Prep Work for The Fundamental Theorem of Calculus
Just like we dened a derivative function, we can dene an integral function, provided we spec-
ˆ
Given a function
f (x),
x
f (t) dt. This is illustrated in the
0
following animation found at the University of Wisconsin-Eau Claire Math Department website:
ify a starting point.
dene
g(x) =
http://tinyurl.com/intfcn-gif.
(NOTE that we are changing the name of the
x-axis
to the
t-axis
and
x
refers to a specic number on
that axis.)
1. If
f (x) > 0,
2. Is
g
what does the function
still dened even if
The function
g
Y2 ,
is not always
f
in
> 0?
Y1 .
MATH, 9
X,0,X)
press
then press
3. Press
give us?
can be graphed on the TI calculator using the following steps:
1. Enter the function
2. For
f
g
WINDOW
to select
f nInt(.
Then press
to select an X-range, then
VARS, Y-VARS, Function..., Y1 ,
ZOOM/ZoomFit
to select a Y-range (NOTE: It
will take some time to graph, so be patient!)
Use the technique above to graph the following pairs of functions. What do you notice?
ˆ
1.
x
2x dx; F (x) = x2
g(x) =
(HINT: put
Y1 = 2X, Y2 = f nInt(Y1 , X, 0, X),
and
Y3 = X 2
0
ˆ
2.
x
3x2 dx; F (x) = x3
g(x) =
0
ˆ
3.
x
g(x) =
0
√
x dx; F (x) =
2 3/2
x
3
The starting point of your integral function does not have to be 0. Graph the following functions;
how are they alike, and how are they dierent?
ˆ
1.
x
3x2 dx; F (x) = x3
g(x) =
2
1