Grade 12 AP Calculus MCV 4 UE Chapter

Grade 12 (AP Calculus -MCV 4 UE) Chapter 4: Applications of Derivatives Extreme Values of Functions Mr. Choi © 2019 E. Choi – MCV 4 UE - All Rights Reserved

Absolute maximum and minimum points The highest and the lowest points in the figure. The highest and the lowest points in the domain. In the figure, D = absolute max. point; A = absolute min. point Also known as Global extreme values. Local maximum and minimum points The turning points. The highest and the lowest points in a particular interval. In the figure, B & D = local max. points; C = local min. point Also known as Relative extreme values. End points • The extreme points in the figure, usually given in the form • In the figure, A & E = extreme points (usually we set the denominator of f’(x)=0 and solve for x) Critical numbers generate the x-coordinates of the turning points. Extreme Values of Functions © 2019 E. Choi – MCV 4 UE - All Rights Reserved

(usually we set the denominator of f’(x)=0 and solve for x) 3) The Local maximum/minimum points are turning points occurring between increasing and decreasing of function or vice versa. Extreme Values of Functions © 2019 E. Choi – MCV 4 UE - All Rights Reserved

Test the Critical numbers and the Endpoints Critical numbers of f ‘(x) Absolute Min None Absolute Max Absolute maximum value is 62 occurs when x = 5 Absolute minimum value is -19 occurs when x = 2 Extreme Values of Functions © 2019 E. Choi – MCV 4 UE - All Rights Reserved

a) b) c) d) Find the critical numbers. Find the intervals of increase and decrease. Find the local maximum and local minimum points. Sketch the graph None Extreme Values of Functions © 2019 E. Choi – MCV 4 UE - All Rights Reserved

Extreme Values of Functions © 2019 E. Choi – MCV 4 UE - All Rights Reserved

Example 3: Vertical Tangent or Cusps Repeat Example 1 for None Extreme Values of Functions © 2019 E. Choi – MCV 4 UE - All Rights Reserved

Example 4: Determine the function Find values for a, b, and c such that the graph of relative maximum at (3, 12) and crosses the y-axis at (0, 1). has a -) Extreme Values of Functions © 2019 E. Choi – MCV 4 UE - All Rights Reserved

Example 5: Increasing and decreasing function & its local extreme values Given: a) b) c) d) e) Determine the domain Find the intervals of increase and decrease. Find the local maximum and local minimum points. Determine the intercepts Sketch the graph Extreme Values of Functions © 2019 E. Choi – MCV 4 UE - All Rights Reserved

Example 6: Determine the maximum and minimum values of the trig. function Determine the maximum and minimum values of the function on the interval Test the Critical numbers and the Endpoints Maximum value is 1 when None Extreme Values of Functions Minimum value is 0 when © 2019 E. Choi – MCV 4 UE - All Rights Reserved

Example 7: Increasing and decreasing function & its local extreme values Given: a) Find the intervals of increase and decrease. b) Find the local maximum and local minimum points. c) Sketch the graph x None Extreme Values of Functions 0. 5 0. 4 0. 6 0. 7 0. 8 0. 75 0. 74 0. 739 cosx 0. 877583 0. 921061 0. 825336 0. 764842 0. 696707 0. 731689 0. 738469 0. 745174 0. 739142 © 2019 E. Choi – MCV 4 UE - All Rights Reserved

Homework Textbook: P. 193 #2 -30, 35 -41 Cal & Vectors (Optional) P. 169 #1, 3, 5, 7, 8 P. 178 #5, 7, 8, 9, 11, 15 P. 233 #12 P. 245 #2, 12 P. 256 #6, 7, 11 P. 260 #5 -7 Extreme Values of Functions © 2019 E. Choi – MCV 4 UE - All Rights Reserved

End of lesson Extreme Values of Functions © 2019 E. Choi – MCV 4 UE - All Rights Reserved