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

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2
votes
1answer
46 views

Surface of revolution of an ellipse

I have been working on this question, but I end up getting the wrong answer overtime: The ellipse $$\frac{x^2}{a^2}+ \frac{y^2}{b^2} = 1$$ where $a>b$ is rotated about the $x$-axis to form a ...
3
votes
1answer
46 views

Rectangle circumscribed to an ellipse of max area/perimeter

I could solve the classical problem of maximizing the area (fixing the perimeter) or maximizing the perimeter (fixing the area) of an inscribed rectangle, but I don't know how to solve ...
1
vote
2answers
64 views

Questionable Power Series for $1/x$ about $x=0$

WolframAlpha states that The power series for $1/x$ about $x=0$ is: $$1/x = \sum_{n=0}^{\infty} (-1)^n(x-1)^n$$ This is supposedly incorrect, isnt it? This is showing the power series about ...
0
votes
2answers
23 views

Chain rule and implicit dfferentiation

We are given that $y(x)=e^{z(x)}$. I want to show $$z''(x) = \frac{y''(x)}{y(x)}-\left(\frac{y'(x)}{y(x)}\right)^{2}$$ But I can't seem to get to this result. Since $y=e^{z}$ then $z=\ln{(y)}$, so ...
2
votes
1answer
57 views

power series of function

I am studying on summation theory on power series of functions. My question is to find the sum of power series $$e^{-n} \sum_{k=0}^{\infty} \frac{n^k\frac{k}{k+1}}{k!}.$$ I tried apply ...
3
votes
1answer
37 views

Prove with Cauchy's limit definition ($\epsilon, \delta$) that $\lim_{x \rightarrow 0} \frac{x^2-8}{x-8}=1$

Prove with Cauchy's limit definition ($\epsilon, \delta$) that $$\lim_{x \rightarrow 0} \frac{x^2-8}{x-8}=1$$ Got really troubled with the proper technique of solving this. Any assistance will be ...
2
votes
2answers
51 views

Wallis Product for $n = \tfrac{1}{2}$ From $n! = \Pi_{k=1}^\infty (\frac{k+1}{k})^n\frac{k}{k+n} $

How does $$\Pi_{k=1}^\infty \sqrt{\frac{k+1}{k}}\frac{k}{k+\tfrac{1}{2}} = \frac{\sqrt{\pi}}{2} = \frac{\sqrt{2(\tfrac{\pi}{2})}}{2} = \frac{1}{2}\sqrt{2 \Pi_{k=1}^\infty ...
0
votes
2answers
58 views

How to find the derivative of $F(x)=\int_{x^2}^{4x^2} \sin \sqrt t\;\;dt$?

For a real number $t>0$, let $\sqrt t$ denote the positive square root of t. For a real number $x>0$, let $F(x)=\int_{x^2}^{4x^2} \sin \sqrt t\;\;dt$. If $F'$ is the derivative of $F$, then ...
5
votes
3answers
91 views

Putnam definite integral evaluation $\int_0^{\pi/2}\frac{x\sin x\cos x}{\sin^4 x+\cos^4 x}dx$

Evaluate $$\int_0^{\pi/2}\frac{x\sin x\cos x}{\sin^4 x+\cos^4 x}dx$$ Source : Putnam By the property $\displaystyle \int_0^af(x)\,dx=\int_0^af(a-x)\,dx$: $$=\int_0^{\pi/2}\frac{(\pi/2-x)\sin ...
-4
votes
0answers
65 views

Determine $1^{x}+2^{x}+3^{x}+4^{x}+5^{x}+6^{x}+7^{x}+8^{x}+9^{x}=10^{x}$ [on hold]

there is only one real solution : $$1^{x}+2^{x}+3^{x}+4^{x}+5^{x}+6^{x}+7^{x}+8^{x}+9^{x}=10^{x}$$
1
vote
1answer
27 views

Solution for a complexed equation

Find $z$ for the equation $e^z + e^{-z} = 0$. So $$e^z + e^{-z} = 0 \iff e^z = -e^{-z} \iff e^z = e^{\pi i - z} \iff z = \pi i -z + 2\pi ik$$ I understand all expect the $2\pi ik$. Can you ...
1
vote
2answers
34 views

