0. Functions

Find the exact length of the curve given by $x=\frac{y^4}{8}+\frac{1}{4y^2}$, for $1\le y\le 3$.

Is the constant function $y\left(t\right)=-4$ a solution to the differential equation $y^{\prime }=t^2(y+4)$?

Which of the following is a power series representation for the function $f\left(x\right)=x^6\arctan \left(x^3\right)$ centered at $x=0$?

Which of the following is a power series representation for the function $f\left(x\right)=x^4\arctan \left(x^3\right)$ centered at $x=0$?

Which of the following is a power series representation for the function $f\left(x\right)=x^2\arctan \left(x^3\right)$ centered at $x=0$?

Find the exact length of the curve defined by $x=\frac{1}{3}y(y-3)$ for $9\le y\le 16$.

Given the boundary-value problem $y^{″}-2y^{\prime }+2y=2x-2$, with $y\left(0\right)=0$ and $y\left(\pi \right)=\pi$, which of the following is the correct solution for $y\left(x\right)$?

For the function $f$ whose graph is given, which of the following best describes the domain of $f$?

Which of the following is the correct definition of a function in calculus?

Given the polar curve $r=\cos \left(2\theta \right)$, what is the exact length of one petal of the curve?

Find the exact length of the polar curve $r=2$, for $0\le \theta \le 4\pi$.

Find the length of the curve given by $r\left(t\right)=(6t,t^2,\frac{t^3}{9})$ for $0\le t\le 1$.

Evaluate the triple integral over the region E, where E lies above the cone $z=\frac{r}{3}$ and below the sphere $z^2+r^2=1$, of the function $y^2{z}^2$ $dV$. Which of the following is the correct value of the integral?

Evaluate the line integral of the vector field $F(x,y)=(2x,3y)$ along the curve $C$ given by the vector function $r\left(t\right)=(t,t^2)$, where $t$ goes from $0$ to $1$.

Given the power series $\sum \underset{n=1}{}\stackrel{\infty }{}\frac{{(}^{-1}n}{x^n}/9^n$, find the radius of convergence, $r$, of the series.