0
votes
1answer
16 views

Proving If $\int^{\frac{\pi}{2}}_{0} f(x) \cos x dx \lt f \left(\frac\pi2\right),$ then $ \int^{\frac{\pi}{2}}_{0} f'(x) \sin x dx \gt 0. $

how do i tackle such problems? Let f(x) be a function differentiable on the interval$ $$\Bigl[$$ 0, \frac\pi2 $$\Bigr]$$ $ such that f'(x) is integrable on this interval. Prove the following ...
0
votes
1answer
7 views

Lorenz curve and Gini index using PDFs

I've been given that $f(w) = \frac{1}{4\sqrt{w}} $ for $0<w \le4$, and $F(w)$ is the associated CDF and represents the fraction of the population with income less than w. I know that the lorenz ...
2
votes
0answers
30 views

L'Hopital quicky

suppose L'Hopital applies and $$\lim_{x\to\infty}\frac{f(x)}{g(x)} = \lim\frac{f'(x)}{g'(x)}$$ under what conditions is it true then that $$\lim_{x\to\infty}\frac{\frac{f(x)}{g(x)} }{ ...
1
vote
1answer
33 views

A question about a continuous function that satisfies the property $\forall x\in\mathbb{R},\exists x<y\in\mathbb{R},f(x)<f(y)$

I got this question: Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function that satisfies the property: forall $x\in\mathbb{R}$ there exists $y \in\mathbb{R}$ such that $x < y$ and ...
1
vote
2answers
28 views

Find the equation of the line tangent to the following curve at x=1. Write your answer in y=mx+b format.

Find the equation of the line tangent to the following curve at $x=1$. Write your answer in $y=mx+b$ format. The curve is defined by $y=2x^2+6x-4.$ Please help with step by step instructions...I ...
1
vote
1answer
43 views

Prove that this matrix is not diagonalizable WITHOUT determinants

I have this matrix: $ \left( \begin{array}{cccc} 22 & 23 & 10 & -98\\ 12 & 18 & 16 & -38\\ -15 & -19 & -13 & 58 \\ 6 & 7 & 4 & -25 \end{array} \right) ...
2
votes
1answer
24 views

Determine the values of real parameters …

If you have an idea, please, do not leave the page, just write it, I will be very thankful. We have the function $$f:R\setminus \{-1 \}\to{R}$$ ...
0
votes
0answers
14 views

Find the values of the parameters for which the function admits an oblique asymptote…

can you please help me solve this exercise: Find the values of real parameters $a$ and $b$ so that the function $$\color{maroon}{f(x)={(ax^3+bx^2)}^{1/ 3}}$$ admits an oblique asymptote: ...
0
votes
3answers
25 views

Find all points on the curve $y=2x+x^{-1}$ which have a tangent parallel to the x-axis

Find all the points on the curve $y=2x+x^{-1}$ which have a tangent parallel to the $x$-axis.
0
votes
0answers
18 views

Finding the distance from a parabola (ballistic trajectory) to a point (for use in collision detection)

I need to have some form of collision detection / prevention for an object moving along a ballistic trajectory and a second stationary object on the same plane plane. The ballistic trajectory is ...
0
votes
1answer
39 views

how does this converges? Sequence and series convergence

Consider the following problem- Converges or Diverges? $$(1-2)-(1-2^{1/2})+(1-2^{1/3})-(1-2^{1/4})+....$$ I said it converges but then my work i showed in paper got wrong How would you prove that ...
0
votes
3answers
50 views

what are the equilibrium points of the following: [on hold]

where $x$ represents susceptible individuals, $y$ represents infected individuals. Find the two biologically meaningful equilibria. $$ \frac{\mathrm{d}x}{\mathrm{d}t} =12−3xy−3x $$ $$ ...
1
vote
1answer
28 views

Ho To Perform U-Substitution On Given Intergal

$\int{x^2\sqrt{2+x}}{dx}$ I haven't been able to find what u should be in this intergal, where should I start? I've gotten as far as: let $u = 2 + x$; $du=\frac{1}{x}dx$
1
vote
1answer
32 views

A question about a continuous function that satisfy certain limits at $\pm\infty$

I got this question: Let $f:\mathbb{R}\to\mathbb{R}$ be a continuous function such that $\lim_{x\to\infty}\frac{f(x)}{x^2}$ and $\lim_{x\to -\infty}\frac{f(x)}{x^2}$ exist and are real numbers. ...
-2
votes
0answers
28 views

Equilibrium question [on hold]

Consider the differential equation $$x' = x^3 − x^2 − 6x.$$ (a) Find all equilibria. (b) Determine the stability of each equilibrium analytically (not from the phase line diagram). (c) Sketch ...
1
vote
1answer
30 views

Continuous function $f:\mathbb{R}\to\mathbb{R}$ that got no extrema must be one to one