Matrix representation of the derivative of a smooth function

Let $V:\mathbb R^n\to\mathbb R$ be a smooth function and define the Hamiltonian function $H:\mathbb R^n\times\mathbb R^n\to\mathbb R$ (kinetic plus potential energy) by $$H(x,y):=\frac ...
1
vote
1answer
26 views

find roots in the complexes

Find the roots of: $$ z^2 -3z +4iz = 1-5i $$ Rearranging the terms: $z^2 + z(4i-3) + 5i - 1 $ Solving by using the quadratic formula: $$z_{1,2} = \frac{3-4i\pm \sqrt{(4i-3)^2 -4(5i-1)}}{2}$$ ...
6
votes
1answer
102 views

Prove that $ ax^2+bx+c=0 $ has at least one root in $(0,1)$ if $10a+12b+15c=0$

If $10a+12b+15c=0$, Prove that $$ ax^2+bx+c=0 $$ has at least one root in $(0,1)$. Progress I tried to solve this by Rolle`s theorem ($f'$ has a root between any two roots of $f$), but could not ...
-1
votes
1answer
35 views

Interval of converge of $\sum_{n=1}^{\infty} \frac{n!(x+1)^n}{(2n-1)!}$

Find the interval of converge of: $$\sum_{n=1}^{\infty} \frac{n!(x+1)^n}{(2n-1)!}$$ I will use the ratio test. Let $\displaystyle a_n = \frac{n!(x+1)^n}{(2n-1)!}$ $\displaystyle a_{n+1} = ...
2
votes
1answer
20 views

$ \int_{ABC} f = \int_{CDA} f $

Problem from this year's MIT-PRIMES application: Let $f$ be a continuous function on the plane. In any rectangle $ABCD$ so that $AB$ is parallel to the $x$-axis and $B$ has a greater ...
0
votes
0answers
13 views

Arc length and curvature for logistic curves [on hold]

How can arc length and curvature for logistic or sigmoid curves can be calculated? Consider the logistic curve given by $y = \frac{y_i-y_f}{1 + \left(\frac{x}{C}\right)^{1/B}} + y_f$ where, ...
0
votes
1answer
35 views

Interchange of derivatives

Given Euler-Lagrangian equation $$\frac{d}{dt}\frac{\partial L}{\partial \dot q}-\frac{\partial L}{\partial q}=0$$ Can I equivalently write as $$\frac{\partial \dot L}{\partial \dot q}-\frac{\partial ...
3
votes
3answers
93 views

$ \lim_{n \to \infty} \int_0^{\frac{\pi}{2}} \sum_{k=1}^{n} \left( \frac {\sin kx}{k} \right)^2 \, \mathrm{d}x $

Here is a problem in calculus shared by a friend. Compute $$ \lim_{n \to \infty} \displaystyle\int_{0}^{\frac{\pi}{2}} \displaystyle\sum_{k=1}^{n} \left( \frac {\sin kx}{k} \right)^2 \, \mathrm{d}x. ...
0
votes
1answer
28 views

Express the limit in terms of $f'(x_{0})$

Find the following limit in terms of $f'(x_{0})$: $$ \lim_{h \to 0} \frac{f(x_{0} - 3h) - f(x_{0})} {h} $$ Any help would be appreciated.
0
votes
4answers
51 views

Power series for $f(x) = \frac{4}{x+2}$

Find the power series $f(x) = 4/(x+2)$ We know the geometric series: $$\sum_{n=1}^{\infty} x^{n-1} = \frac{1}{1-x}$$ $(x+2) = 1 - (-x - 1)$ So: $$\sum_{n=1}^{\infty} (-1)^{n-1}\cdot(x + 1)^{n-1} ...
0
votes
0answers
39 views

Velocity & Acceleration (Calculus)

Suppose a UFO takes off from the surface of the Earth and rises with a vertical velocity given by v(t) = $\frac{3t^2}{t^2+1}$ (in km/s) where t is the number of seconds that have passed since ...
1
vote
0answers
21 views

