Additional Applications of the Derivative
Chaper Three
§3.1 Increasing and Decreasing Function
Increasing and Decreasing Function Let f(x) be a functiondefined on the interval a<x<b, and let x1 and x2 be two numbers in the interval, Then f(x) is increasing on the interval if f(x2)>f(x1) whenever x2>x1
f(x) is decreasing on the interval if f(x2)<f(x1) whenever x2 >x1
Monotonic increasing 单调递增Monotonic decreasing 单调递减
§3.1 Increasing and Decreasing Function
Tangent line with negative slope f(x) will be decreasing
( ) 0f x
Tangent line with positive slope f(x) will be increasing
( ) 0f x
§3.1 Increasing and Decreasing Function
Intermediate value property A continuous function cannot change sign without first becoming 0.
If for every x on some interval I,then f(x) is increasing on the intervalIf for every x on some interval I,then f(x) is decreasing on the intervalIf for every x on some interval I, then f(x) is constant on the interval
( ) 0f x
( ) 0f x
How to determine all intervals of increase and decrease for a function ? How to find all intervals on which the sign of the derivative does not change.
0)( xf
§3.1 Increasing and Decreasing Function
Procedure for using the derivative to determine intervals of increase and decrease for a function of f.
Step 2. Choose a test number c from each interval a<x<b determined in the step 1 and evaluate . Then If the function f(x) is increasing on a<x<b. If the function f(x) is decreasing on a<x<b( ) 0f c
( ) 0f c )(cf
Step 1. Find all values of x for which or isnot continuous, and mark these numbers on a number line.This divides the line into a number of open intervals.
)(xf ( ) 0f x
Example. Find the intervals of increase and decrease for the function
Solution:
The number -2 and 1 divide x axis into three open intervals. x<-2, -2<x<1 and x>1
(0) 0f
(2) 0f Rising f is increasing 2x>1
Falling f is deceasing 0-2<x<1
Rising f is increasing -3x<-2
Direction
of graph
Conclusion Test
number
Interval
)(cf
3 2( ) 2 3 12 7f x x x x
2( ) 6 6 12 6( 2)( 1)f x x x x x
Which is continuous everywhere, with where x=1 and x=-2( ) 0f x
0)3( f
§3.1 Relative Extrema Relative (Local) Extrema The Graph of the function f(x) is said to be have a relative maximum at x=c if f(c) f(x) for all x in interval a<x<b containing c. Similarly the graph has a relative Minimum at x=c if f(c) f(x) on such an interval. Collectively, the relative maxima and minima of f are called its relative extrema
Peaks: C,E,(Relative maxima)Valleys: B, D, G(Relative minima)
§3.1 Critical Points
Relative extrema can only occur at critical points!
Critical Numbers and Critical Points A number c in the domain of f(x) is called a critical number if either or does not exist. The corresponding point (c,f(c)) on the graph of f(x) is called a critical point for f(x).
)(cf ( ) 0f c
§3.1 Critical PointsNot all critical points correspond to relative extrema!
Figure. Three critical points where f’(x) = 0: (a) relative maximum, (b) relative minimum (c) not a relative extremum.
§3.1 Critical PointsNot all critical points correspond to relative extrema!
Figure Three critical points where f’(x) is undefined: (a) relative maximum, (b) relative minimum (c) not a relative extremum.
§3.1 The First Derivative Test
( ) 0f x
The First Derivative Test for Relative Extrema Let c be a critical number for f(x) [that is, f(c) is defined and either or does not exist]. Then the critical point (c,f(c)) is
)(cf
A relative maximum if to the left of c and to the right of c( ) 0f x
( ) 0f x
( ) 0f x A relative minimum if to the left of c and to the right of c( ) 0f x
Not a relative extremum if has the same sign on both sides of c
)(xf
c 0f 0f
c 0f 0f c 0f 0f
c 0f 0f
Example
Solution
Find all critical numbers of the function
and classify each critical point as a relative maximum, a relative minimum, or neither
4 2( ) 2 4 3f x x x
3( ) 8 8 8 ( 1)( 1)f x x x x x x
The derivative exists for all x, the only critical numbers are Where that is, x=0,x=-1,x=1. These numbers divide that x axis into four intervals, x<-1, -1<x<0, 0<x<1, x>1
( ) 0f x
1 1 15( 5) 960 0 ( ) 3 0 ( ) 0 (2) 48 0
2 4 8f f f f
Choose a test number in each of these intervals
-1 min-------- ++++++ -------- +++++
+0 max 1 min
Thus the graph of f falls for x<-1 and for 0<x<1, and rises for -1<x<0 and for x>1 x=0 relative maximumx=1 and x=-1 relative minimum
§3.1 Sketch the graphA Procedure for Sketching the Graph of a Continuous Function f(x) Using the Derivative
Step 1. Determine the domain of f(x).( )f x Step 2. Find and each critical number, analyze the sign of
derivative to determine intervals of increase and decrease for f(x).
