Additional Applications of the Derivative

Mr. ABHISHEK SINGH

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

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

Increasing and Decreasing Function

If for every x on some interval I , then f(x) is increasing on the interval

If for every x on some interval I , then f(x) is decreasing on the interval

If for every x on some interval I , then f(x) is constant on the interval

( ) 0f x

( ) 0f x

0)( xf

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

Absolute(Global) Maximum Absolute(Global) Minimum

Let f(x) be a function with domain D. Then f(x) has an

absolute maximum value on D at a point c if f(x) ≤ f(c) for all x in D and absolute minimum value on D at a point c if f(x) ≥ f(c) for all x in D.

Relative Extrema Relative (Local) Extrema : A function f(x) has a relative maximum value at an interior point c of its domain if f(x) ≤ f(c) for all x in some open interval containing c. A function f(x) has a relative minimum value at an interior point c of its domain if f(x) ≥ f(c) for all x in some open interval containing c.

Critical Points : An interior point c in the domain of f(x) is called a critical point if either or undefined. The corresponding point (c,f(c)) on the graph of f(x) is called a critical point for f(x).

)(cf ( ) 0f c

The First Derivative Theorem

If f(x) has a local maximum or minimum value at an interior point c of its domain and if is defined at c , then

)(xf

( ) 0f c

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.

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.

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 02 4 8

f 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 tt

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) 63

R

7 Max

++++++ -------- t0 63

Concavity The graph of differentiable function y=f(x) is

(1) concave up on an open interval I if is increasing on I

(2) concave down on an open interval I if is decreasing on I.

)(xf

)(xf

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.

The Second Derivative Test for Concavity Let y = f(x) be twice-differentiable on an interval I. 1. If on I , the graph of f(x) over I is concave up. 2. If on I , the graph of f(x) over I is concave down. Second Derivative Procedure for Determining Intervals of

Concavity for a Function f. Step 1. Find all values of x for which or is not continuous, and mark these numbers on a number line. This divides the line into a number of open intervals. 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.

0)( xf0)( xf

0)( xf )(xf

)(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 concavitySign 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 concavitySign 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