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

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0
votes
3answers
100 views

About $\sin 2\theta+\sqrt{3}\cos 2\theta=-\frac{\sqrt{3}}{2}$.

$0\leq\theta<2\pi$. When $\theta$ satisfies $\sin 2\theta+\sqrt{3}\cos 2\theta=-\frac{\sqrt{3}}{2}$, solve $\alpha+\beta$ ( $\alpha$:= minimum $\theta$, $\beta$:= maximum $\theta$). From the graph ...
1
vote
1answer
95 views

Can't solve indefinite integral

Can you help me with this, please? $$\int\sqrt{a\sin(bx)+c} \, dx$$ I thank you in advance for your help.
12
votes
4answers
373 views

Evaluating $ \int x \sqrt{\frac{1-x^2}{1+x^2}} \, dx$

I am trying to evaluate the indefinite integral of $$\int x \sqrt{\frac{1-x^2}{1+x^2}} \, dx.$$ The first thing I did was the substitution rule: $u=1+x^2$, so that $\displaystyle x \, ...
1
vote
3answers
87 views

Find $ \int \sqrt{\frac{1}{\theta^2}+ \frac{1}{\theta^4}} d\theta$

Any Ideas ! is this integrable function
2
votes
1answer
62 views

How to find $g'(0)$ if $g(x)=\sin(f(x^2+x)-2x)$ and $f$ satisfies $|f(x)|\le x^2$ for all $x$?

The problem goes as follows: Let $f(x)$ be a function such that $|{f(x)}|\le x^2$ $\forall x \in [-1,1/7]$. The first part of the problem is prove that $\lim_{x\rightarrow0} ...
2
votes
1answer
85 views

Integral of a cosine function

I search the integral of this function : $$\int_{-\infty}^{+\infty}\! \frac{\cos(x^3)}{x^3} \, \mathrm{d}x.$$ Thank you.
0
votes
1answer
41 views

Finding area between two functions

Find the area of the region bounded by the two functions $f(x)=2$ and $g(x)=\displaystyle \frac{x^2}{x+4}$. My try: \begin{align*} \int_{-2}^4 2 - \frac{x^2}{x+4} \, dx &= \int_{-2}^4 2 \, dx - ...
2
votes
3answers
83 views

indeterminate limit where applying L'Hopitals Rules directly doesn't help and using ln gives wrong answer

I am trying to determine the limit $\displaystyle{ \lim_{x \to 0^-}{\frac{-e^{1/x}}{x}}}$. Plugging in $x$ directly, yields $0/0$ which is indeterminate. Applying L'Hopitals rule does not simplify ...
2
votes
0answers
47 views

Special properties of bounded functions

I have a problem understanding the reasons as to why under some circumstances a term can be omitted due to it being a part of a bounded function, and I hoped to get some clarity to this here. There is ...
3
votes
2answers
83 views

Are there some functions that cannot be optimized using calculus?

I've been working on a project to maximize a functions output using a genetic algorithm. However, from the limited calculus I know I thought there were methods to find the maximum of a mathematical ...
3
votes
4answers
324 views

Does the recursive sequence $a_1 = 1, a_n = a_{n-1}+\frac{1}{a_{n-1}}$ converge?

Does the recursive sequence $a_1 = 1, a_n = a_{n-1}+\frac{1}{a_{n-1}}$ converge? Since the function $x+1/x$ is strictly monotonic increasing for all $x>1$, I don't think that the limit converges, ...
1
vote
3answers
155 views

Computing the sum of an infinite series

I am confused as to how to evaluate the infinite series $$\sum_{n=1}^\infty \frac{\sqrt{n+1}-\sqrt{n}}{\sqrt{n^2+n}}.$$ I tried splitting the fraction into two parts, i.e. ...
3
votes
2answers
65 views

Let $C_n =\int^{1/n}_{1/(n+1)}\frac{\tan^{-1}(nx)}{\sin^{-1}(nx)}dx $

Let $$C_n =\int^{1/n}_{1/(n+1)}\frac{\tan^{-1}(nx)}{\sin^{-1}(nx)}dx\textrm{.} $$ Then $\lim_{n \to \infty} n ^2C_n$ equals : a) $1$ b) $0$ c) $-1$ d) $\frac12$ I am not getting any clue ...
1
vote
1answer
185 views

Taylor series and Lagrange's remainder f(x)=$e^x$

In my textbook the Lagrange's remainder which is associated with the Taylor's formula is defined as: $R_{n}(x)= \frac{(x-a)^n}{n!} f^{(n)}(a + \vartheta (x-a))$, for some $\vartheta$ $\in$ <0 ,1> ...
6
votes
4answers
259 views

How to maximize or minimize $f(x)=ax^2+bx$?

I am trying to self-study calculus from the Internet. I have learnt things mostly from MIT OCW site and also from other sites. However, I am stuck on this simple problem: Find the maximum/minimum ...
2
votes
1answer
42 views

limit of trigonometric series

Given $$a_n = \frac 1{n^2}\sum_{k=1}^n {(2k+1)\sin\left(\frac{k^2 \pi}{n^2} \right)}$$ find $\lim_{n \to \infty} a_n$. My try: Because $k = 1,2,\dots, n$, $$0\le\frac {k^2}{n^2} \le 1$$ hence ...
2
votes
0answers
123 views

Line integral $\int_{\gamma}e^xcos(y)dx+e^xsin(y)dy$

First of all, I'm having trouble evaluating $$\int_{\gamma}e^xcos(y)dx+e^xsin(y)dy$$ where $\gamma$ is the triangle with vertices $(0,0), (1,0), (1,\frac\pi2)$ This is what I've done so far: 1) With ...
6
votes
4answers
211 views

How to find $\int_{0}^{\pi/2} \log ({1+\cos x}) dx$ using real-variable methods?

How do you find the value of this integral, using real methods? $$I=\displaystyle\int_{0}^{\pi/2} \log ({1+\cos x}) dx$$ The answer is $2C-\dfrac{\pi}{2}\log {2}$ where $C$ is Catalan's constant.
1
vote
1answer
86 views

Tough integral with many radicals

I am completed baffled with this integral $$\int\left[\dfrac{1}{x^{1/3}+x^{1/4}}+\dfrac{\ln(1+x^{1/6})}{x^{1/3}+x^{1/2}}\right]\mathrm dx$$ Any tips?
-1
votes
2answers
57 views

supremum of function

I've got some homework which I need to give an example to functions that makes this: $$ \sup f(g(x)) < \sup f(x) $$ I tries some ways but they are all lead that this is what happens: $\sup f(g(x)) ...
2
votes
3answers
192 views

Struggling to evaluate this limit: $\;\lim_{x\to 0} {\sqrt{2+x^2}-\sqrt{2-x^2}\over x^2}$

How do I solve this limit? I'm stuck. $$\lim_{x\to 0} {\sqrt{2+x^2}-\sqrt{2-x^2}\over x^2}$$
0
votes
2answers
48 views

Finding $\lim_{x\to0}\frac{1}{x^4}\Big(\frac{6}{6+x^2}-\frac{x}{\sinh(x)}\Big)$

Evaluate the limit $$\lim_{x\to0}\frac{1}{x^4}\Big(\frac{6}{6+x^2}-\frac{x}{\sinh(x)}\Big).$$ I realise we could write this as one fraction and apply L'Hospital a lot of times, but this is very ...
5
votes
1answer
66 views

how to determine the existence of double limit?

Let $f(x,y)$ be a function of two variables. Are there any criterions to determine the existence of double limit $$ \lim_{(x,y)\to(x_0,y_0)} f(x,y)? $$ If for all $y\in(y_0-\delta,y_0+\delta)$, ...
2
votes
2answers
124 views

$\int_0^\infty \frac{\sin^2(x)}{x^2(x^2+1)} dx$ =?

After reading articles on differentiation under the integral sign, I hit this post from mit, where after introducing the power tool, it challenges reader to do $$\int_0^\infty ...
0
votes
1answer
77 views

Proving that a specific function isn't continuous

Assume the following: $d(x)$ is the Dirichlet Function, $f(0) = 1$, $f(0)$ is continuous at $x = 0$, and $g(x) = d(x)f(x)$. I need to prove (in two ways: with $\delta$ and $\epsilon$, and with ...
2
votes
1answer
63 views

direction limits and double limit

Let $f(x,y)$ be a function of two variables. What is the counterexample that there exists $A$ s.t. for all $\theta$, $$\lim_{r\to 0+}f(r\cos \theta,r\sin \theta)=A$$ but double limit $$ ...
0
votes
2answers
52 views

Show that for all $(\tau, \xi) \in \mathbb R^{n+1}$ we have $|(\tau-ia)^2 - |\xi|^2| \ge a(\tau ^2+|\xi|^2+a^2)^{1/2}$

Show that, for all $(\tau, \xi) \in \mathbb R^{n+1}$, $|(\tau-ia)^2 - |\xi|^2| \ge a(\tau ^2+|\xi|^2+a^2)^{1/2}$ This is the exercise 7.4 in the book by Francois Treves. It is just a fundamental ...
4
votes
3answers
167 views

Does the integral $\int_{a}^{b}\frac{dx}{\sqrt{(x-a)(x-b)}}$ exist?

What is the result of this integral $\displaystyle\int_{a}^{b}\dfrac{dx}{\sqrt{(x-a)(x-b)}}$ ? I have tried many possibilities like letting $\sqrt{(x-a)(x-b)}$=u or trying to make the denominator ...
0
votes
1answer
32 views

Are all singular functions of bounded variation?

Let $f$ be a function of bounded variation on $[a,b]$. Then there exist a unique pair (up to adding a constant) of absolute continuous function $g$ and singular function $h$ (i.e., $h'=0$ a.e.) such ...
0
votes
2answers
35 views

How do I expand this summation?

So I just started doing these today and this has me stuck (It's a beginner question and i'm upset I'm stumped). So I have $\sum_{k=1}^{4}9k\sin(\frac{k\pi}{2})$ which I turn into ...
3
votes
3answers
107 views

Using the chain rule with a composite function

I'm a little confused on this homework problem and I could use some explanation if anyone has seen something like it before. The question is: Use the Chain Rule to find $\frac{dy}{dt}$ at $t = 9$ ...
0
votes
2answers
49 views

Finding a value R that maximizes the flux a vector field over half a sphere of radius R

Sorry for the bad title, couldn't think of a less convoluted way of writing it. I have to find $ R\in \mathbb{R}$ so that the flux of $$F(x,y,z) = (xz - x\cos(z), -yz +y\cos(z), -4 - (x^2 + y^2)) $$ ...
0
votes
3answers
48 views

Find the derivative of $\frac{{(x^3)^{4/3}}}{(2-x)^{4/3}}$

I tried to solve it using the chain rule first and then doing the quotient rule after. However, I end up with $\frac{24x^2-8x^3(x^3)^{4/3}}{3(2-x)^{7/3}}$ My professor said it's wrong. Kindly explain ...
0
votes
1answer
103 views

Find an equation for the conic that satisfies the given conditions.

Find an equation for the conic that satisfies the given conditions. parabola, focus $(−10, 0)$, directrix $x = 4$ I know how to do this type of problems and found the equation $y^2=-28(x+1)$ ...
3
votes
2answers
68 views

Let $f: \mathbb{R}-\{0\} \rightarrow \mathbb{R}, f(x)=\left| \cos(\frac{1}{x}) \right|$ Why is f not differentiable in $\frac{2}{\pi}$?

Let $f: \mathbb{R}-\{0\} \rightarrow \mathbb{R}, f(x)=\left|\cos(\frac{1}{x}) \right|$ Why is f not differentiable in $\frac{2}{\pi}$? If we take the difference quotient: $$\lim_{h \rightarrow ...
3
votes
1answer
70 views

Derivative: $f_x, f_y, f_{xy}$ of function - $f(x,y)$

Let's say $f(x,y) = x^2 + 2xy +y^2$ $f'_x = 2x + 2y$ $f'_y = 2y + 2x$ $f'_{xy} = 2x + 2y$ ? Am I right about the third?
4
votes
1answer
467 views

How to find the tenth derivative of an exponential function?

I have this equation, $f(x) = e^{-x^2}$. My question is how should I find $f^{(10)} (0)$, ie the tenth derivative of this equation. I have tried differentiating to get a formula, and I get ...
2
votes
2answers
212 views

A limit two variables

How can I compute or prove that $\displaystyle\lim_{(x,y)\to(0,0)}\dfrac{\mathrm{e}^{xy}-1}{\sqrt{x^2+y^2}}=0$?
11
votes
0answers
226 views

A closed form for $\sum_{k=1}^\infty \psi^{(1)} (k+a)\psi^{(1)} (k+b)$

The following result $$ \sum_{k=1}^\infty\left(\psi^{(1)} (k)\right)^2 = 3\zeta(3) $$ where $\psi^{(1)}$ is the polygamma function makes me think there is a nice sum for the series $$ ...
3
votes
4answers
136 views

Evaluate $\int{\frac{xe^x}{(1+x)^2} dx}$

How would I evaluate this integral? $$\int{\frac{xe^x}{(1+x)^2} dx}$$ I know I need to use parts but I ended up getting a very complicated expression to integrate the second time.
2
votes
0answers
71 views

Indefinite integral $\int \sqrt{(e^{2x}+e^{-2x})(e^{2x}-e^{-2x}+4) }\, dx$

The goal is to integrate $$\int \sqrt{(e^{2x}+e^{-2x})(e^{2x}-e^{-2x}+4) } dx$$ My approach would be to factorize the expression beneath the square root and then take the squareroot of that. ...
3
votes
3answers
156 views

Convergent or divergent $\sum_{n=1}^{\infty} \frac{e^nn!}{n^n}$?

Any suggestion/hint, not the whole solution, how to determine convergence/divergence of $$ \sum_{n=1}^{\infty}\dfrac{e^n \cdot n!}{n^n} $$ I'm currently stuck.
3
votes
3answers
70 views

Evaluate $\int x\sqrt{(a^2 - x^2)}dx$

I need to find $$\int x\sqrt{(a^2 - x^2)}dx$$ I tried putting $x=a cos(t)$ but I ended up getting a very complicated expression, so any tips?
1
vote
3answers
98 views

Evaluate $\frac{\partial^2}{\partial t^2} \left[ \prod_{j=1}^k (1+t+\dots+t^{d_j -1}) \right]$ at $t=1$

I need to find a "nice" formula for the evaluation of $\frac{\partial^2}{\partial t^2} \left[ \prod_{j=1}^k (1+t+\dots+t^{d_j -1}) \right]$ at t=1, where $d_j \in \mathbb{N}$. I have already proved ...
6
votes
4answers
151 views

If $f\left(x-\frac{2}{x}\right) = \sqrt{x-1}$, then what is the value of $f'(1)$

Find $f'(1)$ if $$f\left(x-\frac{2}{x}\right) = \sqrt{x-1}$$ My attempt at the question: Let $(x-\dfrac{2}{x})$ be $g(x)$ Then $$f(g(x)) = \sqrt{x-1} $$ Differentiating with respect to x: ...
3
votes
0answers
71 views

How to construct a diffeomorphism with $p_k \mapsto q_k$?

How to prove the following property? I cannot do anything. Let $M$ be a connected paracompact smooth manifold of dimension $m\geq 2$. Let $(p_k), (q_k)_{k\in \mathbb{N}}$ be sequences on $M$ which ...
0
votes
1answer
68 views

Find the volume inside

Find the volume inside the torus $\rho=\sin\phi$. First of all how can $\rho=\sin\phi$ represent a torus? I can't even visualise that. All Ideas are welcome, this looks like a 'food for thought ...
0
votes
1answer
56 views

Find $\int_0^1 \int_{3x}^3 (x^2+y^2)\sqrt{9-y^2}\hspace{1mm}dy dx$ [closed]

You can use a calculator after simplification if its not possible by hand All Ideas will be appreciated Also If you could find $$\int_0^1 \int_{3x}^3 x(x^2+y^2)\sqrt{9-y^2}\hspace{1mm}dy dx$$ ...
1
vote
2answers
70 views

How to prove that the sequence is decreasing $a_{n}=\frac{ln(n)}{n^2}$

Is my way/proof good and completely mathematically rigorous? $a_{n}=\frac{ln(n)}{n^2}$ --> $a_{n+1}=\frac{ln(n+1)}{(n+1)^2}$ $\frac{ln(n)}{n^2} > \frac{ln(n+1)}{(n+1)^2}$ ...
1
vote
2answers
110 views

Summation of infinite series, $\sum((3n+1)^{-1}-(3n+2)^{-1})$ [closed]

Find the sum of: $$\sum_{n=0}^\infty \left(\frac{1}{3n+1}- \frac{1}{3n+2}\right) $$ Anwser given was $\dfrac{\pi}{3\sqrt{3}}$. Thanks in advance.