Applications of Derivatives 4.2 Relative and Absolute Extreme Points
USING THE NUMINTGL PROGRAM TO SIMPLIFY AREA APPROXIMATIONS
5.2 Accumulation Functions
In this section of the Guide, we see how to use the calculator’s definite integral function and
use it along with the calculator’s numerical derivative to illustrate the Fundamental Theorem in action.
DEFINITE INTEGRAL NOTATION AND CALCULATOR NOTATION Recall that
nDeriv is the calculator’s numerical derivative and provides, in most cases, a very good approximation for the instantaneous rate of change of a function when that rate of change exists. As is the case with the numerical derivative, your calculator does not give a formula for accumulation functions. However, it does give an excellent numerical estimate for the definite integral of a function between two specific input values when that integral exists.
The TI’s numerical integral is called fnInt, and the correspondence between the mathemati- cal notation f x dx
a b
( )
z
and the calculator’s notation fnInt(f(x), x, a, b) is illustrated in Figure 6.Figure 6
We illustrate the use of the calculator’s definite integral function by finding the area from
x = 0 to x = 1 between the function f(x) = 4 2 x
− and the horizontal axis. You can enter f in Y1 and refer to it
in the fnInt expression as Y1. Access fnInt with MATH 9 [fnInt(]. (Any function location can be used. Recall that in the Y= list, you must use X as the input variable name.)
Or, you can enter the function directly on the home screen as shown to the right. When you type a function formula on the home screen, you can use any letter as the input variable symbol.
•
The calculator’s fnInt function yields the same result (to 3 decimal places) as that found in the limit of sums investigation on page 66 of this Guide.5.3 The Fundamental Theorem of Calculus
Intuitively, this theorem tells us that the derivative of an antiderivative of a function is the function itself. Let us view this idea both numerically and graphically. The correct syntax for the calculator’s numerical integrator is
fnInt(function, name of input variable, left endpoint for input, right endpoint for input) Consider the function f(t) = 3t2 + 2t – 5 and the accumulation function F(x) = x f t dt( )
1
z
.The Fundamental Theorem of Calculus tells us that F′(x) = d
dx f t dt x ( ) 1
z
F
H
IK
= f(x); that is, F′(x) is f evaluated at x.Input f in Y1 and F′in Y2 (remember that the calculator requires that you use X as the input variable in the Y= list).
Access fnInt with MATH 9 [fnInt(] and nDeriv with MATH 8 [nDeriv(]. (Turn off any stat plots that are on.)
Have TBLSET set to ASK and press 2ND GRAPH (TABLE). Input several different values for X.
Other than occasional roundoff error because the calculator is approximating these values, the results are identical.
Find a suitable viewing window such as the one set with ZOOM 6 [ZStandard]. Without changing the window (that is, draw the graphs by pressing GRAPH ), turn off Y2 and draw the graph of Y1. Then turn off Y1 and draw the graph of Y2. (Note: The graph
of Y2 takes a while to draw.) Turn both Y1 and Y2 on and draw the graph of both functions. Only one graph is seen in each case.
EXPLORE: Enter several other functions in Y1 and do not change Y2 except possibly for the
left endpoint 1 in the fnInt expression. Perform the same explorations as above. Confirm your results with derivative and integral formulas.
DRAWING ANTIDERIVATIVE GRAPHS Recall when using fnInt(f(x), x, a, b) that a and
b are, respectively, the lower and upper endpoints of the input interval. Also remember that
you do not have to use x as the input variable unless you are graphing the integral or
evaluating it using the calculator’s table.
Unlike when graphing using nDeriv, the calculator will not graph a general antiderivative; it only draws the graph of a specific accumulation function. Thus, we can use x for the input at
the upper endpoint when we want to draw an antiderivative graph, but not for the inputs at both the upper and lower endpoints.
All of the antiderivatives of a specific function differ only by a constant. We explore this idea using the function f(x)= 3x2 – 1 and its general antiderivative F(x) = x3 – x + C. Because
we are working with a general antiderivative in this illustration, we do not have a starting point for the accumulation. We therefore choose some value, say 0, to use as the starting point for the accumulation function to illustrate drawing antiderivative graphs. If you choose a different lower limit, your results will differ from those shown below by a constant.
Enter f in Y1, fnInt(Y1, X, 0, X) in Y2, and F in Y3, Y4, Y5, and Y6, using a different number for C in each function location.
(You can use the values of C shown to the right or different
values.)
Find a suitable viewing window and graph the functions Y1 through Y6. The graph to the right was drawn with −3 ≤x≤ 3
and −20 ≤y≤ 20.
It appears that the only difference in the graphs of Y2 through Y6 is the y-axis intercept. But, isn’t C the y-axis intercept of each of
these antiderivative graphs?
Clear Y4, Y5, and Y6. Turn off Y1 and change the 1 in Y3 to 0. Press GRAPH and draw the graphs of Y2 and Y3. You should see only one graph.
Set the calculator TABLE to ASK and enter some values for x. It
appears that Y2 and Y3 are the same function.
CAUTION: The methods for checking derivative formulas that were discussed in Sections
4.3.2b and 4.3.2c are not valid for checking general antiderivative formulas. Why not? Because to graph an antiderivative using fnInt, you must arbitrarily choose values for the constant of integration and for the input of the lower endpoint. However, for most of the rate-of-change functions where f(0) = 0, the calculator’s numerical integrator values and your antiderivative
formula values should differ by the same constant at every input value where they are defined.