Transcript Limits and Derivatives

Chapter 4 – Applications of
Differentiation
4.3 How Derivatives Affect the Shape of a Graph
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4.3 How Derivatives Affect the Shape of a Graph
Applications of Calculus
Because the first derivative represents the slope of a curve y=f(x)
at the point (x, f(x)), it tells us the direction in which the curve
proceeds at each point. So, it is reasonable to expect that
information about the first derivative will provide us with
information about the function.
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4.3 How Derivatives Affect the Shape of a
Graph

To see how the derivative of f can tell us where a function is
increasing or decreasing, look at the graph below.

Between A and B and between C and D, the tangent lines have
positive slope and so f (x) > 0.
Between B and C the tangent lines have negative slope and so
f (x) < 0. Thus it appears that f increases when f (x) is positive and
decreases when f (x) is negative.

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4.3 How Derivatives Affect the Shape of a
Graph
The Increasing/Decreasing Test
(a)
(b)
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If f’(x)>0 on an interval, then f is increasing
on that interval.
If f’(x)<0 on an interval, then f is decreasing
on that interval.
4.3 How Derivatives Affect the Shape of a
Graph
Example 1
Find the intervals on which the function is
increasing or decreasing.
f ( x)  x  2 x  3
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4.3 How Derivatives Affect the Shape of a
Graph
The First Derivative Test
Suppose that c is a critical number of a
continuous function f.
(a) If f’ changes from positive to negative at c,
then f has a local maximum at c.
(b) If f’ changes from negative to positive at c,
then f has a local minimum at c.
(c) If f’ does not change sign at c, then f has no
local minimum or maximum.
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4.3 How Derivatives Affect the Shape of a
Graph
The First Derivative Test

It is easy to remember the First Derivative Test
by visualizing diagrams such as those shown
Local maximum
Local minimum
below.
No maximum or
minimum
No maximum or
minimum
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4.3 How Derivatives Affect the Shape of a
Graph
Example 2
Find the local minimum and maximum values of
the function.
f ( x)  x  2 x  3
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4.3 How Derivatives Affect the Shape of a
Graph
Definition
If the graph of f lies above all of its tangents on
an interval, then it is called concave up on that
interval.
If the graph of f lies below all of its tangents on
an interval, then it is called concave down on
that interval.
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4.3 How Derivatives Affect the Shape of a Graph
Concavity Test
(a)
(b)
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If f”(x)>0 for all x in an interval, then the
graph of f is concave up on that interval.
If f”(x)<0 for all x in an interval, then the
graph of f is concave down on that interval.
4.3 How Derivatives Affect the Shape of a
Graph
Definition
A point P on a curve y=f(x) is called an
inflection point if f is continuous there and the
curve changes from concave up to concave
down or concave down to concave up at P.
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4.3 How Derivatives Affect the Shape of a
Graph
Example 3
Find the intervals of concavity and the inflection
points.
f ( x)  x  2 x  3
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4.3 How Derivatives Affect the Shape of a
Graph
Example 4
f ( x )  sin x  cos x
(a)
(b)
(c)
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0  x  2
Find the intervals on which the function is
increasing or decreasing.
Find the local minimum and maximum values
of the function.
Find the intervals of concavity and the
inflection points.
4.3 How Derivatives Affect the Shape of a
Graph
The Second Derivative Test
Suppose f” is continuous near c.
(a) If f’(c)=0 and f”(c)>0, then f has a local
minimum at c.
(b) If f’(c)=0 and f”(c)<0, then f has a local
maximum at c.
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4.3 How Derivatives Affect the Shape of a
Graph
Example 5
Sketch the graph of a function that satisfies all of
the given conditions
f '(0 )  f '(2 )  f '(4 )  0
f '( x )  0 if x  0 o r 2  x  4
f '( x )  0 if 0  x  2 o r x  4
f "( x )  0 if 1  x  3
f "( x )  0 if x  1 o r x  3
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4.3 How Derivatives Affect the Shape of a
Graph
Example 6
h ( x )   x  1  5 x  2
5
(a)
(b)
(c)
(d)
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Find the intervals of increase or decrease.
Find the local minimum and maximum
values.
Find the intervals of concavity and the
inflection points.
Use the information from the above parts to
sketch the graph.
4.3 How Derivatives Affect the Shape of a
Graph
Example 7
f ( x) 
(a)
(b)
(c)
(d)
(e)
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x
2
 x  2
2
Find the vertical and horizontal asymptotes.
Find the intervals of increase or decrease.
Find the local minimum and maximum values.
Find the intervals of concavity and the inflection
points.
Use the information from the above parts to
sketch the graph.
4.3 How Derivatives Affect the Shape of a
Graph