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

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0
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1answer
32 views

Using bordered Hessian matrix to determine non-degeneracy and type of constrained extremum

I have the following problem: $\def\f{f(x_1,x_2,x_3)}\def\1{x_1}\def\2{x_2}\def\3{x_3}\def\n{\nabla}\def\g{g(x_1,x_2,x_3)}\def\l{\lambda}\def\q{\begin{pmatrix}}\def\p{\end{pmatrix}}$ Find the ...
1
vote
1answer
39 views

Local minimum of the function:

Find the local minimum of the function: $$\def\f{f(x_1,x_2,x_3)}\def\1{x_1}\def\2{x_2}\def\3{x_3}\def\n{\nabla}$$ $$\f=\1^2-2\1\2+2\2^2+\3^2 \text{ in } \mathbb{R}^3$$ $\n\f=(2\1-2\2,-2\1+4\2,2\3) ...
0
votes
1answer
48 views

A conjectured identity with Hölder conjugate exponents: if $|x|^p+|y|^p=1$, then $|x+y|^{p^*}+|x-y|^{p^*}=2$

Let $1\leq p<\infty$ and $x,y\in\mathbb{R}$ such that $|x|^p+|y|^p=1$. If $\frac{1}{p}+\frac{1}{p^*}=1$, prove that $|x+y|^{p^*}+|x-y|^{p^*}=2$. Actually I do not know if this is true. I think so, ...
2
votes
1answer
43 views

Sum involving integer part and cosine function

How to find the close form of sum and eliminate $k$? $$ \sum_{k=1}^{n} \frac{n \left[ \cos \left( \frac{n}{k}- \left[\frac{n}{k} \right]\right) \right]}{k} $$
1
vote
1answer
52 views

Proof on $F(x)=\sum_{n=1}^{\infty}f\left(\frac{x}{n}\right)$ uniform convergence and differentiability

Let $f$ be a function of $C^{\infty}$ class, such that $f(0)=0=f'(0)$. Prove that if $x\in\mathbb{R}$ and $$F(x)=\sum_{n=1}^{\infty}f\left(\frac{x}{n}\right)\ ,$$ then $F(x)\in\mathbb{R}$ and $F$ is ...
1
vote
1answer
27 views

Calculate a limit where $\sin n_k$ increases monotonically to 1.

Let us consider numbers $n_k$ ($k=0,1,2,3,...$), where $\sin n_k$ increases monotonically to 1. Calculate $$\lim_{k\rightarrow \infty} n_k\left(1-\sin n_{k}\right)-n_{k+1}\left(1-\sin n_{k+1}\right)$$ ...
0
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1answer
39 views

Linearizing systems about critical points.

$$\def\q{\begin{pmatrix}}\def\p{\end{pmatrix}}\def\l{\lambda}\def\f{\frac{\sqrt{11}}{2}}$$ Find all the critical points of the following systems and derive the linearised system about each ...
2
votes
1answer
58 views

Contour integral in complex plane (tricky)

Let U be a simply connected domain with a simple closed boundary curve C oriented anticlockwise, and define for all w ∈ C \ C $$ g(w)=\oint_C \frac{e^zdz}{(z-w)^2}$$ Find a formula for g(w) which does ...
2
votes
1answer
29 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.
0
votes
1answer
38 views

A rigorous proof of continuous differentiability

This small step comes from my reading on saddle point approximation: suppose $$ w=\text{sign}(s)\sqrt{2(s K'(s)-K(s))}\tag{*} $$ where $K$ is convex with and continuously differentiable for all orders ...
1
vote
2answers
45 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?
1
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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 $$\displaystyle \int_{0}^{1}\left(\sin x+a\cdot \cos x\right)^3dx-\frac{4a}{\pi-2}\int_{0}^{1}x\cdot \cos xdx = 2.$$ ...
0
votes
0answers
17 views

How to compute using integration the areas of the dodecagons (i.e. twelve-sided polygons) inscribed and circumscribed around a unit circle?