Surface area of cylindrical surface using double integrals

Please help lead me in the right direction for this question, I'll give a description of my progress so far. My understanding is that the formula for the surface area is given by this equation: ...
0
votes
1answer
28 views

Volume of solid, calculus II [on hold]

Find the volume of the solid generated by revolving the region bounded by the curve $y=e^{x+1}$, the $x$-axis, the $y$-axis, and the line $x=-1$, about the line $y=e$. Please help me...it would be ...
0
votes
4answers
19 views

Find the domain, co-domain and range of a function

The function is $$g:\Bbb R\setminus\{0\}\to\Bbb R\setminus\{1\}\;,$$ where $$g(x) = x-\frac1x\;.$$ Please pardon my formatting as I am new to this. I know what a function is of course and their ...
0
votes
2answers
86 views

How would I compute this sum?

So I would to compute this integral which is coupled by a sum: $$ \int_{x = 0}^{x = \lambda} \sum_{k=-\infty}^\infty e^{-( \frac{x-k \lambda}{\sigma} )^2} dx$$ I was thinking about using parseval's ...
1
vote
0answers
23 views

Finding the normal vector of a surface (Flux of a vector field n*dS expression)

This problem is practice for a final exam. Let S be the closed surface whose bottom face B is the unit disc in the $xy$-plane and whose upper surface U is the paraboloid $ z = 1 − x^2 − y^2 , z \geq ...
0
votes
2answers
33 views

Finding the mass of a cone using triple integral

I have a density $\rho(x,y,z) = 3-z$ and have converted my given information to form a triple integral equation for finding the volume of my cone in cylindrical coordinates and have found the volume ...
1
vote
1answer
37 views

Derive a formula for the volume of the wedge in terms of the constants a, b, c.

Derive a formula for the volume of the wedge in terms of the constants a, b, c. Seeing a similar triangle, I see that $\frac{x}{y}=\frac{c}{b}$, $y$ being the distance from the $a$ line to the ...
2
votes
1answer
44 views

Differential Equation - Water evaporation

Given that a glass of water is filled to its fullest, $10\,cm$ in height, and that after three days the water level is at $9\,cm$ in height. Find when the glass will be empty. The water is ...
2
votes
2answers
45 views

Find the geodesics on the cylinder $x^2+y^2=r^2$ of radius $r>0$ in $\mathbb{R}^3$.

Find the geodesics on the cylinder $x^2+y^2=r^2$ of radius $r>0$ in $\mathbb{R}^3$. I know that the geodesics for cylinders are helices, circles, lines, and points, but i do not know how to ...
0
votes
3answers
44 views

Work done to fill up a spherical tank

A spherical tank of radius $12$ feet is $40$ feet above the ground. How much work is done in pumping water into the tank until it is full? I obtained $$ w= \int_{16}^{40}[12^2-(40-y)^2y] \, dy. ...
0
votes
1answer
46 views

find the possible values of z

given two complex number $z,w$ such number that $|z|\le1,|w|\le1$ and $|z+iw|=|z-i\overline{w}|=2$, then find the possible values of $z$ i tryed to use triangular inequality and got that ...
1
vote
2answers
20 views

Volume of solid of revolution by shell method

consider the region bounded by $ \displaystyle y=4{{x}^{2}}$ and $ \displaystyle 2x+y=6$. What is the volume of solid of revolution about $\displaystyle x$-axis. What is thought about setting the ...
14
votes
3answers
192 views

Sum of $k$-th powers

Given: $$ P_k(n)=\sum_{i=1}^n i^k $$ and $P_k(0)=0$, $P_k(x)-P_k(x-1) = x^k$ show that: $$ P_{k+1}(x)=(k+1) \int^x_0P_k(t) \, dt + C_{k+1} \cdot x $$ For $C_{k+1}$ constant. I believe a proof by ...
0
votes
3answers
24 views

Derivative with Logarithm Problem

I'm not sure how to approach this problem and solve it. $$y=\log_5\ln(x^3+6)^4$$
5
votes
4answers
48 views

Proving limit through definition

Prove $$\lim_{x\to 2}\frac{x^2+4}{x+2}=2$$ through definition. My solution: Fix $\epsilon >0$ and find $\delta$ \begin{align} 0<|x-2|<\delta &\Rightarrow \left| \frac{x^2+4}{x+2}-2 ...
1
vote
2answers
64 views

Intuitive but hard question about an integral?

