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

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5
votes
2answers
406 views

How prove this $x+y=0$ if $\left(\sqrt{y^2-x^3}-x\right)\left(\sqrt{x^2+y^3}-y\right)=y^3$

Question: let $x,y$ are real numbers,and such $$\left(\sqrt{y^2-x^3}-x\right)\left(\sqrt{x^2+y^3}-y\right)=y^3$$ show that $$x+y=0\tag{1}$$ before I have solve following problem: if ...
3
votes
1answer
103 views

Calculate $\lim_{n\to\infty} \sum_{k=1}^{n} \Big(\frac{k}{n}\Big)^{\alpha k}$

Let $\alpha$ be a positive number. Calculate $$\lim_{n\to\infty} \sum_{k=1}^{n} \Big(\frac{k}{n}\Big)^{\alpha k}$$ Edit: I have deleted my attempt, it didn't seem to lead me anywhere and I discovered ...
2
votes
1answer
47 views

If $y^{\frac{1}{m}} + y^{\frac{-1}{m}}=2x$, show that $x^2y_{n+2}+(2n+1)xy_{n+1} + (n^2-m^2)y=0$

My try : I have taken $y_1,y_2$ and tried to get a recursive relation between them but couldn't find any pattern. Please help.
2
votes
3answers
312 views

Gradient of modulus of vector.

I came across this in my lecture notes: This is using index notation, non-bold r is the modulus of r, and the partials are with respect to the components of r. I understand most of the steps, but ...
1
vote
2answers
48 views

If $a_{1}\;a_{2},a_{3}$ are the Roots of cubic eq. , Then $1000\left(a^2_{1}+a^2_{2}+a^2_{3}\right)$

If $a_{1}\;a_{2},a_{3}$ are three real values of $a$ which satisfy the equation $$\displaystyle \int_{0}^{1}\left(\sin x+a\cdot \cos x\right)^3dx-\frac{4a}{\pi-2}\int_{0}^{1}x\cdot \cos xdx = 2.$$ ...
3
votes
4answers
1k views

Find maximum and minimum value of function $f(x,y,z)=y+z$ on the circle

Find maximum and minimum value of function $$f(x,y,z)=y+z$$ on the circle $$x^2+y^2+z^2 = 1,3x+y=3$$ We have that $$y=3-3x$$ So we would like find minimum and maximum value of function ...
1
vote
2answers
57 views

Reversing the chain rule

I'm pretty new to calculus, but is there a way to reverse the chain rule so I can take the antiderivative of 1/(x^3+1) without using partial fractions?
4
votes
6answers
339 views

Why does $\lim_{n \to \infty} \sqrt{\frac{n}{n+1}} = 1$?

Why does $\displaystyle\lim_{n \to \infty} \sqrt{\frac{n}{n+1}} = 1$? Shouldn't it be undetermined?
2
votes
1answer
65 views

What is an intuitive way to see $\frac{d}{dx}\sin^{-1}x+\frac{d}{dx}\cos^{-1}x=0$?

Without calculation, explain why $\frac{d}{dx}\sin^{-1}x+\frac{d}{dx}\cos^{-1}x=0$?
3
votes
2answers
279 views

How to prove that the following function is convex?

I want to prove convexity of the following function: $$f(x) = log_x \left(1 + \frac{(x^a-1)(x^b - 1)}{x-1}\right)$$ for any fixed $a, b \in (0, 1)$ and: $x\in(0,1)$ $x\in(1, \infty)$ I'm trying ...
5
votes
2answers
181 views

How to choose the integration method for integrals involving powers and quotients of trigonometric functions?

I need help on these three integrals. Any hints on which method to use are greatly appreciated. $$1)\ \int \frac{1}{\cos^4 x}\tan^3 x\mathrm{d}x$$ $$2)\ \int \frac{1}{\sin 2x}(3\cos x + 7\sin ...
4
votes
1answer
110 views

How to evaluate $\int \frac{\mathrm{dx}}{x^4[x(x^5-1)]^{1/3}}$

How to evaluate: $$\int \frac{\mathrm{dx}}{x^4[x(x^5-1)]^{1/3}}$$ I have done a substantial work on it: Let $x^5z^3=x^5-1$. So $$x^5(z^3-1)=1\implies ...
3
votes
1answer
122 views

Integration by parts for Matrices

I understand how to do integration by parts for individual functions. I am trying to apply integration by parts to matrices/vectors where the order of terms is important. So say I have a matrix A ...
4
votes
4answers
109 views

