0. Functions

Evaluate the line integral of the vector field $F(x,y)=(2x,3y)$ along the path $C$ given by $r\left(t\right)=(t,t^2)$ for $0\le t\le 1$.

Consider the geometric series with first term $2$ and common ratio $\frac{1}{2}$. Write out the first four terms of the series: $2,1,\frac{1}{2},\frac{1}{4}$. What is the sum of the infinite series?

Evaluate the integral $\int \frac{5x^2\sqrt{x}-4}{x^2} dx$.

Which of the following is a power series representation for the function $f\left(x\right)=x^8\arctan \left(x^3\right)$ centered at $x=0$?

If $f\left(x\right)=x^2$, which of the following is the value of $f(-3)$?

Express the radius of a sphere as a function of the sphere’s surface area. Then express the surface area as a function of the volume.

Let $r\left(t\right)$ be the vector-valued function defined by $r\left(t\right)=(3t,\frac{1}{t+1},e^{-t})$. Evaluate the definite integral from $t=0$ to $t=1$ of $r\left(t\right)dt$.

State the inputs and outputs of the following relation. Is it a function? {$\left(-3,5\right),\left(0,2\right),\left(3,5\right)$}

State the inputs and outputs of the following relation. Is it a function? {$\left(2,5\right),\left(0,2\right),\left(2,9\right)$}

Find the domain and range of the following graph (write your answer using interval notation).

Decide whether $f$, $g$, or both represent one-to-one functions. <IMAGE>

Daylight function for 40 °N Verify that the function $D(t)=2.8\sin(\frac{2\pi}{365}(t-81))+12$ has the following properties, where t is measured in days and D is the number of hours between sunrise and sunset. It has a period of 365 days.

Daylight function for 40 °N Verify that the function $D(t)=2.8\sin(\frac{2\pi}{365}(t-81))+12$ has the following properties, where t is measured in days and D is the number of hours between sunrise and sunset.

Its maximum and minimum values are 14.8 and 9.2, respectively, which occur approximately at $t=172$  and $t=355$, respectively (corresponding to the solstices).

Daylight function for 40 °N Verify that the function $D(t)=2.8\sin(\frac{2\pi}{365}(t-81))+12$ has the following properties, where t is measured in days and D is the number of hours between sunrise and sunset.

$D\left(81\right)=12$ and $D\left(264\right)\approx 12$  (corresponding to the equinoxes).

The population of a small town was 500 in 2018 and is growing at a rate of 24 people per year. Find and graph the linear population function p(t) that gives the population of the town t years after 2018. Then use this model to predict the population in 2033.