0. Functions

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.

Let $f$ be the function defined by $f\left(x\right)=\frac{6}{1+x^2}$. Which of the following is the domain of $f$?

Evaluate the line integral of the vector field $F(x,y)=(x+6y)dx+x^{2}dy$ along the curve $C$, where $C$ is the line segment from $(0,0)$ to $(2,1)$. Which of the following is the value of the integral?

Evaluate the triple integral ${\int }_{}^{}{\int }_{}^{}{\int }_{}^{}z{d}_V$, where E is the region bounded by the parabolic cylinder $y=x^2$, the planes $z=0$, $z=4$, and $y=4$.

Which of the following best describes the contour lines (level curves) of the function $f(x,y)=x^2+7y^2$?

Consider the function $z=e^x\cos \left(y\right)$. Which of the following best describes the domain of this function?

If $f\left(x\right)=2x^2+1$, what is the value of $f\left(3\right)$?

Given the function $f\left(x\right)=x^2$ defined on the interval $0\le x\le 2$, determine the y-coordinate of the centroid of the region bounded by the graph of $f\left(x\right)$, the x-axis, and the lines $x=0$ and $x=2$.

Find the third-degree Taylor polynomial, $t_3\left(x\right)$, for the function $f\left(x\right)=\frac{1}{x}$ centered at $a=2$.