Shorter way to integrate $\int \frac{x^9}{(x^2+4)^6} \, \mathrm{d}x$

$$ I=\int \frac{x^9}{(x^2+4)^6}\mathrm{d}x $$ Yeah I know, I can substitute: $$t=x^2+4\text{ or }2\tan\theta$$ So that: $$I=\frac12\int\frac{(t-4)^4}{t^6}\mathrm{d}t\text{ or } ...
1
vote
1answer
48 views

Double Integral $\int_{0}^{4} \int_{\sqrt{x}}^{2} \frac{1}{1+y^3} \mathrm{d}y\;\mathrm{d}x$

I am having trouble computing the double integral: $$ \int_{0}^{4} \int_{\sqrt{x}}^{2} \frac{1}{1+y^3} \mathrm{d}y\,\mathrm{d}x $$ I computed the inner integral: $$ \left [ \frac{1}{3}\ln|y + 1| - ...
4
votes
1answer
128 views

Is it true that a complex function has a global antiderivative if and only if it integrates to zero over every closed curve?

I am somehow thinking that these properties must be equivalent, unfortunately I do not know a theorem that says it: $f$ has a global antiderivative iff the line integral $ \int_{\gamma}f$ over every ...
0
votes
3answers
144 views

What are other unexpected results of integration?

I have integral of $\dfrac{1}{t^2 + 1}$ and integral of $\dfrac{t}{t^2 + 1}$ whose output is $\arctan(t)$ and $\dfrac12\ln(t^2 + 1)$ respectively. Are there any similar unexpected results when we ...
0
votes
2answers
95 views

Repeated differentiation of $\frac{1}{1+x^2}$

Let $g(x)=\frac{1}{1+x^2}$. I want to calculate the n-th derivative of $g(x)$ at $x=0,x=1$. For $x=0$, I wrote $g(x)=\sum_{n=0}^\infty (-1)^n x^{2n}$ from the geometric series. This says that ...
3
votes
2answers
333 views

How prove this integral $ \int\limits_0^1{\int\limits_0^1{\ln\Gamma\left({x+{y^3}}\right)}}dxdy =-\frac{7}{{16}}+\frac{1}{2}\ln 2\pi$

show that $$ I=\int\limits_0^1{\int\limits_0^1{\ln\Gamma\left({x+{y^3}}\right)}}dxdy =-\frac{7}{{16}}+\frac{1}{2}\ln 2\pi$$ where $$\Gamma{(a)}=\int_{0}^{\infty}x^{a-1}e^{-x}dx$$ then ...
5
votes
2answers
147 views

Evaluation of a simple limit with Taylor Series

I would like to evaluate $$\lim_{x\to0} \frac{e^{\sin x} - \sin^2x - 1}{x}$$ using Taylor Series expansion in a completely rigorous way. What would a rigorous version of the following argument look ...
2
votes
2answers
65 views

Find the Derivative

I'm currently studying the product rule and have come across a section of questions that seems to make no sense. I'm sure there's just one little thing that I'm missing but I am unable to spot it. ...
1
vote
0answers
87 views

Proving $\frac\pi{22}\cos\frac\pi{22}+\frac{2\pi}{11}\cos\frac{5\pi }{22}+\frac{2\pi}{ 11}\cos\frac{9\pi}{22}+\frac\pi{22}\cos\frac{5\pi}{11}<\cdots$

$$(\frac{\pi}{22}) \cos (\frac{\pi}{22}) +(\frac{2\pi}{11}) \cos (\frac{5\pi }{22}) + (\frac{2\pi}{ 11}) \cos (\frac{9\pi}{22}) + (\frac{\pi}{22}) \cos(\frac{5\pi}{11}) < (\frac{\pi}{26}) ...
3
votes
2answers
314 views

Nested Radicals with multiplication

I think this one goes to section of nested radicals, I was trying to solve if for a couple of days now. Maybe you have some nice solution to this one. $$\sqrt{1\sqrt{2\sqrt{3\sqrt{4\sqrt{...}}}}}$$ ...
0
votes
1answer
27 views

What is the value of the unknown parameter so that the given area condition holds?

