1. 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

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

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

4. 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

5. 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 = , 2.35
But
 d2 V = 24h – 120
dh2 so
V”(2.35) < 0
V”(7.64) > 0
 a local max
 a local min

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

7. 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 :

8. Mini-Lesson
Optimization Problem Solution Method:

9. Mini-Lesson
Optimization Problems – Example:

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

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