I got this question: Prove that if $f:\mathbb{R}\to\mathbb{R}$ is a continuous function that got no extrema then $f$ is one to one. I tried to prove it but I don't know how to proceed. I started by ...
1
vote
0answers
13 views

Prove with Lebesgue’s Criterion for integrablility that the composition $f\circ g$ is integrable

I have this homework question regarding Lebesgue's criterion for integrability and could use a bit of help. I'm not sure if my proof is entirely correct or formal enough. Here is said question: ...
0
votes
0answers
6 views

Non-monotonic function but Homothetic function

Is it possible for a function to be non-monotonic, but still homothetic? Thank you for your explanations.
-2
votes
1answer
33 views

Calculate Summation of series

One of my homework: Compute $\sum_{i=0}^{k-1}\alpha_i - (k-1) $ given $\alpha_i$ is $\frac{k-1}{k}$ Answer given is zero. So i suspect that $\sum_{i=0}^{k-1}\frac{k-1}{k}=k-1$? But how can i get ...
1
vote
2answers
30 views

Jacobian matrix with two equations

Evaluate the Jacobian for: $$f(x,y)=(x^2+x+y, yx+x^2)$$ at the point $(1,2)$.
0
votes
0answers
19 views

How can I write this in Divergence form

Consider the PDE $u_{xx}-(yu_y)_x-y(u_x)_y+yu_y+(y^2+\frac{1}{H^2(x)})u_{yy}$ I need to write this in divergence form. That is, I need to write it in the form $\sum_{i,j}\frac{\partial}{\partial ...
-4
votes
2answers
32 views

Find the coordinates of all points that satisfy certain conditions. [on hold]

Find the coordinates of all points whose distance from $(-3,6)$ is $\sqrt{13}$ and whose distance from $(2,7)$ is $\sqrt{13}$.
2
votes
2answers
27 views

Show that the difference quotient of $1/x^n$ exists

Let $n>0$ be a positive integer. For all $x\not=0$, prove that $f(x) = 1/x^n$ is differentiable at $x$ with $f^\prime(x) = -n/x^{n+1}$ by showing that the limit of the difference quotient ...
0
votes
3answers
41 views

Alternating p series. given that summation

Given that $$\sum_{k=1}^\infty{\frac{1}{k^2}} = \frac{\pi^2}{6}\ $$ Show that $$\sum_{k=1}^\infty{\frac{(-1)^{k+1}}{k^2}} = \frac{\pi^2}{12}\ $$
0
votes
1answer
27 views

Convergence parameter: Find the value of $p>0$ for which the series converge

For the sum for $k=2$ to infinity: $$\frac{\ln k}{k^p}\ $$ The textbook says the answer is $p>1$.
2
votes
1answer
37 views

Determine whether the series converge (adding fractions)

$$\frac{1}{1 \cdot 3} + \frac{1}{3 \cdot 5} + \frac{1}{5 \cdot 7} + ... $$ Help convert to summation. Not sure what test to use.
0
votes
1answer
28 views

Use comparison or limit comparison test to determine whether the series converge [on hold]

Summation symbol $$\frac{(k^2+1)^{1/3}}{(k^3+2)^{1/2}} \ .$$
2
votes
3answers
70 views

Lagrange multipliers from hell

I was asked to solve this question, decided to try and solve it with lagrange multipliers as I see no other way: "Find the closest and furthest points on the circle made from the intersection of the ...
-1
votes
0answers
29 views

Finding the solutions of an equation

I've got a huge problem that I haven't been able to understand , no matter how hard I tried . Let $f(x)$ be a random real function , and $f(x)=m$ an equation. The question is : To which interval does ...
0
votes
3answers
30 views

How to get from $3\int_{-1}^0 (x^3-x) dx \,\,\,- \,\,\, 3\int_0^1 (x^3-x) dx$ to $6\int_{-1}^0(x^3-x)dx$?

Homework problem: Set up the definite integral that gives the area of the region. Two functions are given: $y_1 = 3(x^3-x)$ $y2 = 0$ The graph of $y1$ runs from x=-1 to x=1. I've gotten this ...
0
votes
1answer
28 views

Related rates and velocity

A pedestrian $2~\text{m}$ tall walks directly away from a street light $6~\text{m}$ above the ground at $80~\text{m}/\text{min}$. When he is $8~\text{m}$ away from the post, determine the velocity of ...
0
votes
3answers
43 views

How to use Rolle's Theorem or the Mean Value Theorem to prove particular intersection points are the only intersection points

$$f(x) = x^{3}$$ and $$g(x) = \sqrt{x}$$ Find all of the intersection points between the graphs of $f$ and $g$. Show that these are the only intersection points I have found the intersection points ...
-4
votes
0answers
26 views

differentiate the given function. Simplify your answers [closed]

In Exercise 1 through 28, differentiate the given function. Simplify your answers y=√2X
1
vote
1answer
60 views

Vector Calculus Surface Integral (Limits of Integration)

