# Subsection1.4.1Exercises

The function $z$ shown in Figure 1.4.1 was generated by the formula $\fe{z}{x}=2+4x-x^2$.

 $h$ $\frac{\fe{z}{4+h}-\fe{z}{4}}{h}$ $-0.1$ $-3.9$ $-0.01$ $-0.001$ $\phantom{-}0.001$ $\phantom{-}0.01$ $\phantom{-}0.1$
##### 1

Simplify the difference quotient for $z$.

##### 2

Use the graph to find the slope of the secant line to $z$ between the points where $x=-1$ and $x=2$. Check your simplified difference quotient for $z$ by using it to find the slope of the same secant line.

##### 3

Replace $x$ with $4$ in your difference quotient formula and simplify the result. Then copy Table 1.4.2 onto your paper and fill in the missing values.

##### 4

As the value of $h$ gets closer to $0$, the values in the $y$ column of Table 1.4.2 appear to be converging on a single number; what is this number?

##### 5

The value found in Exercise 1.4.1.4 is called the slope of the tangent line to $z$ at $4$. Draw onto Figure 1.4.1 the line that passes through the point $\point{4}{2}$ with this slope. The line you just drew is called the tangent line to $z$ at $4$.

Find the difference quotient for each function showing all relevant steps in an organized manner.

##### 6

$\fe{f}{x}=3-7x$

##### 7

$\fe{g}{x}=\dfrac{7}{x+4}$

##### 8

$\fe{z}{x}=\pi$

##### 9

$\fe{s}{t}=t^3-t-9$

##### 10

$\fe{k}{t}=\dfrac{(t-8)^2}{t}$

Suppose that an object is tossed directly upward in the air in such a way that the elevation of the object (measured in ft) is given by the function $\fe{s}{t}=150+60t-16t^2$ where $t$ is the amount of time that has passed since the object was tossed (measure in seconds).

##### 11

Find the difference quotient for $s$.

##### 12

Use the difference quotient to determine the average velocity of the object over the interval $\cinterval{4}{4.2}$ and then verify the value by calculating $\frac{\fe{s}{4.2}-\fe{s}{4}}{4.2-4}$.

Several applied functions are given below. In each case, find the indicated quantity (including unit) and interpret the value in the context of the application.

##### 13

The velocity, $v$, of a roller coaster (in fts) is given by \begin{equation*}\fe{v}{t}=-100\fe{\sin}{\frac{\pi t}{15}}\end{equation*} where $t$ is the amount of time (s) that has passed since the coaster topped the first hill. Find and interpret $\frac{\fe{v}{7.5}-\fe{v}{0}}{7.5-0}$.

##### 14

The elevation of a ping pong ball relative to the table top (in m) is given by the function $\fe{h}{t}=1.1\abs{\fe{\cos}{\frac{2\pi t}{3}}}$ where $t$ is the amount of time (s) that has passed since the ball went into play. Find and interpret $\frac{\fe{h}{3}-\fe{h}{1.5}}{3-1.5}$.

##### 15

The period of a pendulum (s) is given by $\fe{P}{x}=\frac{6}{x+1}$ where $x$ is the number of swings the pendulum has made. Find and interpret $\frac{\fe{P}{29}-\fe{P}{1}}{29-1}$.

##### 16

The acceleration of a rocket (mphs) is given by $\fe{a}{t}=0.02+0.13t$ where $t$ is the amount of time (s) that has passed since lift-off. Find and interpret $\frac{\fe{a}{120}-\fe{a}{60}}{120-60}$.