The Fundamental Theorem Of Calculus; Drawing Antiderivative Graphs - Texas Instruments TI-84 Plus Manual

•
The calculator's
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) = 3t
The Fundamental Theorem of Calculus tells us that F ' (x) =
is f evaluated at x.
and F ' in
Input f in
Y1
that you use
X
Access with
fnInt
Turn off any stat plots that are on.)
[nDeriv(]. (
Have
TBLSET
Input several different values for
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
Without changing the window (that is, draw the
6 [ZStandard].
graphs by pressing
. Then turn off
Y1
of takes a while to draw.) Turn both
Y2
the graph of both functions. Only one graph is seen in each case.
EXPLORE: Enter several other functions in
left endpoint 1 in the
results with derivative and integral formulas.
DRAWING ANTIDERIVATIVE GRAPHS Recall when using
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
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
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.
74
function yields the same result (to 3 decimal places) as that found in
fnInt
2
+ 2t – 5 and the accumulation function F(x) =
(remember that the calculator requires
Y2
as the input variable in the
and
MATH 9 [fnInt(]
set to and press
ASK 2ND
.
X
), turn off
GRAPH
and draw the graph of
Y1
expression. Perform the same explorations as above. Confirm your
fnInt
, the calculator will not graph a general antiderivative;
nDeriv
2
= 3x – 1 and its general antiderivative F
f(x)
F
z
H
d
dx
list).
Y=
with
nDeriv MATH 8
GRAPH (TABLE).
ZOOM
and draw the graph of
Y2
. (Note: The graph
Y2
and on and draw
Y1 Y2
and do not change
Y1
Copyright © Houghton Mifflin Company. All rights reserved.
Chapter 5
z
x
( )
f t dt
1
I
x
K
= f(x); that is, F ' (x)
( )
f t dt
1
except possibly for the
Y2
that a and
fnInt(f(x), x, a, b)
3
= x – x + C. Because
(x)
.