Example: When describing where a function is increasing, decreasing or constant we use the x- axis values.

Transcription

1 Business Calculus Lecture Notes (also Calculus With Applications or Business Math II) Chapter 3 Applications o Derivatives 31 Increasing and Decreasing Functions Inormal Deinition: A unction is increasing when the graph moves upward as increases A unction is decreasing when the graph moves downward as increases Formal Deinition: A unction is increasing on an interval when, or any two numbers 1 and in the interval, 1 > implies (1) > ( A unction is decreasing on an interval when, or any two numbers 1 and in the interval, 1 > implies (1) < ( A unction is constant on an interval when, or any two numbers 1 and in the interval, 1 > implies (1) = ( Eample: When describing where a unction is increasing, decreasing or constant we use the - ais values (1,3) (5,3) (7,0) The ollowing statements reer to the above unction: is increasing on (,1 ) and ( on (,1 ) is constant on and ( on is decreasing on ( on (1,5) (5,7) and 0 (1,5) (5,7) Deinition o Critical Numbers I is deined at =c, then c is a critical number i (c)=0 or i (c) is undeined Note: A critical number must be in the domain o the unction Identiy the critical numbers or the unctions below Their derivatives are given 3 ( 4 6 /3 4( ) ( ( 4 ; ( ( 6) ( 1 1/ 3 1 3( 4 ( ; ( 3 ( 3) 1 1 1( 1) 1, 1 =0,1 are critical numbers because the derivative is 0 at those values 4( ) 1/ 3 3( 4 4( ) 0 3( 4 1/ 3 4 ( 4) = is a critical number because the derivative is 0 at = =0,4 are critical numbers because the derivative is undeined at those values ( 6) 0 ( 3) ( 6), 6 =0,6 are critical numbers because the derivative is 0 at those values =3 is NOT a critical number because it is not in the domain o ( 1

2 Consider the graphs o the above unctions 3 /3 ( 4 6 ( ( 4 ( Critical Numbers =0,1 Critical Numbers =0,,4 3 Critical Numbers =0,6 horizontal tangent lines at =0 and = (derivative = 0) horizontal tangent lines at = (derivative = 0); and notice the graph has sharp turns at =0 and = (derivative undeined) horizontal tangent lines at =0 and =6 (derivative = 0) See the vertical asymptote at =3? Note: Critical numbers are important because they can show us where a graph changes rom increasing to decreasing or decreasing to increasing A graph does not always turn around at a critical number but we will see that critical numbers are the only place it can turn around Find the critical numbers o the ollowing: 3 a) ( 3 18 b) ( 1 ( 9 36 ( 1)( 1) ( 9 36 ( 1) 9 4 ( 1)( 1) ( 1),4 are the critical numbers o Since the derivative is deined or all real 1, 1 are the critical numbers o numbers, we get no critical numbers rom the Because they make the derivative equal 0 derivative being undeined So 0,4 are the only critical numbers 1/ 3 c) ( 1 ( 1) 1/ 3 d) ( 1 ( 1) Find the derivative using product rule: ( 3 / 3 ( 3( 1) 1 The critical numbers are =0,-1,1 =0 makes the The critical numbers are / 3, 1 = -/3 derivative=0 and = 1 makes the derivative makes the derivative=0 and = -1 makes the undeined derivative undeined

3 3 For the unction: ( 1 Graph ( with your calculator and see what s happening at the values Can you see why these values are critical? At = -1 and =1 the tangent line is vertical (slope undeined) At =0, the tangent line is horizontal (slope=0) Notice that the graph does not turn at =1 Test or Increasing and Decreasing Functions: There is a relationship between the sign o the derivative and whether the graph is increasing or decreasing Let be dierentiable on the interval (a,b) 1 I ( >0 or all in (a,b), then is increasing on (a,b) I ( <0 or all in (a,b), then is decreasing on (a,b) 3 I ( =0 or all in (a,b), then is constant on (a,b) To Apply the Increasing/Decreasing Test Do the Following: 1 Find the derivative Identiy the critical numbers to determine the test intervals 3 Test the sign o ( at a random number in each test interval 4 Use the test to determine i ( is increasing or decreasing on each interval For a continuous interval, the graph can only change direction at a critical number Determine the open intervals where each unctions is increasing and/or decreasing 3 EXAMPLE 1: ( 3 Find the derivative and set = 0 ( 6 18 Solutions to this equation are =0, =3 (critical numbers) Intervals (-, 0) (0, 3) (3, ) Test Point k Sign o ( (-1) >0 (1) <0 (4) >0 Conclusion o ( is increasing is decreasing is increasing is increasing on (-, 0) U (3, ) and is decreasing on (0, 3) Graph the unction to veriy the results 3 3 EXAMPLE : ( Find the derivative and set = 0 ( 3 3 Critical Numbers are =0, =1 Intervals (-, 0) (0, 1) (1, ) Test Point k -1 1/ Sign o ( (-1) >0 (1/) <0 () >0 Conclusion o ( is increasing is decreasing is increasing is increasing on (-,0) ( 1, ) and decreasing on (0,1) Graph the unction to veriy the results 3

4 EXAMPLE 3: / 3 ( ( 4) 4 Find where derivative is zero or undeined ( 1/ 3 3( 4) Critical numbers are =0,-, Intervals (-, -) (-, 0) (0,) (, ) Test Point k Sign o ( (-3) < 0 (-1) > 0 (1) < 0 (3) >0 Conclusion o ( is decreasing is increasing is decreasing is increasing is increasing (-,0) (, ) and decreasing (-,-) (0,) Graph the unction to veriy the results EXAMPLE 4: y 5 3 ( 3 No Critical numbers since -3 can't equal 0 Since the derivative is always negative, the graph is strictly decreasing EXAMPLE 5: ( ( ) ( ( ) Interval(s) (, ) (, ) Test Value -3 0 Sign o ( 3) ( 0) Conclusion or decreasing increasing The unction is decreasing on (, ) and is increasing on (, ) EXAMPLE 6: y 4 ( 4 The critical values are,, but the domain o the unction is [-,] We can't go outside o that interval to test Interval(s) (,0) ( 0,) Test Value -1 1 Sign o ( 1) ( 1) Conclusion or increasing decreasing The unction is decreasing on ( 0,) and increasing on (,0) 4

5 EXAMPLE 7: 3 ( 3 18 ( ( 4) Critical numbers are =0, =4 Interval(s) Test Value Sign o Conclusion or increasing decreasing increasing The unction is decreasing on and increasing on (,0) (4, ) (,0) ( 1) (0,4) (0,4) ( 1) ( 4, ) ( 5) EXAMPLE 8: ( 1 3 ( 1 Critical numbers are = -/3 and = -1, but the domain is [ 1, ) ( 1, / 3) ( / 3, ) Interval(s) Test Value Sign o ( 09) ( 0) Conclusion or decreasing increasing The unction is decreasing on ( 1, / 3) and increasing on ( / 3, ) EXAMPLE 9: ( 1 ( 1)( 1) ( ( 1) Critical numbers are =1, and = -1 Interval(s) (, 1) ( 1,1 ) ( 1, ) Test Value - 0 Sign o ( ) ( 0) ( 0) Conclusion or decreasing increasing decreasing The unction is decreasing on (, 1) (1, ) and increasing on ( 1,1 ) 5