While it's easy to see that the formula \(\frac{\fe{f}{x_1}-\fe{f}{x_0}}{x_1-x_0}\) gives the slope of the line connecting two points on the function \(f\text{,}\) the resultant expression can at times be awkward to work with. We actually already saw that in Example 1.2.2.
The algebra associated with secant lines (and average velocities) can sometimes be simplified if we designate the variable \(h\) to be the run between the two points (or the length of the time interval). With this designation we have \(x_1-x_0=h\) which gives us \(x_1=x_0+h\text{.}\) Making these substitutions we get (1.3.1). The expression on the right side of (1.3.1) is called the difference quotient for \(f\text{.}\)
\begin{equation}
\frac{\fe{f}{x_1}-\fe{f}{x_0}}{x_1-x_0}=\frac{\fe{f}{x_0+h}-\fe{f}{x_0}}{h}\tag{1.3.1}
\end{equation}
Let's revisit the function \(\fe{f}{x}=3+2x-x^2\) from Example 1.2.2. The difference quotient for this function is derived in Example 1.3.1.
Example1.3.1Calculating a Difference Quotient
\begin{align*}
\frac{\fe{f}{x_0+h}-\fe{f}{x_0}}{h}&=\frac{\left[3+2\left(x_0+h\right)-\left(x_0+h\right)^2\right]-\left[3+2x_0-x_0^2\right]}{h}\\
&=\frac{3+2x_0+2h-x_0^2-2x_0h-h^2-3-2x_0+x_0^2}{h}\\
&=\frac{2h-2x_0h-h^2}{h}\\
&=\frac{h\left(2-2x_0-h\right)}{h}\\
&=2-2x_0-h\text{ for }h\neq 0
\end{align*}
Please notice that all of the terms without a factor of \(h\) subtracted to zero. Please notice, too, that we avoided all of the tricky factoring that appeared in Example 1.2.2!
For simplicity's sake, we generally drop the variable subscript when applying the difference quotient. So for future reference we will define the difference quotient as follows:
Definition1.3.2The Difference Quotient
The difference quotient for the function \(y=\fe{f}{x}\) is the expression \(\frac{\fe{f}{x+h}-\fe{f}{x}}{h}\text{.}\)
Completely simplify the difference quotient for each of the following functions. Please note that the template for the difference quotient needs to be adapted to the function name and independent variable in each given equation. For example, the difference quotient for the function in Exercise 1.3.1.1 is \(\frac{\fe{v}{t+h}-\fe{v}{t}}{h}\text{.}\)
Please make sure that you lay out your work in a manner consistent with the way the work is shown in Example 1.3.1 (excluding the subscripts, of course).
1
\(\fe{v}{t}=2.5t^2-7.5t\)
2
\(\fe{g}{x}=3-7x\)
3
\(\fe{w}{x}=\dfrac{3}{x+2}\)
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 \(f\) defined by \(\fe{f}{t}=40+40t-16t^2\) where \(t\) is the amount of time that has passed since the object was tossed (measured in s).
4
Simplify the difference quotient for \(f\text{.}\)
5
Ignoring the unit, use the difference quotient to determine the average velocity over the interval \(\cinterval{1.6}{2.8}\text{.}\)
HintUse \(t=1.6\) and \(h=1.2\text{.}\) (Make sure that you understand why!)
6
Calculate \(\fe{f}{1.6}\) and \(\fe{f}{2.8}\text{.}\) What unit is associated with the values of these expressions? What is the contextual significance of these values?
7
Use the expression \(\frac{\fe{f}{2.8}-\fe{f}{1.6}}{2.8-1.6}\) and the answers to Exercise 1.3.1.6 to verify the value you found in Exercise 1.3.1.5. What unit is associated with the values of this expression? What is the contextual significance of this value?
8
Ignoring the unit, use the difference quotient to determine the average velocity over the interval \(\cinterval{0.4}{2.4}\text{.}\)
Moose and squirrel were having casual conversation when suddenly, without any apparent provocation, Boris Badenov launched anti-moose missile in their direction. Fortunately, squirrel had ability to fly as well as great knowledge of missile technology, and he was able to disarm missile well before it hit ground.
