3.2 Solución del Problema
3.2.2 Fase de Implantación
3.3 Exponential and Logarithmic Rate-of-Change Formulas
The calculator only approximates numerical values of slopes – it does not give a slope in formula form. You also need to remember that the CALCULATOR calculates the slope (i.e.,
the derivative) at a specific input value by a different method than the one we use to calculate the slope.
DERIVATIVE NOTATION AND CALCULATOR NOTATION In addition to learning
when your calculator gives an acceptable answer for a derivative and when it does not, you also need to understand the differences and similarities in mathematical derivative notation and calculator notation. The notation that is used for the calculator’s numerical derivative is nDeriv(f(x), x, x). The correspondence between the mathematical derivative notation df x( )
dx and the calculator’s notation nDeriv(f(x), x, x) is illustrated in Figure 5.
Figure 5
We illustrate another use of the calculator’s derivative by constructing Table 3.7 in Section 3.3 of Calculus Concepts. The table lists x, y = f(x) = ex, and y′ =df
dx for 4 different inputs. We next evaluate this function at these 4 and several other inputs.
You can evaluate the CALCULATOR numerical derivative on the home screen or in the table. We choose to use the table. Enter the function f in the Y= list, say in Y1. In Y2 enter the numerical derivative evaluated at a general input X.
Press 2nd WINDOW (TBLSET) and choose ASK in the Indpnt: location. Access the table with 2ND GRAPH (TABLE) and delete or type over any previous entries in the X column. Enter the values for X that are shown on the screen to the right.
It appears that the derivative values are the same as the function outputs. In fact, this is a true statement for all inputs of f – this function is its own derivative.
You can use the methods discussed on pages 35 and 35 of this Guide to find the values used in Table 3.9 in Section 3.3 of Calculus Concepts to numerically estimate
h h h → − 0 2 1 lim . Instead, we explore an alternate method of confirming that d
dx
x
(2 )
Press Y= and edit Y1 to be the function g(x) = 2x.
Access the statistical lists, clear any previous entries from L1, L2, L3, and L4. Enter the x-values shown above in L1. Highlight L2 and enter Y1(L1). Remember to type L1 using 2ND 1 (L1). Press ENTER to fill L2 with the function outputs. Then, highlight L3 and type Y2(L1).
Press ENTER to fill L3 with the derivative of Y1 evaluated at the inputs in L1. Note that these values are not the same as the function outputs.
To see what relation the slopes have to the function outputs, press ► and highlight L4. Type L3 ÷ L2 ENTER .
It appears that the slope values are a multiple of the function output. In fact, that multiple is ln 2 ≈ 0.693147. Thus we confirm this slope formula: If g(x) = 2x, then dg
dx= (ln 2) 2 x.
4.3.2a CALCULATING dydx AT SPECIFIC INPUT VALUES The previous two sections of this
Guide examined the calculator’s numerical derivative nDeriv(f(x), x, a) and illustrated that it gives a good approximation to the slope of the tangent line at points where the instantaneous rate of change exists. You can also evaluate the calculator’s numerical derivative from the graphics screen using the CALC menu. However, instead of being named nDeriv( in that menu, it is called dy/dx. We illustrate this use with the function in part a of Example 2 in Section 3.3.
Clear all previously-entered functions in the Y= list. Enter f(x)
= 12.36 + 6.2 ln x in Y1.
We want to draw a graph of f. Realize that x > 0 because of
the log term. Choose some value for Xmax, say 5. Then use ZOOM ▲ [ZoomFit] to set the height of the graph. With the graph on the screen, press
2ND TRACE (CALC) 6 [dy/dx]. Use ◄ or ► to move to some point on the graph. Press ENTER and the slope of the function is calculated at the input of this point.
To find the derivative evaluated at a specific value of X, you could just type in the desired input instead of pressing the arrow keys. Press CLEAR 2ND TRACE (CALC) 6 [dy/dx] 3 ENTER .
The slope dy/dx= 2.0666667 appears at the bottom of the screen. Return to the home screen and press X,T,θ,n . The calculator’s X memory location has been updated to 3. Now type the numerical derivative instruction (evaluated at 3) as shown to the right. This is the dy/dx value you saw on the graphics screen.
You can use the ideas presented above to check your algebraic formula for the derivative. We next investigate this procedure.