How to compute the areas of the dodecagons (i.e. twelve-sided polygons) inscribed and circumscribed around the unit circle centered at the origin using the methods of the integral calculus?
2
votes
1answer
60 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 $\frac{d}{dx}\sin^{-1}x+\frac{d}{dx}\cos^{-1}x=0$?
-2
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0answers
22 views

the sum of trigonometric functions and harmonic numbers [closed]

The first equation in a) gives a sum of 1 and the second equation starts with a sum equal to $\pi$. By removing $\sqrt x$ in b) the value of y is still almost the same.What is the exact value of y ...
2
votes
1answer
92 views

Show: $ f(a) = a,\ f(b) = b \implies \int_a^b \left[ f(x) + f^{-1}(x) \right] \, \mathrm{d}x = b^2 - a^2 $

If $a,b$ are fixed points of $f$, then $$ \int_a^b \left[ f(x) + f^{-1}(x) \right] \, \mathrm{d}x = b^2 - a^2 $$ In the words of 2014 MIT Integration Bee Champion (Carl Lian), the above property ...
5
votes
1answer
80 views

solution of $y' + y^2 = \varphi^2(x)$

I need to solve differential equation in the interval $[-\pi/2,\pi/2]$ \begin{eqnarray} y''(x) = y(x)\sin^2x \end{eqnarray} Trying $y(x) = \exp(\psi(x))$ yields, \begin{eqnarray} \zeta'(x) + ...
0
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3answers
41 views

Show that $u(t) \leq u(a) e^{\int_a^t f(s) ds}$

Let u(t) be a continuously differentiable function on [a,b] and the following inequality holds $\forall t \in [a,b]$ such that $u'(t) \leq f(t)u(t).$ Show that $u(t) \leq u(a) e^{\int_a^t f(s) ds}$ I ...
3
votes
1answer
34 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 ...
5
votes
1answer
72 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 ...
4
votes
1answer
37 views

Locating a radar in a plane

Given two located targets at $(x, y)=(- 2.0)$ and $(x, y)=(2,0)$. A radar, located in an unknown location of the $XY$ plane, and sends a pulse and in return receives pulses from the two targets. ...
2
votes
3answers
24 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 ...
5
votes
2answers
120 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 ...
3
votes
4answers
70 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
36 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| - ...
0
votes
2answers
76 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 ...
4
votes
2answers
73 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 ...
0
votes
1answer
22 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 ...
2
votes
2answers
176 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. $$\sqrt{1\sqrt{2\sqrt{3\sqrt{4\sqrt{...}}}}}$$ ...
3
votes
1answer
89 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 ...
4
votes
3answers
60 views

Differentiate with product rule

Question: differentiate $x(x^2 +3x)^3$ I have gotten to the point where i've used the product rule and i've gotten $$(x^2 + 3x)^3 + x\cdot(3x+9)(x^2 + 3x)^2$$ but now that it comes to the ...
0
votes
1answer
29 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 :: ...
1
vote
0answers
62 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}) ...
2
votes
2answers
59 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
1answer
19 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
50 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 ...
2
votes
1answer
35 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
50 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 ...
2
votes
4answers
61 views

For what $p$ does this series converge

"Find the values of $p$ s.t. the following series converges: $\sum_{n=2}^{\infty} \frac{1}{n^p \ln(n)}$" I am trying to do this problem through using the Integral Test to find the values of $p$. I ...
0
votes
1answer
18 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)),\ ...
17
votes
4answers
365 views

How to find ${\large\int}_1^\infty\frac{1-x+\ln x}{x \left(1+x^2\right) \ln^2 x} \mathrm dx$

Please help me to find a closed form for this integral: $$I=\int_1^\infty\frac{1-x+\ln x}{x \left(1+x^2\right) \ln^2 x} \mathrm dx$$
4
votes
1answer
74 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?
0
votes
2answers
51 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 " . ...
2
votes
1answer
87 views

How to find $\int_0^1 \frac {\mathrm dx}{\left \lfloor{1-\log_2(1-x)}\right \rfloor}$

We want to evaluate; $$\int_0^1 \frac {\mathrm dx}{\left \lfloor{1-\log_2(1-x)}\right \rfloor}$$ The $\left \lfloor{x}\right \rfloor$ is the floor function. I have made no progress so far.
2
votes
3answers
70 views

If $\ln x$ is defined via an integral and $e$ defined from $\ln x$, how would you prove that $\ln x$ is the inverse of $e^x$?

This is a somewhat technically specific question about the relationship between $\ln x$ and $e^x$ given one possible definition of $\ln x$. Suppose that you define $\ln x$ as $$\ln x = ...
5
votes
0answers
33 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 ...
0
votes
0answers
35 views

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

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 ...
6
votes
0answers
104 views
+50

Hands of the clock, Revisited.

It has already been answered (here) that it is impossible for the (continuously moving) hands of a clock to trisect the face of said clock. Even ideally the hour, minute, and second hand can never ...
1
vote
1answer
42 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. ...
-1
votes
0answers
27 views

Maximizing the profit of a monopolist, given the joint cost function and demand functions for two products

A monopolist produces quantities x and y of two goods, X and Y , respectively and the inverse demand functions for these are given by $$p_X =4−2x \ \text{ and } \ p_Y =2−2y$$ where $p_X$ and ...