The elevation (ft) of the tip of the missile \(t\) seconds after it was launched is given by the function \(\fe{h}{t}=-16t^2+294.4t+15\text{.}\)
9
Calculate the value of \(\fe{h}{12}\text{.}\) What unit is associated with this value? What does the value tell you about the flight of the missile?
10
Calculate the value of \(\frac{\fe{h}{10}-\fe{h}{0}}{10}\text{.}\) What unit is associated with this value? What does this value tell you about the flight of the missile?
11
The velocity (ft⁄s) function for the missile is \(\fe{v}{t}=-32t+294.4\text{.}\) Calculate the value of \(\frac{\fe{v}{10}-\fe{v}{0}}{10}\text{.}\) What unit is associated with this value? What does this value tell you about the flight of the missile?
Timmy lived a long life in the 19th century. When Timmy was seven he found a rock that weighed exactly half a stone. (Timmy lived in jolly old England, don't you know.) That rock sat on Timmy's window sill for the next 80 years and wouldn't you know the weight of that rock did not change even one smidge the entire time. In fact, the weight function for this rock was \(\fe{w}{t}=0.5\) where \(\fe{w}{t}\) was the weight of the rock (stones) and \(t\) was the number of years that had passed since that day Timmy brought the rock home.
12
What was the average rate of change in the weight of the rock over the 80 years it sat on Timmy's window sill?
13
Ignoring the unit, simplify the expression \(\frac{\fe{w}{t_1}-\fe{w}{t_0}}{t_1-t_0}\text{.}\) Does the result make sense in the context of this problem?
14
Showing each step in the process and ignoring the unit, simplify the difference quotient for \(w\text{.}\) Does the result make sense in the context of this problem?
Truth be told, there was one day in 1842 when Timmy's mischievous son Nigel took that rock outside and chucked it into the air. The velocity of the rock (ft⁄s) was given by \(\fe{v}{t}=60-32t\) where \(t\) was the number of seconds that had passed since Nigel chucked the rock.
15
What, including unit, are the values of \(\fe{v}{0}\text{,}\) \(\fe{v}{1}\text{,}\) and \(\fe{v}{2}\) and what do these values tell you in the context of this problem? Don't just write that the values tell you the velocity at certain times; explain what the velocity values tell you about the motion of the rock.
16
Ignoring the unit, simplify the difference quotient for \(v\text{.}\)
17
What is the unit for the difference quotient for \(v\text{?}\) What does the value of the difference quotient (including unit) tell you in the context of this problem?
Suppose that a vat was undergoing a controlled drain and that the amount of fluid left in the vat (gal) was given by the formula \(\fe{V}{t}=100-2t^{3/2}\) where \(t\) is the number of minutes that had passed since the draining process began.
18
What, including unit, is the value of \(\fe{V}{4}\) and what does that value tell you in the context of this problem?
19
Ignoring the unit, write down the formula you get for the difference quotient of \(V\) when \(t=4\text{.}\)
20
Copy Table 1.3.3 onto your paper and fill in the missing values.
Look for a pattern in the output and write down enough digits for each entry so that the pattern is clearly illustrated; the first two entries in the output column have been given to help you understand what is meant by this direction.
| \(h\) |
\(y\) |
| \(-0.1\) |
\(-5.962\ldots\) |
| \(-0.01\) |
\(-5.9962\ldots\) |
| \(-0.001\) |
|
| \(\phantom{-}0.001\) |
|
| \(\phantom{-}0.01\) |
|
| \(\phantom{-}0.1\) |
|
Table1.3.3\(y=\frac{\fe{V}{4+h}-\fe{V}{4}}{h}\)
21
What is the unit for the \(y\) values in Table 1.3.3? What do these values (with their unit) tell you in the context of this problem?
22
As the value of \(h\) gets closer to \(0\text{,}\) the values in the \(y\) column of Table 1.3.3 appear to be converging to a single number; what is this number and what do you think that value (with its unit) tells you in the context of this problem?
