi \sqrt{1+t^2}} = 0$

I'd like to show that $$ \lim_{t \to \infty} \frac{\int_{a}^{b}e^{\frac{-x^2}{1 + t^2}} dx}{\sqrt\pi \sqrt{1+t^2}} = 0 $$ with $a, b $ constant. It seems pretty intuitive to me that this is ...
1
vote
2answers
60 views

How to calculate this?$S=\sum\limits_{i=1}^\infty(-1)^i(i-1)!x^i$

How to calculate this equation $S=\sum\limits_{i=1}^\infty(-1)^i(i-1)!x^i$ ?
0
votes
4answers
3k views

Find the equations that are tangent to $x^2 + 4y^2 = 16$ that also pass through $(4,6)$

How would I go about solving the following question? Find the equations of the lines that are tangent to the ellipse $x^2 + 4y^2 = 16$ and that also pass through the point $(4,6)$ Please provide ...
3
votes
0answers
45 views

Need $f:[0,\infty)\to[0,\infty)$ such that $f$ is not convex but $f(x^p)$ is for $p>1$.

To be more specific I need to find an $f:[0,+\infty)\to[0,+\infty)$ which satisfies the following: (somewhat trivial stuff) The function $f$ is continuous, nondecreasing, there exists $k>0$ such ...
3
votes
4answers
90 views

Hyperbolic cosine

I have an A level exam question I'm not too sure how to approach: a) Show $1+\frac{1}{2}x^2>x, \forall x \in \mathbb{R}$ b) Deduce $ \cosh x > x$ c) Find the point P such that it lies on ...
2
votes
0answers
145 views

Extremal condition calculus of variations

if I have a functional with a Lagrangian $L(t,x(t),y(t),x'(t),y'(t))$, meaning two functions x and y of one parameter t. And want to solve the minimization problem $$ \int_0^t L \, dt. $$ Then I get ...
1
vote
2answers
100 views

Question about Maclaurin Series for $\cos x$

I understand how to get the proper maclaurin series representation for $\cos x$, but I'm having trouble understanding the following part conceptually: I get $\cos x$ as $\sum_{n=0}^\infty ...
2
votes
3answers
77 views

convergence of $\sum\frac{a_{n}}{n}$ if $\sum_{k=1}^{n}a_{k}\le M*n^{r}$ where $r<1$

Show that if the partial sums $s_{n}$ of the series $\sum_{k=1}^{\infty}a_{k}$ satisfy $|s_{n}|\le M*n^{r}$ for some $r<1$, then the series $\sum_{n=1}^{\infty}\frac{a_{n}}{n}$ converges.
6
votes
1answer
101 views

if $\sum a_{n}$ is a convergent series, what about $\sum \frac{a_{n}}{1+|a_{n}|}$?

Suppose that $\sum a_{n}$ is convergent series of real numbers. Either prove that $\sum b_{n}$ converges, or give a counter-example, when we define $b_{n}$ by $\frac{a_{n}}{1+|a_{n}|}$.
0
votes
1answer
47 views

Check Logic for Induction Of T(n) Equation

I've just finished an induction equation, however i'm a little hit and miss about whether its actually right on not. Mainly in my working out, I'd really appreciate if you could have a look at it and ...
0
votes
1answer
59 views

How to find $du$ where $u=f(t,t^2,t^3)$?

I have to find $du$ if $u=f(x,y,z)$, $x=t$, $y=t^2$, and $z=t^3$. So this means that I have to find $du$ where $u=f(t,t^2,t^3)$. Where do I go from here?
2
votes
2answers
201 views

Limit of Fraction of Polynomial Proof

Let $p$ be a polynomial of degree $\ge 1$. I need help proving the following statement: $$\lim_{x \to \infty} \frac{1}{p(x)} = 0$$ I'm complete lost and don't know how to approach it.
1
vote
1answer
172 views

Find taylor polynomial that approximates e^x with accuracy at least 1.

Find Taylor polynomial at $x=0$ which approximates $e^x$ with accuracy at least $1$ for each $x \in [-2,2]$. I dont undestand these questions that involve the $n^{th}$ remainder. I know I need to ...
1
vote
1answer
75 views

How to find $\frac{dz}{dx}$ and $\frac{dz}{dy}$ if $z=\ln(e^x+e^y),$ and $y=x^3$

I have to find $\frac{dz}{dx}$ and $\frac{dz}{dy}$ if $$z=\ln(e^x+e^y), \quad y=x^3.$$ This seems to be a special type of derivative, looks like a compound one - so to find $\frac{dz}{dx}$ I have to ...
4
votes
3answers
92 views

Finding the solution for $Ax=0$

Find the solution for $Ax=0$ for the following $3 \times 3$ matrix: $$\begin{pmatrix}3 & 2& -3\\ 2& -1&1 \\ 1& 1& 1\end{pmatrix}$$ I found the row reduced form of that ...
-2
votes
1answer
62 views

Linear Approximation Problem [closed]

Suppose, $f$ is a function satisfying $f(1) = 2$ and $f'(2) = 2$. Using a linear approximation to $f$ at $x = 2$, find out an approximation to $f (0.95)$.
1
vote
2answers
54 views

Radius of convergence in a series. Ratio test.

I am having a hard time with this question. $$\sum_{k=0}^{\infty} \frac{-(1)^k (4^k -3)x^{2k}}{k^4+3}$$ I used the ratio test and got stuck here: $$x^2 \lim_{k\to\infty} \frac ...
0
votes
0answers
51 views

Inverse functions of two summations

I am trying to find the inverse functions of these summations $$\large \displaystyle f(x) = \lim_{z\to\infty} \; \sum_{i=1}^x \left(\left|\cos \left(\frac{\pi}{a} i - \frac{\pi ...
0
votes
2answers
32 views

Are all defined points relative maxima?

$f(c)$ is a relative maximum in the interval $(a,b)$ iff $f(c) \ge f(x)$ for all $x$ in $(a,b)$ Doesn't this imply all points are relative maximums by taking $a = c$ and $b=c$?
1
vote
1answer
186 views

Is the Folium of Descartes a smooth curve?

The Folium of Descartes, $$x^3+y^3=3axy$$ differentiatiing gives $$\frac{\mathrm dy}{\mathrm dx}=\frac{ay-x^2}{y^2-ax}$$ Is it possible that $y^2 = ax$? I looked up on wikiapedia that under the ...
1
vote
1answer
117 views

Is there an easier way to solve this solid of revolution problem?

We just learned about solids of revolution today and one of the homework problems asks to find the volume obtained by revolving the region bounded by $x^2$ and $x^5$, and the revolution is around the ...
3
votes
1answer
106 views

Optimization Problem.

I'm working on some calculus homework, which deals with optimization problems, we have the solution posted for us and when looking over it I got a bit confused. Here's the question: An open ...