( ) 0f x ( ) 0f x
Step 3. Plot the critical point P(c,f(c)) on a coordinate plane, with a “cap” at P if it is a relative maximum or a “cup” if P is a relative minimum. Plot intercepts and other key points that can be easily found.
( ) 0f x
Step 4 Sketch the graph of f as a smooth curve joining the criticalpoints in such way that it rise where , falls where and has a horizontal tangent where
Example
Solution
Sketch the graph of the function 4 3 2( ) 8 18 8f x x x x
3 2 2( ) 4 24 36 4 ( 3)f x x x x x x
( 5) 80 0 ( 1) 16 0 (1) 64 0f f f
The derivative exists for all x, the only critical numbers are Where that is, x=0, x=-3. These numbers divide that x axis into three intervals, x<-3, -3<x<0, x>0. Choose test number in each interval (say, -5, -1 and 1 respectively)
( ) 0f x
-3 neither
-------- ++++++--------
0 min
Thus the graph of f has a horizontal tangents where x is -3 and 0, and it is falling in the interval x<-3 and -3<x<0 and is rising for x>0
f(-3)=19 f(0)=-8
Plot a “cup” at the critical point (0,-8)
Plot a “twist” at (-3,19) to indicate a galling graph with a horizontal tangent at this point . Complete the sketch by passing a smooth curve through the Critical point in the directions indicated by arrow
Example
Solution
The revenue derived from the sale of a new kind of motorized skateboard t weeks after its introduction is given by
2
2
63( )
63
t tR t
t
million dollars. When does maximum revenue occur? What isthe maximum revenue
Critical number t=7 divides the domain into two intervals x<=t<7 and 7<t<=63 0 63t
2
2
63(7) (7)(7) 3.5
(7) 63R
7 Max
++++++ -------- t0 63
§3.2 ConcavityIncrease and decrease of the slopes are our concern!
Figure The output Q(t) of a factory worker t hours after coming to work.
§3.2 Concavity Concavity If the function f(x) is differentiable on the interval a<x<b then the graph of f is Concave upward on a<x<b if is increasing on the interval Concave downward on a<x<b if is decreasing on the interval
)(xf )(xf
§3.2 ConcavityA graph is concave upward on the interval if it lies above all its tangent lines on the interval and concave downward on an Interval where it lies below all its tangent lines.
Note Don’t confuse the concavity of a graph with its “direction”(rising or falling). A function may be increasing or decreasing onan interval regardless of whether its graph is concave upward or concave downward on the interval.
§3.2 Concavity and the second Derivative
How to characterize the concavity of the graph of function f(x)in terms of the second derivative?
A function f(x) is increasing where its derivative is positive. Thus, the derivative function must be increasing where its derivative is positive. Similarly, on interval a<x<b, where , the derivative will be decreasing.
)(xf 0)( xf
)(xf )(xf
§3.2 Concavity and the second
Derivative
Second Derivative Procedure for Determining Intervals of Concavity for a Function f. Step 1. Find all values of x for which or isnot continuous, and mark these numbers on a number line.This divides the line into a number of open intervals.
0)( xf )(xf
Step 2. Choose a test number c from each interval a<x<b determined in the step 1 and evaluate . Then If , the graph of f(x) is concave upward on a<x<b. If the graph of f(x) is concave downward on a<x<b
)(cf
0)( cf
0)( cf
to be continued
-1
--------++++++ -------- ++++++
0 1
Type of concavity
Sign of
§3.2 Inflection points
-------- -------- ++++++
0No inflection
1inflection
Type of concavity
Sign of
to be continued
--------
++++++
0inflection
Type of concavity
Sign of
Note: A function can have an inflection point only where it is continuous.!!
§3.2 Behavior of Graph f(x) at an inflection point P(c,f(c))
-1.5min
-------- ++++++ 1
Neither
++++++
to be continued
-------- ++++++
-2/3inflection
1inflection
Type of concavity
Sign of ++++++
to be continued
b. Find all critical numbers of the function
c. Classify each critical point as a relative maximum, a relative minimum, or neither
e. Find all inflection points of function
a.
d.
Review
to be continued
-------- ++++++
--------++++++
to be continued
++++++-------- -------- ++++++
to be continued
§3.2 The Second Derivative Test
to be continued
3 Max
++++++ -------- t0 4
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