NUMERICALLY CHECKING SLOPE FORMULAS It is always a good idea to check
your answer. Although your calculator cannot give you an algebraic formula for the derivative function, you can use numerical techniques to check your algebraic derivative formula. The basic idea of the checking process is that if you evaluate your derivative and the calculator’s numerical derivative at several randomly chosen values of the input variable and the outputs are basically the same values, your derivative is probably correct.
These same procedures are applicable when you check your results (in the next several sections) after applying the Sum Rule, the Chain Rule, or the Product Rule. We use the fun- ction in part c of Example 2 in Section 3.3 of Calculus Concepts to illustrate.
Enter m(r) = 8 12
r − r in Y1 (using X as the input variable). Compute m′(r) using pencil and paper and the derivative rules.
Enter this function in Y2. (What you enter in Y2 may or may not be the same as what appears to the right.)
Enter the calculator’s numerical derivative of Y1 (evaluated at a general input X) in Y3. Because you are interested in seeing if the outputs of Y2 and Y3 are the same, turn off Y1.
Press 2ND WINDOW (TBLSET) and choose ASK in the Indpnt: location. Access the table with 2ND GRAPH (TABLE) and delete or type over any previous entries in the X column. Enter at least three different values for X.
The table gives strong evidence that that Y2 and Y3 are the same function.
GRAPHICALLY CHECKING SLOPE FORMULAS When it is used correctly, a graphi-
cal check of your algebraic formula works well because you can look at many more inputs when drawing a graph than when viewing specific inputs in a table. We illustrate this use with the function in part d of Example 2 in Section 3.3 of Calculus Concepts.
Enter j(y) = 17 1
(
+0.02512)
12y in Y1, using X as the input variable. Next, using pencil and paper and the derivative rules, computej′ (y). Enter this function in Y2.
Enter the calculator’s numerical derivative of Y1 (evaluated at a general input X) in Y3. Before proceeding, turn off the graphs of Y1 and Y3.
To graphically check your derivative formula answer, you now need to find a good graph of Y2. Because this function is not in a context with a given input interval, the time it takes to find a graph is shortened if you know the approximate shape of the graph. Note that the graph of the function in Y2 is an increasing exponential curve.
Start with ZOOM 4 [ZDecimal] or ZOOM 6 [ZStandard]. Neither of these shows a graph (because of the large coefficient in the function), but you can press TRACE to see some of the output values. Using those values, reset the window. The graph to the right was drawn in the window [−10, 10] by [0.331, 0.545], but any view that shows the graph will do.
Now, press Y= , turn off Y2 and turn on Y3. (Recall that Y3 holds the formula forthe derivative of Y1 as computed
numerically by the calculator.) Press GRAPH . Note that you are drawing the graph of Y3 in exactly the same window in which
you graphed Y2.
If you see the same graph, your algebraic formula (in Y2) is very likely correct.
3.4 The Chain Rule
You probably noticed that checking your answer for a slope formula graphically is more diffi- cult than checking your answer using the table if you have to spend a lot of time finding a window in which to view the graph. Practice with these methods will help you determine which is the best to use to check your answer. (These ideas also apply to Section 3.5.)
SUMMARY OF CHECKING METHODS Before you begin checking your answer, make
sure that you have correctly entered the function. It is very frustrating to miss the answer to a problem because you have made an error in entering a function in your calculator. We sum- marize the methods of checking your algebraic answer using the function in Example 3 of Section 3.4 of Calculus Concepts.
Enter the function P(t) = 84 4 1 33 6 0 484
. . .
+ e− t
in the Y1 location, your answer for P′ in Y2, and the CALCULATOR derivative in Y3. Turn off Y1.
Go to the table, which has been set to ASK mode. Enter at least 3 input values. (Because this problem is in a context, read the problem to see which inputs make sense.)
It seems that the answer in Y2 is probably correct.
If you prefer a graphical check, the problem states that the equa- tion is valid between 1980 and 2001 (and the input is the number of years after 1980). So, turn off Y3, set Xmin = 0, Xmax = 21, and draw a graph of Y2 using ZoomFit. Then, using the same window, draw a graph of Y3 with Y2 turned off. The graphs are the same, again suggesting a correct answer in Y2.