The graphs of $f(x) \colon= x^2$ and $g(x) \colon= cx^3$, where $c > 0$, intersect at the points $(0,0)$ and $(1/c, 1/c^2)$. What is the value of $c$---and how to compute this value---so that the ...
1
vote
1answer
66 views

Order of $\{x\in\mathbb {Z}, |x|+|3x-1|<5\}$

There is a multiple choices which says what is the order of $\{x\in\mathbb {Z}, |x|+|3x-1|<5\}$? a. 1 b. 3 c. 2 d. empty I know that by considering certain cases, for example when $x<0$ or ...
7
votes
2answers
3k views

Some basic practical applications of Calculus

I am currently studying Calculus on my own for fun. I enjoy different components of math and how they can be used to solve so many problems. Many people, however, think I am crazy because I am ...
0
votes
1answer
34 views

Definite Integral theorem validity :- $\int_{0}^{L} \left( \int_{s}^{L}p(t)\ dt \right) \ ds =\int_{0}^{L} \ p(s) \ ds$?

Can we write $\int_{0}^{L} \left( \int_{s}^{L}p(t)\ dt \right) \ ds =\int_{0}^{L} \ p(s) \ ds\tag 1$ ? In other words, is this result valid? If so, could you help me to get the proof it NB :: ...
7
votes
1answer
177 views

Prove $\int_0^1 \frac{\ln(1+t^{4+\sqrt{15}})}{1+t}\mathrm dt= -\frac{\pi^2}{12}(\sqrt{15}-2)+\ln (2) \ln(\sqrt{3}+\sqrt{5})+\ln(\phi) \ln(2+\sqrt{3})$

Prove that: \begin{equation} \int_0^1 \frac{\ln\left(1+t^{4+\sqrt{15}}\right)}{1+t}\mathrm dt= -\frac{\pi^2}{12}(\sqrt{15}-2)+\ln (2) \ln(\sqrt{3}+\sqrt{5})+\ln(\phi) \ln(2+\sqrt{3}) ...
0
votes
2answers
302 views

Definite integral-dot product

I have an integral equation containing dot product $$\int_{0}^{L} \left(\frac{a}{L}.b(s)\right)\mathrm ds\tag 1$$ Data Given a is a constant vector of size 3 b(s) is a varying vector of size 3 " . ...
7
votes
4answers
156 views

How to find $\int \frac{x\ln(x+\sqrt{1+x^2})}{\sqrt{1+x^2}}\mathrm dx$

$$I=\int x.\frac{\ln(x+\sqrt{1+x^2})}{\sqrt{1+x^2}}\mathrm dx$$ Try 1: Put $z= \ln(x+\sqrt{1+x^2})$, $\mathrm dz=1/\sqrt{1+x^2}\mathrm dx$ $$I=\int \underbrace{x}_{\mathbb u}\underbrace{z}_{\mathbb ...
4
votes
2answers
60 views

Differentiability for the uniform limit of a uniformly bounded sequence of functions

Let a sequence $\{f_n\}\subset C^1(\mathbb{R})$ and $f\in C(\mathbb R)$ such that $f_n \to f$ uniformly and $f_n, f'_n$ are uniformly bounded. Question : is $f \in C^1(\mathbb R)$ ?
5
votes
3answers
1k views

What does it mean when an integral cannot be solved in terms of elementary functions?

My calculus teacher told our class that in integral calculus, they teach you how to integrate all kinds of functions by various methods, but in the end tell you that there are infinitely many ...
29
votes
4answers
1k views

Evaluation of $\int_0^{\pi/3} \ln^2\left(\frac{\sin x }{\sin (x+\pi/3)}\right)\,\mathrm{d}x$

I plan to evaluate $$\int_0^{\pi/3} \ln^2\left(\frac{\sin x }{\sin (x+\pi/3)}\right)\, \mathrm{d}x$$ and I need a starting point for both real and complex methods. Thanks ! Sis.
1
vote
1answer
98 views

Show that if $\sup\big\{\sum\lvert\, f(a)\rvert\big\} < \infty$, then $\{ a \in A : f(a) > 0\}$ is countable.

Let $f:A \to \mathbb R$ and suppose that $$ \sup\Big\{\sum_{a\in F}\lvert\, f(a)\rvert : F\text{ is finite subset of }A\Big\} < \infty $$ then the set $\{ a \in A : f(a) > 0\}$ is countable. ...
1
vote
1answer
37 views

finding local maximum and range of function

Consider the function $f(x)= \frac{ln x}{x}$, $0 < x < e^2$. (a) (i) Solve the equation $f'(x) = 0$ here I differenciated and I got $f'(x)= \frac{\frac{x}{ln x}-ln x}{x^2}$ and when I ...
1
vote
3answers
54 views

A simple-looking rational limit

Please help me compute: $$ \lim_{z\to 0}\frac{\sqrt{2(z-\log(1+z))}}{z} $$ I know the answer is 1 because I plugged it into Mathematica. Attempts with L'Hopital's Rule didn't work. This a step in an ...
3
votes
3answers
5k views

