WORKSHEET 27  Fall 1995
1. Below, the graph of
f
(
x
) =
x
2
is given. The line below the graph is the tangent line at
x
= 1. The lines
above the graph connect the points (0
,
0), (1
,
1), and (2
,
4).
1
2
3
4
1
2
a) Find the area bounded by the tangent line, the
x
axis, and the line
x
= 2.
b) Find the area bounded by the two segments above the graph, the
x
axis, and the line
x
= 0. the
graph.
c) Use your answers to a) and b) to venture a guess as to the area between the graph of
f
and the
x
axis lying over the interval [0
,
2].
d) Find a function
g
such that
g
=
f
. What is
g
(2)
−
g
(0)? How close is this to your guess in part
c)?
Creepy, eh?
There are several functions having
x
2
as a derivative. What if you had chosen
another?
2. Let
f
be a continuous function with
f
(
x
)
>
0 for all
x
. Let
A
be the function such that
A
(
x
) equals the
area between the graph of
f
and the
x
axis on the interval [0
, x
].
a) Draw a picture of this for some value of
x
. Label
A
(
x
). For a small value of
h
, also indicate
A
(
x
+
h
)
in your picture. What is
A
(0)?
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 Spring '08
 Grether
 Calculus, Topology, Derivative, Continuous function

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