0. Functions

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.

Throwing a stone Suppose a stone is thrown vertically upward from the edge of a cliff on Earth with an initial velocity of 32 ft/s from a height of 48 ft above the ground. The height (in feet) of the stone above the ground t seconds after it is thrown is $s\left(t\right)=-16t^2+32t+48$ .

d. When does the stone strike the ground?

Demand and elasticity Based on sales data over the past year, the owner of a DVD store devises the demand function $D\left(p\right)=40-2p$ , where D(p) is the number of DVDs that can be sold in one day at a price of p dollars.

a. According to the model, how many DVDs can be sold in a day at a price of $10?

Velocity from position The graph of $s=f\left(t\right)$ represents the position of an object moving along a line at time $t\ge 0$ . <IMAGE>

a. Assume the velocity of the object is 0 when $t=0$ . For what other values of t is the velocity of the object zero?

Velocity from position The graph of $s=f\left(t\right)$ represents the position of an object moving along a line at time $t\ge 0$ . <IMAGE>

c. Sketch a graph of the velocity function.

Fish length Assume the length L (in centimeters) of a particular species of fish after t years is modeled by the following graph. <IMAGE>

b. What does the derivative tell you about how this species of fish grows?

Evaluating functions from graphs Assume ƒ is an odd function and that both ƒ and g are one-to-one. Use the (incomplete) graph of ƒ and g the graph of to find the following function values. <IMAGE>

ƒ(g(4))