1
Section 4.3
Derivatives and the Shape of
Curves
• Goals
– Apply the Mean Value Theorem
to finding
where functions are increasing and
decreasing
– Discuss the
• first derivative test
and
• second derivative test
for local max/minima
Mean Value Theorem
• This theorem is the key to connecting
derivatives with the shape of curves:
Geometric (cont’d)
Geometric (cont’d)
• Thus Equation 1 of the Theorem says that
there is at least one
point
P
(
c
,
f
(
c
)) on the
graph where
– the slope of the tangent
line is the same as
– the slope of the secant
line
AB
.
• In other words, there is a point
P
where
the tangent line is parallel
to the secant
line
AB
.
Example
• If an object moves in a straight line with
position function
s
=
f
(
t
) , then the average
velocity between
t
=
a
and
t
=
b
is
and the velocity at
t
=
c
is
f
′
(
c
) .
• Thus the Mean Value Theorem says
that…
( )
()
−
−
fb
fa
ba
Example (cont’d)
• …at some time
t
=
c
between
a
and
b
the
– instantaneous
velocity
f
′
(
c
) is equal to the
– average
velocity.
• For instance, if a car traveled 180 km in
2 h, then the speedometer must have read
90 km/h at least once
.
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Increasing/Decreasing Functions
• Earlier we observed from graphs that a
function with a positive derivative is
increasing.
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 Spring '08
 ZhoraManseur
 Critical Point, Derivative, Mean Value Theorem, 90 km/h, 180 km

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