Daily Challenge Please solve the following without a GDC: None None Increases for x > 0, never decreases Concave down for x > 0, never concave up

Questions, Questions Essential Question: How can the Calculus solve real world problems? Guiding Question: How are derivatives used to optimize real world applications?

Homework Review “Problem” homework problems: 22G p585 #1c – f; 2a, c, e, g

Mini-Lesson Finding Optimal Solutions: An optimal solution to a problem may involve finding a function’s maxima or minima Recall the Math vs Monsters project 12 x 18 Cardboard Maximize volume

Mini-Lesson Finding Optimal Solutions (Continued): V = h(12 – 2h)(18 – 2h)  d V = 12(h2 – 10h + 18) dh And, setting the first derivative = 0 h = 5  √7 = 7.64, 2.35 But  d2 V = 24h – 120 dh2 so V”(2.35) < 0 V”(7.64) > 0  a local max  a local min

Mini-Lesson Finding Optimal Solutions (Continued): Note:

Mini-Lesson Graph It! Test for Conditions: Given: near x = a, f ‘(a) = 0; First Derivative Sign Diagram Test : Or, if you can, Graph It! Second Derivative Test :

Mini-Lesson Optimization Problem Solution Method:

Mini-Lesson Optimization Problems – Example:

Group Work Please solve the following without a GDC: 0 ≤ x ≤ 200/ P = 2r + 2L 400 = 2x + 2L or L = 200 - x But 0 ≤ L ≤ 200 Since L > 0, x < 200  0 ≤ x ≤ 200/ A = r2 + 2xL = x2 + 2xL = x2 + 2x(200 - x) = x2 + 400x - 2x2  A = 400x - x2 A’(x) = 400 - 2x Setting = 0, x = 200/  and L = 0. Since A ‘(x): + → - around 200/  Max is a circle!

Homework Textbook: Please read 22H p587 – 596 Please do exercises: Journal: Please answer the Guiding Question.