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

learn more… | top users | synonyms

0
votes
1answer
24 views

Integrate $dx/(4x^2-1)^{3/2}$

I have trouble using trig sub. After I get that x = 2x+1, should I substitute back into the original problem's $4x^2$ with $(4(2x+1)^2)$?
1
vote
1answer
28 views

How to find integral of $\int \cot^2(x) \operatorname{dx}$

How do you find the antiderivative of $\cot^2$? My steps to find it First $$ \csc^2 x = \cot^2 x+ 1 $$ because of Pythagorean Identities, so $$ \cot^2 x= \csc^2 x-1$$ so $$ \int \cot^2 x\, ...
2
votes
2answers
39 views

Integrate $\int \csc^6(2x)\, dx$

I know to use the identity $1+\cot^2(2x)$. I'm not sure how to use $u$-substitution to substitute the $2x$ from the problem. I would have to use a $u$-substitution and then another $w$-substitution. ...
0
votes
1answer
23 views

If $f$ s twice differentiable and satisfies the following constraints, prove $f'(0)>-\sqrt 2$

Let $f$ be a twice differentiable function on the open interval $(-1,1) $such that $f(0)=1$. Suppose $f$ also satisfies $f(x) \ge 0, f'(x) \le 0 $and $f''(x) \le f(x)$, for all $ x\ge 0$. Show that ...
0
votes
2answers
49 views

Why doesn't $\ln (x)$ have an asymptote since its derivative is $1/x$?

My understanding is that the derivative gives the gradient of the function at that point. So for the function $x^2$, its gradient at point $x=10$ is equal to $20$. Extrapolating this to $\ln (x)$, ...
7
votes
2answers
95 views

Why doesn't it work when I calculate the second order derivative?

Let $y=y(x)$ be determined by the equation \begin{align*}\begin{cases} x=t-\sin{t}\\ y=1-\cos{t}.\end{cases} \end{align*} I understand the solution: ...
2
votes
1answer
37 views

How to prove $\lim\limits_{t \to 1^-} \frac{\sqrt{1-t^2}}{2\pi}\int_{S^1}\frac{f(x,y)}{1-tx}ds=f(1,0)$?

$f(x,y)$ is a continuous function defined on unit circle $\ S^1 :$ $x^2+y^2=1$, prove $$\lim\limits_{t \to 1^-} \frac{\sqrt{1-t^2}}{2\pi}\int_{S^1}\frac{f(x,y)}{1-tx}ds=f(1,0)$$ I have tried to ...
-1
votes
2answers
22 views

Maximal value, several variables [on hold]

Let $x_i>0, \quad n=1,...,n, \quad \sum_{i=1}^nx_i=1$. Show that the function $\sum_{i=1}^nx_i\log_2\frac{1}{x_i}$ attains a maximal value at $x_i=\frac{1}{n}, \quad i=1,...,n$. Give me a hint, ...
2
votes
2answers
45 views

Proof : If $f$ continuous in $[a,b]$ and differentiable in $(a,b)$ and there is $c \in (a,b)$ so $(f(c)-f(a))(f(b)-f(c))<0$

I need to proof this : If $f$ continuous in $[a,b]$ and differentiable in $(a,b)$ and there is $c \in (a,b)$ so $(f(c)-f(a))(f(b)-f(c))<0$ then there is $d \in (a,b)$ so $f'(d)=0$. I'm not sure ...
3
votes
1answer
27 views

Volume when rotated about the line $y=-1$

Find the volume when the region enclosed by $y=x^2$, $y=4$ is revolved around the line $y=-1$ My teacher has given the following answer: I assume she has done this through the method of shells, ...
1
vote
0answers
60 views

Integrals and f(x)dx

Suppose $$\int_0^2 f(x)\,dx=3$$ $$\int_0^5 f(x)\,dx=8$$ Compute $$\int_2^5 f(x)\, dx$$ $$\int_0^2 f(2x)\,dx$$ For the first one, I know that by subtraction $$\int_2^5 f(x)\,dx = \int_0^5 ...
0
votes
0answers
8 views

Logistic model. Did I set up the differential equation $(1)$ correctly?

Update: I fixed it. The major mistake I made was that originally put $I(t) = \beta\cdot(P-y(t))$ while it of course is supposed to be $I(t) = \beta\cdot y(t)$. NB: I came up with this problem ...
0
votes
1answer
18 views

How to find the integration bounds when calculating area

To calculate an area between curves, I need to integrate with respect to x between the curve $y=\sqrt{2x}$, the x-axis and the line $y=\frac{4x-12}{5}$ My understanding, using google to display plot ...
3
votes
1answer
54 views

Factorial identity $\left(\tfrac{1}{2}\right)!$ to get Waallis

I asked the wrong question here, my fault :( How does one see, using $n! = \prod_{k=1}^\infty \left(\frac{k+1}{k}\right)^n \frac{k}{k+n}$, that $$\left(\frac{1}{2}\right)! = \frac{}{} ...
0
votes
0answers
18 views

Searching for a constant transformation in $ \mathbb C$

I am having a continous transformation: $f: \mathbb C \to \mathbb C $ with a set $B \subseteq \mathbb C $, which is bounded. Now I want to proove that $ A = f^{-1} (B)$ is NOT bounded! I know it ...
1
vote
1answer
38 views

When integrating, can only one term of an equation be integrated or must entire equation be integrated to maintain equality?

Is integration considered a basic operation in the sense you have to do it to all parts of the equation? $y dy - x dx = 0$ Is it valid to do $\int y dy - \int x dx = \int 0$ but invalid to leave out ...
2
votes
1answer
40 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
30 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
60 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
50 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
36 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
41 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
55 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
81 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 ...
-5
votes
0answers
62 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
26 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
32 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
25 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
97 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 ...
0
votes
0answers
23 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
12 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
34 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 ...
2
votes
2answers
80 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
37 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
17 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
27 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
82 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
21 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
27 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
35 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
43 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
18 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 ...