0. Functions

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?

In Exercises 19–32, find the (a) domain and (b) range.

𝔂 = |x| - 2

{Use of Tech} A family of superexponential functions Let ƒ(x) = (a + x)ˣ , where a > 0.

b. Describe the end behavior of f (near the left boundary of its domain and as x→∞).

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?

In each exercise, a function and an interval of its independent variable are given. The endpoints of the interval are associated with points $P$ and $Q$ on the graph of the function.

a. Sketch a graph of the function and the secant line through $P$ and $Q$.

b. Find the slope of the secant line in part (a), and interpret your answer in terms of an average rate of change over the interval. Include units in your answer.

After $t$ seconds, an object dropped from rest falls a distance $d=16t^2$, where $d$ is measured in feet and $2\(\leq{t}\]\leq{5}\)$.

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?

Functions and Graphs

Graph the following equations and explain why they are not graphs of functions of x.

a. |x| + |y| = 1

Functions and Graphs

Find the domain of y = (x + 3) / (4 − √(x² − 9)).

101–104. Proving identities Prove the following identities.

cosh (x + y) = cosh x cosh y + sinh x sinh y

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.

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 $x=0$?

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

Slope functions Determine the slope function S (x) for the following functions

$\text{ƒ}\left(x\right)=\mid x\mid$

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