Let $f \colon [0,1]\rightarrow \mathbb{R}$ be a function with continuous derivative such that $f(1)=1$. Evaluate $$\lim_{y\rightarrow \infty}\int_0^1yx^yf(x)dx$$
5
votes
1answer
59 views

If $f(0)=0$ and $f(1)=1$, prove that $\int_0^1 \left |f'(x)-f(x) \right |dx\geq e^{-1}$

Let $f$ be a differentiable function on $[0,1]$ such that $f(0)=0$ and $f(1)=1$. If $f'$ is continuous, prove that $$\int_0^1 \left |f'(x)-f(x) \right |dx\geq e^{-1}$$ Progress I let ...
0
votes
1answer
9 views

Is there a closed form expression for the Taylor series of (1- a X - b Y - c XY )^ (-1)?

Is there a closed form expression for the Taylor series of f(X , Y ) = (1- a X - b Y - c XY )^ (-1) ? a, b and c are constants X and Y are thank you
1
vote
1answer
27 views

If a function has asymptote and the derivative does not, then its second derivative is not bounded

Let $f\colon\mathbb{R} \rightarrow \mathbb{R}$ be a function with second derivative everywhere in is domain. Prove that if $\lim_{x\rightarrow\infty}f(x)=b \in \mathbb{R}$ and ...
0
votes
1answer
12 views

Get the closed form of Taylor series with Maple

Is it possible to get the closed form of Taylor series with Maple? The series command can give any given number of terms, but the question is about the closed form ...
2
votes
1answer
42 views

Logarithmic Differentiation - Always possible?

Logarithm functions in basic calculus classes are defined only for positive real numbers. But whenever we find an expression of the form $f(x)^{g(x)}$ we try to use logarithmic differentiation, we ...
1
vote
0answers
28 views

Second derivative of the position vector in a spherical coordinate system

In a spherical coordinate system my unit vectors are: $\vec{e_r}=\begin{pmatrix}\sin\theta\cdot \cos\phi \\ \sin\theta \cdot \sin\phi \\ \cos\theta \end{pmatrix}$; ...
1
vote
2answers
45 views

Prove that for $ f(x+2\cdot\pi)=f(x), x \in \mathbb{R} $, there exists $ x_0 $ so $ f(x_0+\pi)=f(x_0) $

Prove that for $ f(x+2\cdot\pi)=f(x), x \in \mathbb{R} $, there exists $ x_0 $ so $ f(x_0+\pi)=f(x_0) $. This is supposed to be question about continuity, but I’m not sure exactly what they mean, ...
1
vote
1answer
18 views

Double solutions and plotting transcendental equations

I have the following transcendental equation: $y^2 - \log(y)^2 = 4\cdot\log(x) + 4/x + C$ and I aim to plot the equation in the positive, real quadrant, with $x>0$ (actually in the $0 < x ...
23
votes
3answers
1k views

Is it OK to evaluate improper integrals this way?

Today in class we learned that when you have an improper integral like this one: $$\int_{-\infty}^\infty {f(x)} \: dx$$ you must split it before you do the limits (like so): $$\lim_{a \to \infty} ...
0
votes
0answers
24 views

To check the differentiability of following functions and my attempt

Hello i am posting my attempt .kindly please confirm that it is correct or incorrect .Kindly please suggest . I Need to check differentiability at origin of following
1
vote
2answers
67 views

How to solve ${\int_{\pi/4}^{\pi/2} x\cos x\,dx}$ using integration by parts?

$${\int_{\pi/4}^{\pi/2} x\cos x\,dx}$$ Would the method to solve this be integration by parts?
0
votes
3answers
32 views

Optimization Problem multivariable calculus or single variable

Problem is that a right circular cylinder is inscribed in a sphere of radius a .What is height of cylinder when its volume is maximal ? As per suggested by answer i attempted Any hints please ? ...