Limits: Limits and the Infinitely Large
Chapter 2 Describing Change: Rates
2.3 Derivative Notation and Numerical Estimates
CALCULATING PERCENTAGE RATE OF CHANGE Percentage rate of change =
rate of change at a point
100%
value of the function at that point⋅ . We illustrate calculating the percentage rate of change with the example found on page 134 in Section 2.3 of Calculus Concepts. Suppose the
growth rate of a population is 50,000 people per year and the current population size is 200,000 people.
What is the percentage rate of change of the population? The answer is 25% per year.
Suppose instead that the current population size is 2 million. What is the percentage rate of change? The answer is 2.5% per year, which is a much smaller percentage rate of change.
Using your calculator to find slopes of tangent lines does not involve a new procedure. How- ever, the techniques that are discussed in this section allow you to repeatedly apply a method of finding slopes that gives quick and accurate results.
NUMERICALLY ESTIMATING A RATE OF CHANGE Finding the slopes of secant
lines joining the point at which the tangent line is drawn to increasingly close points on a function to the left and right of the point of tangency is easily done using your calculator. We illustrate this technique using the function in Example 5 of Section 2.3. Suppose we want to numerically estimate the slope of the tangent line at t = 5 to the graph of the function that gives
the value of an investment given by: y = 32 1.12
( )
t billion dollars aftert years.
Enter the equation in the Y1 location of the Y= list.
We now evaluate the slopes of secant lines that join close points to the left of t = 4 with t = 4.
On the home screen, type in the expression shown to the right to compute the slope of the secant line joining the close point where t = 3.99 and the point of tangency where t = 4.
Record on paper each slope, to at least 1 more decimal place than the desired accuracy, as it is computed. You are asked to find the nearest whole number that these slopes are approaching, so record at least one decimal place in your table of slopes. Press 2ND ENTER (ENTRY) to recall the last entry, and then use the arrow keys to move the cursor over a 9 in the “3.99”. Press 2ND DEL (INS) and press 9 to insert another 9 in both places that 3.99 appears. Press ENTER to find the slope of the secant line joining t = 3.999 and t = 4.
Continue in this manner, recording each result on paper, until you can determine to which value the slopes from the left seem to be getting closer and closer.
It appears that the slopes of the secant lines from the left are approaching 5.706 billion dollars per year.
We now evaluate the slopes joining the point of tangency and nearby close points to the right of t = 4.
Clear the screen, recall the last expression with 2ND ENTER (ENTRY), and edit it so that the nearby point is t = 4.001. Press
ENTER to calculate the secant line slope.
Continue in this manner as you did when calculating slopes to the left, but each time insert a 0 before the “1” in two places in the close point. Record each result on paper until you can determine the value the slopes from the right are approaching. It appears that the slopes of the secant lines from the right are approaching 5.706 billion dollars per year.
When the slopes from the left and the slopes from the right approach the same number, that number is the slope of the tangent line at the point of tangency. In this case, we estimate the slope of the tangent line to be 5.706 billion dollars per year.
NUMERICALLY ESTIMATING SLOPES USING THE TABLE
The process dis-
cussed beginning on page 49 of this Guide can be done in fewer steps and with fewer
keystrokes when you use the calculator TABLE. The point of tangency is t = 4, y = Y1(4), and let’scall the close point (t, Y1(t)). Then, slope = riserun = Y1(4) 4 - X−Y1(X). We illustrate numeri- cally estimating the slope using the TABLE with y = 32 1.12
( )
tHave y = 32 1.12
( )
t in the Y1 location of the Y= list. Enter the above slope formula in another location, say Y2. (Alsoremember to enclose both the numerator and denominator of the slope formula in parentheses.)
Turn off Y1 because we are only considering the output from Y2. Press 2ND WINDOW (TBLSET) and choose ASK in the Indpnt: location. (See page A-8 for more specific instructions.)
Access the table with 2ND GRAPH (TABLE) and delete or type over any previous entries in the X column. Enter values for X, the input of the close point, so that X gets closer and closer to 8 from the left.
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Notice that as you continue to enter numbers, the calculator displays rounded output values so that the numbers can fit on the screen in the space allotted for inputs and outputs of the table. You should position the cursor over each output value and record on paper asmany decimal places as are necessary to determine the limit to the desired degree of accuracy.