Math 1a: The derivative function (Solutions)
September 30, 2009
1.
Let
f
(
x
) =
x
(
x

1)(
x

2) =
x
3

3
x
2
+ 2
x.
Determine
f
0
(
x
) using the limit definition. Sketch
f
(
x
) and
f
0
(
x
), and compare the graphs.
Solution.
Recall that
f
0
(
x
) is defined to be the slope of the tangent to the graph of
f
at the point (
x, f
(
x
)).
We start by choosing a nearby point on the graph, say (
x
+
h, f
(
x
+
h
)), compute the slope of the secant joining
(
x, f
(
x
)) and (
x
+
h, f
(
x
+
h
)) and take the limit as the nearby point approaches (
x, f
(
x
)), or equivalently
as
h
approaches 0. Let’s work it out!
f
0
(
x
) = lim
h
→
0
Slope of the line joining (
x, f
(
x
)) and (
x
+
h, f
(
x
+
h
))
.
= lim
h
→
0
f
(
x
+
h
)

f
(
x
)
h
= lim
h
→
0
(
x
+
h
)
3

3(
x
+
h
)
2
+ 2(
x
+
h
)

(
x
3

3
x
2
+ 2
x
)
h
= lim
h
→
0
x
3
+ 3
x
2
h
+ 3
xh
2
+
h
3

3
x
2

6
xh

h
2
+ 2
x
+ 2
h

x
3
+ 3
x
2

2
x
h
= lim
h
→
0
3
x
2
h
+ 3
xh
2
+
h
3

6
xh

h
2
+ 2
h
h
= lim
h
→
0
3
x
2

6
x
+ 2 + 3
xh
+
h
= 3
x
2

6
x
+ 2
.
I will leave the sketch of
f
0
(
x
) to you.
2.
Let
h
(
x
) =

x

1

+

x
+ 1

. On what intervals is
h
(
x
) continuous? What about differentiable? Sketch
h
(
x
) and its derivative.
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 Fall '09
 BenedictGross
 Derivative, lim, lim Slope

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