1. Material Taken From: Mathematics for the international student Mathematical Studies SL Mal Coad, Glen Whiffen, John Owen, Robert Haese, