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

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3
votes
1answer
46 views

A question of Integration by parts

Consider $\int_a^b f'(x)g(x)dx$. Then the integration by parts gives $$ \int_a^b f'(x)g(x)dx = \left[ f(x)g(x) \right]_{a}^b - \int_a^b f(x) g'(x) dx.$$ In the case that $f(a), g(a), f(b), g(b)$ are ...
3
votes
3answers
222 views

Necessary condition for an improper integral to converge

I am working on this problem from a past examination: Let $f:[0,\infty)\rightarrow\mathbb R$ be a continuous, non-negative and non-increasing function such that the improper integral ...
3
votes
1answer
254 views

A tricky integral (flux of a point charge through a disk)

The integrals: $$ \oint \frac{r\,dr\,d\phi}{\left(L^2+r^2+h^2+2Lr\cos\phi\right)^{3/2}}\\ \oint \frac{dx\,dy}{\left((L+x)^2+y^2+h^2\right)^{3/2}} $$ If we have a point charge at the origin and we ...
0
votes
1answer
72 views

$\int_0^{\pi/6}\frac{1-\cos2{(x/2)}}{2} dx$

I have worked up to this stage of the question : $$\int_0^{\pi/6}\frac{1-\cos2{(x/2)}}{2} dx$$ So that's where I worked up to. Can someone please show me how to finish it off?
0
votes
2answers
83 views

Trigonometric function, with integration of definited integrals

I have worked up to this stage of the question : $$\int_0^{\pi/6}\frac{1-\cos2{(x/2)}}{2} dx$$ so that's where I worked up to. can someone please show me how to finish it off
1
vote
4answers
1k views

Integral of $\sin^2 \pi x$

Evaluate $$\int_0^{1/4} \sin^2 \pi x \; dx$$ Can someone please explain what to do if theres a power and how to do it in general thanks
2
votes
2answers
204 views

Injective function $f(x) = x + \sin x$

How can I prove that $f(x) = x + \sin x$ is injective function on set $x \in [0,8]$? I think that I should show that for any $x_1, x_2 \in [0,8]$ such that $x_1 \neq x_2$ we have $f(x_1) \neq f(x_2)$ ...
0
votes
0answers
308 views

Derivative Riccati-Bessel function

I have found two derivatives of the so-called Riccati-Bessel functions in a textbook $$ (x j_n(x))'=xj_{n-1}(x)-nj_{n}(x)$$ and $$ (x h_n^{(1)}(x))'=x h_{n-1}^{(1)}(x)-n h_n^{(1)}(x)$$ so $j_n$ is ...
1
vote
1answer
131 views

How to simplify $1-\frac1{a+b+1} $

I have this term with two factors a and b. a and b are positive integer numbers. $$1-\frac1{a+b+1}$$ b is an error in the problem that I want to separate it from the problem. For example such that ...
1
vote
1answer
33 views

Question involving PDE's

Suppose $f$ is a differentiable function of a single variable and $F(x,y)$ is defined by $F(x,y) = f(x^2-y)$. a) show that F satisifies the PDE $\frac{\partial F}{\partial x} + 2x \frac{\partial ...
4
votes
2answers
127 views

what fails in this proof of limits?

I always thought that this theorem was true, but today when asking about the proof of the chain rule in calculus I realized that it is false. I understand the counterexample, but now I don't ...
1
vote
0answers
27 views

How find this $m$ Value range

let $a\ge\dfrac{2^{m-1}-1}{m-1}$and such $$\left(\dfrac{\dfrac{3}{4}(\dfrac{3}{4}+a)(\dfrac{3}{4}+2a)\cdots(\dfrac{3}{4}+(m-1)a)}{(1+a)(1+2a)\cdots ...
1
vote
1answer
95 views

Existence of a function in one real variable

Does there exist a function $f: \mathbb{R} \to \mathbb{R}$ such that $f(f(x)) \neq x$ for all $x \in \mathbb{R}$ and for every $a \in \mathbb{R}$ there exists a sequence $\{x_n\}$ such that $$\lim_{n ...
2
votes
2answers
147 views

Is there a general family of curves that satisfies the following conditions?

Is there a general family of curves $f(x,c)$ that satisfies the following conditions? $f(x,c)$ is strictly increasing for $x \ge 0$ $f(0,c) = 0$ $f(1,c) = 1$ $f(x,c) \to \infty$ as $x \to \infty$ ...
2
votes
2answers
111 views

2 limits logarithms $ \lim_{t\to \infty} t-t^2\ln(\frac{t+1}{t}) $

I have problems with the following limits logarithms, and I can not use L'Hopital or power series (I know the results of the problems, using these methods), so I need solutions that do not occupy ...
1
vote
4answers
260 views

Differentiate $\ln(\cos2x)$ With respect to $x$.

I need to differentiate $\,\ln(\cos2x)$. Can someone please explain how to do this question? Thank you.
0
votes
1answer
123 views

Berkeley summer '81

Let $$y(h) = 1 - 2 \sin ^2 (2 \pi h), \quad f(y)= \dfrac{2}{1+\sqrt{1-y^2}} .$$ Justify the statement $$ f(y(h)) = 2 - 4 \sqrt 2 \pi + O(h^2) $$ where $$\limsup _{h \rightarrow 0} ...
5
votes
2answers
290 views

Is there any notable difference between studying the Riemann integral over open intervals and studying it over closed intervals?

(1) A function $f:[a,b]\to\mathbb{R}$ is said to be Riemann integrable on $[a,b]$ . . . (2) A function $f:(a,b)\to\mathbb{R}$ is said to be Riemann integrable on ...
1
vote
2answers
208 views

improper (double) integral: $\int_0^\infty\int_x^\infty\frac{1}{\sqrt{t^{3}+1}}\,\mathrm{d}t\,\mathrm{d}x$

I want to determine if the integral $\,\displaystyle\int_0^\infty\displaystyle\int_x^\infty\frac{1}{\sqrt{t^{3}+1}}\,\mathrm{d}t\,\mathrm{d}x$ converges. I know that ...
0
votes
2answers
251 views

Indefinite integral of normal distribution

How does one calculate the indefinite integral? $$\int\frac1{\sigma\sqrt{2\pi}}\exp\left(-\frac{x^2}{2\sigma^2}\right)dx$$ Where $\sigma$ is some constant. Work so far: Integrating from exp as ...
3
votes
1answer
639 views

How to prepare for Integral Calculus (Calculus 2)

I'm majoring in computer engineering and I have Calculus 2 coming up this semester. From what I understand, Calculus 2 is the most difficult math class in the engineering path. Over the summer, I've ...
1
vote
1answer
112 views

Question about lemma (2) - Spivak' calculus - page 89

I have a question about the proof of lemma (2) in Spivak's calculus, page 89. How does he simplifies $$ \ |y_0|\frac{\epsilon}{2(|y_0|+1)} $$ to get $$ \ \frac{\epsilon}2 $$ Thanks.
1
vote
2answers
123 views

verifying a polynomial is positive on the half-line

Math people: I am running experiments that produce polynomials $P(z)$ that, in every experiment I have run, are always positive on the half-line $\{z \geq 1\}$. I want to prove analytically that the ...
6
votes
3answers
379 views

Prove the sum of squares of two functions equals 1

If you have $f'(x)=g(x)$, $g'(x) = -f(x)$, $f(0)=0$ and $g(0)=1$, how do you prove that $f^2(x)+g^2(x) = 1$?
2
votes
4answers
174 views

a dog tied to a pole by a rope

A square hole of depth $h$ whose base is of length $a$ is given. A dog is tied to the center of the square at the bottom of the hole by a rope of length $L>\sqrt{2a^2+h^2}$ ,and walks on the ...
4
votes
2answers
395 views

proof of the chain rule for calculus

I was comparing my attempt to prove the chain rule by my own and the proof given in Spivak's book but they seems to be rather different. Please tell me if I'm wrong or if I'm missing something. I ...
1
vote
2answers
209 views

Differential of normal distribution

Let $$f(x)=\frac{\exp\left(-\frac{x^2}{2\sigma^2}\right)}{\sigma\sqrt{2\pi}}$$ (Normal distribution curve) Where $\sigma$ is constant. Is my derivative correct and can it be simplified further? ...
2
votes
1answer
91 views

The limit of $\frac{|A_n|}{n^2}$

Let $A_n=\{(i,j)\in\mathbb{Z}^2:\gcd(i,j)=1, \ \ 0 \leq i,j\leq n \}$. How to prove the existence of $\lim_{n\to\infty}\frac{|A_n|}{n^2}$, and how calculate this limit? Thank you!
0
votes
1answer
120 views

Trigonometric Functions. Definite Integrals

Find, correct to one decimal place, the value of $$\int_{0}^{60} 2\sin(x/2) \, dx.$$ Can someone please show me how this question is done. It would be very helpful thanks!
1
vote
0answers
37 views

A problem with bounds of derivatives

If $b,f\in C^4(\overline\Omega)$ and $|u'_\epsilon(y)|\leq C\epsilon^{-1/2}$, $\epsilon$ is a small parameter. If I also know that ...
1
vote
2answers
69 views

Integral calculation problem

![[] Could someone please explain how come the last passage is correct - We don't understand where the 1 went... Thanks
0
votes
1answer
152 views

Show that two integrals are equal:

Can someone show how to get the equality $$\int_0^{\pi/2} \cos^m(x) \sin^m(x) dx = 2^{-m}\int_0^{\pi/2} \cos^m (x) dx$$ with integral substitution? Thanks!
1
vote
1answer
43k views

Slope of line tangent to a curve at a given point, using First Principles

Find, from first principles, the gradient of the tangent to the curve $y = 5 - x^2$ at the point $(1,4)$ on the curve. So I'm currently lost on this question can some one please show me the solutions ...
2
votes
1answer
161 views

Improper Multiple Integral

I am trying to solve the following exam problem: Let $s$ be a real number. Find the condition under which the improper integral $$I:=\iint_{\mathbb R^2} \frac{dxdy}{(x^2-xy+y + 1)^s}$$ converges, ...
2
votes
0answers
44 views

getting PDF from a given Moment Generating Function

if the moment generating function mgf of a random variable w is M(t)=(1-7t)-20 find the i)pdf ii)mean iii)variance of w
5
votes
1answer
155 views

Computing the integral $ \lim\limits_{\epsilon\to 0} \int_{-2}^{0} \frac{e^{1/x(x+2)}}{x+1+i\epsilon} $

I got stuck when calculating of this expression $$ \lim_{\epsilon\rightarrow 0} \int_{-2}^{0} \frac{e^{\frac{1}{x(x+2)}}}{x+1+i\epsilon} $$ I will be grateful for the advice.
1
vote
1answer
70 views

moment generating function of gambling [duplicate]

Suppose a gambler starts with one dollar and plays a game in which he or she wins one dollar with probability $p$ and loses one dollar with probability $1-p$. Let $f_n$ be the probability that he or ...
2
votes
2answers
68 views

Using partial fractions to find an antiderivative of $(x^2+2x)/(x+1)^2$

Evaluate $$ \int\frac{x^2+2x}{(x+1)^2}dx $$ My solution Let $u =x+1$, $ du=dx $. Then $ du(x^2+2x)=(x^2+2x)dx $ and $ x=u-1 $. We get $$ \int\frac{(u-1)^2+2(u-1)}{u^2}du = ...
2
votes
1answer
41 views

Parametric solution of a multiple integral

Let $b$ and $a_i, i = 1,2,3,..., n$ be positive real numbers such that $$ \int_{0}^{\infty}\int_{0}^{\infty}\ldots \int_{0}^{\infty} e^{-b x_1^{a_1}x_2^{a_2}\ldots x_1^{a_2}} dx_1 dx_2 \ldots dx_n ...
-2
votes
1answer
81 views

Find the area of the Polar coordinates [closed]

Find the area of the region that is enclosed by the cardioid $r = 2 + 2 \cos\theta$.
1
vote
3answers
352 views

What must I do to plot the graph of $\sin x=\sin y$

I must represent the domain of the function: $$z=\frac{x-y}{\sin x-\sin y}$$ Therefore, $\sin x\neq\sin y$. So I must plot $\sin x=\sin y$. How do I do this?
1
vote
3answers
135 views

Initial value problem differential equation $y' = (x-1)(y-2)$

$$y' = (x-1)(y-2)$$ $y(2)= 4$ $$\frac{1}{y-2}dy = (x-1)dx$$ $$\int \frac{1}{y-2}dy =\int (x-1)dx$$ $$\ln(y-2) = \frac{x^2}{2} - x + c$$ $$y - 2 = e^{\frac{x^2}{2} - x + c} $$ $$y = ...
4
votes
1answer
151 views

$\frac{1}{N} | \int_1^N e^{2 \pi i b \log x }dx |\rightarrow \frac{1}{ \sqrt{1 + 4\pi ^2 b^2} } $ as $N \rightarrow \infty$

I want to show that $$\frac{1}{N} \left| \int_1^N e^{2 \pi i b \log x }dx \right| \rightarrow \frac{1}{ \sqrt{1 + 4\pi ^2 b^2} } $$ as $N \rightarrow \infty$. This what I have done: First do a ...
0
votes
0answers
47 views

Dynamics of sequence

The sequence $x_n,\, n \in \mathbb{N}$ is defined as follows: $x_1=1, \, x_{n+1}= \frac 1 {2+x_n} +\{\sqrt{n}\}\, $ for $n >1, $ where $ \{x \}$ denotes the fractional part of a real number $x$. ...
1
vote
1answer
60 views

Evaluating a contour integral

Find the contour integral of $$\int_C\frac{(z+a)(z+b)}{(z-a)(z-b)} \mbox{d}z,$$ where the modulus of $a$ and $b$ are less than $1$, and the integral path $C$ is the anticlockwise unit circle ($|z|= ...
1
vote
2answers
70 views

Finding x when slope = 1

I've been working out some problems relating to slope on the points of a curve. I'm having issues with this one: In the curve to which the equation is... $$x^2 + y^2 = 4$$ find the value of $x$ at ...
0
votes
2answers
166 views

Applying the “Cavalieri Principle” to approximate integration

(Apologies for asking another question based on Julian Havil's "The Irrationals") On page 86 of Havil's "The Irrationals" the author outlines how John Wallis approximated integration by applying a ...
2
votes
1answer
253 views

Is exponential function analytic over all complex numbers

In my textbook, I find a text where it says $e^z$ is analytic everywhere (in complex plane). Is it true? If so, what is the proof? I approached using maclaurin series, which gives $e^z= ...
5
votes
3answers
226 views

Dedekind's cut and axioms

What is the importance of 3rd axiom of dedekind's cut? a Dedekind cut is a partition of a totally ordered set into two non-empty parts (A and B), such that A is closed downwards (meaning that for all ...
2
votes
3answers
2k views

To prove the limit of given function does not exist.

Ques: I want to show that a limit of a function $$f(x,y)=\frac{x^{3}+y^{3}}{x-y}$$ does not exist at point $(0,0)$. My try: I am just taking path $y=x-x^{3}$ then $$\lim ...