Relative maximum and minimum of function of three variables

I know that how to find relative maximum and minimum of function of two variables. How can I determine function when $f(x,y,z)=x^2+y^2-z^2 $ has relative maximum or relative minimum? Please give me ...
2
votes
1answer
39 views

Compute a rational limit

I tried to calculate this limit using change of variable: $\displaystyle\lim_{x\to1}\frac{\sqrt[2]{2-x} - 1}{1 + \sqrt[5]{x - 2}}$ But i don't get the result, which is -5/2. I would appreciate if ...
1
vote
2answers
113 views

What is the first step to solving $\cos3x - \sin x = \sqrt{3}(\cos x - \sin 3x)$?

My calculus BC teacher has given us some trig "review". $$\cos3x - \sin x = \sqrt{3}(\cos x - \sin 3x).$$ How do I get rewrite the cos3x and sin3x? Do I just use sum and difference, because it ...
0
votes
1answer
56 views

Derivative of this equation

Our teacher gave us an assignment to find the derivative the equation given, but I have tried for more than 2 months, and yet can't get any way to do it. Are you guys willing to try? $$\frac{d}{dx} ...
4
votes
1answer
93 views

Showing derivative of polynomial has $n$ distinct roots

Let $$f_n(x)=\frac{\mathrm d^n}{\mathrm dx^n}((1-x^2)^n)$$ Any hints on how to show that it has $n$ distinct real roots?
1
vote
1answer
27 views

An uniformly convergent iterated function series

Let $f$ be twice continuously differentiable on $[-1,1]$ with ragne $[-1,1]$, and $$f(0)=0, 0<f'(0)<1/2, |f''(x)|\leq M<1.$$ Denote by $$f_1(x)=f(f(x)), f_n(x)=f(f_{n-1}(x)),\ ...
1
vote
1answer
364 views

Assignment: Find $a$ and $b$ such that a piecewise function is continuous

I'm having trouble solving a problem given in an assignment: If the following function $f(x)$ is continuous for all real numbers $x$, determine the values of $a$ and $b$. $$ ...
1
vote
4answers
53 views

How to solve $(x-3)\left(\frac{\mathrm dy}{\mathrm dx}\right)+y=6e^x, x>0$

Solve $$(x-3)\left(\frac{\mathrm dy}{\mathrm dx}\right)+y=6e^x, x>0$$ I have a very similar problem like this on my homework, and I have no clue how to set it up or even start. How could I set ...
1
vote
1answer
110 views

What is the average velocity of the motorcycle?

The position of a person riding in a motorcycle race is give by $s(t)=4t^2+3t$, where $t$ measures time in seconds since the race began, and position is measured in feet beyond the starting line. ...
5
votes
0answers
48 views

For the exponential operator $e^{f(x)\frac d{dx}}= \sum_{i=0}^\infty F_i(x) \frac{d^i}{dx^i}$, is there a formula for the $F_i$ in terms of $f$?

Consider the operator $$ e^{ f(x) \frac{d}{dx} } = \sum_{i = 0}^\infty \frac{1}{i!} \left(f \frac{d}{dx} \right)^i $$ If one commutes the derivatives with the powers of $ f $, then there are functions ...
1
vote
2answers
43 views

How to find an 'optimum' minimum difference value to identify the closest similar points on a plot?

Data Set I have asked a similar question and the data set is same, here. The Goal In the given plot, you can see there is a loop (all the points in the blue tiles make 1 big loop) i.e. the plot ...
2
votes
4answers
106 views

Does $\int_{-\infty}^\infty \frac{\mathrm dx}{(1+x^2)^\alpha}$ converge?

I'm wondering when the integral $$ \int_{-\infty}^\infty \frac{\mathrm dx}{(1+x^2)^\alpha} $$ converges for the real number $\alpha$.
1
vote
1answer
43 views

How to prove that an integral converges

Let $(a_n)$, $(M_n)$ be sequences of positive real numbers such that ${a_n} \downarrow 0$, ${M_n} \uparrow \infty$ as $n\to\infty$. Let $\alpha>0$ and $\beta>1$. How to prove the following ...
0
votes
0answers
39 views

Evaluation of $\int_{1}^{\infty} x^{-\frac{5}{3}} \cos \left( \left( x-1 \right) h \right)dx$ with Maple [duplicate]

I have calculated the Integral with the aid of some professors here and I get a problem: $$\int_{1}^{\infty} x^{-\frac{5}{3}} \cos \left( \left( x-1 \right) h \right)dx$$ I have done the Integral ...