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

# Activity1.4Supplement¶ permalink

# Subsection1.4.1Exercises

##### 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 ^{ft}⁄_{s}) 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 (^{mph}⁄_{s}) 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 \(\frac{\fe{a}{120}-\fe{a}{60}}{120-60}\).