I'm currently having trouble with the following problem. I believe that I have most of the problem set up, but I am having trouble finding what the limits of integration should be. $\int\limits_S ...
0
votes
2answers
28 views

Line integral over a curve in the II quadrant

I am lost here: $C = x^2 + y^2 = 4$ from $(0,2)$ to $(-2, 0)$. Calculate $ \ \int_c y^2 ds \ \ $ and give reasons the sign is correct. It's obviously the circular arc going counterclockwise from ...
2
votes
2answers
36 views

Center of Mass and Centroid

Find the centroid of the region lying between the graphs of the functions $y=\sin x$ and $y=\cos x$ over the interval $[0,\frac\pi4]$. I approached the question like this: Find the $M$ $$M = ...
0
votes
1answer
39 views

Proof of integral equality

Let $f^{(n)}(x)$ be the $n$-th derivative of $f(x) = \cos(x)$. Prove that : $$ \int_0^{2\pi} f^{(n)}(x) \,\, dx = \int_0^{2\pi} f^{(n)}(kx) \,\, dx, $$ where $n$, $k$ are natural numbers equal or ...
0
votes
0answers
43 views

how far from its starting point (and in which direction!) will the pendulum be after $2.5$ sec?

The Riddler has rigged a pendulum in the clock tower with enough explosives to level the nearby elementary school. Batman has figured out that he must must snip the green wire when the pendulum has ...
1
vote
2answers
56 views

How to integrate: $\int _{-1}^1\left(4x+1\right)\sin \left(x^2+x\right)\cos \left(x^2\right)dx\:$

$$\int _{-1}^1\left(4x+1\right)\sin \left(x^2+x\right)\cos \left(x^2\right)dx\:$$ This was the last question on my calculus final. Although it is useless to know how to do it now, I want to ...
3
votes
2answers
40 views

Nth derivative function

Is there any technique to find the $n$th derivative of $1/(1+x^2)$? I have been trying to find the $n$th derivative but cant.
-2
votes
0answers
18 views

Applied Maximum/Minimum [closed]

Please show work, thank you. A) Find the dimensions of the rectangle with the greatest area that can be built so the base of the rectangle is on the $x$-axis between $0$ and $1$ ( $0 \leq x \leq 1$ ) ...
0
votes
2answers
27 views

At what point on the graph of $y=\frac{1}{2}x^2$ is the tangent line parallel to the line $2x-4y=3$

I found the slope of the second equation to be $x\over 2$, and the derivative of the first equation is simply $x$, I believe.
0
votes
2answers
34 views

Find the parameter M

m(x+1)=e^|x| , m is a real number .Find the interval to which the parameter 'm' belongs , so that the previous equation has exactly two different solutions . Any idea how to approach this kind of ...
1
vote
1answer
17 views

How do I calculate the area under a curve using the midpoints of rectangles?

I figured out how to calculate the area under the curve from the Right endpoint and Left endpoints, but I can't figure out how to calculate it using the midpoints. Especially when it says $M_3$. Ill ...
1
vote
2answers
43 views

Area under the curve $f(x) = \sin x$

Find the area under the curve $f(x) = \sin x$ on the interval $[0, \pi]$ if $\sin x \ge 0$ My handbook give this as $$\int_0^\pi \sin x \space dx = (\cos \pi) - (\cos 0) = (-1) - (-1) = 2$$ what ...
-5
votes
1answer
32 views

Applied Max/Min [closed]

Please show work, thank you. A) Find the dimensions of the rectangle with the greatest area that can be built so the base of the rectangle is on the $x$-axis between $0$ and $1$ ( $0\leq x \leq 1$ ) ...
0
votes
0answers
37 views

Line Integral and Green's Theorem

I have been working on a simple line integral: line integral of $x\,dy+y\,dx$ (I don't know how do write this properly, I'm sorry!) over the closed curve enclosed by the the ellipse $x^2+5y^2=4$ and ...
0
votes
1answer
68 views

Proof of a condition in a proof that $\sqrt{5}$ exists

As a continuation of Prove that $\sqrt{5}$ exists I came across a different proof of $\sqrt{5}$ exists that is different from the answers on the original post and the proof goes something like this: ...
2
votes
1answer
96 views

Determining k: $\int_{6}^{16} \frac{dx}{\sqrt{x^3 + 7x^2 + 8x - 16}} = \frac{\pi}{k}$

I have a calculus II final coming up and this question came up in a past final exam: $$\int_{6}^{16} \frac{dx}{\sqrt{x^3 + 7x^2 + 8x - 16}} = \frac{\pi}{k},$$ where $k$ is a constant. Find $k$. My ...
0
votes
4answers
88 views

Prove that 1 is the only real number which satisfies $|x-1|<\frac{1}{n^2}$ for every $n \in N$

Prove that 1 is the only real number which satisfies $|x-1|<\frac{1}{n^2}$ for every $n \in N$ Here's how I would do this problem: First I noticed that $|x-1|\geq 0$ for all $x \in R$ So suppose ...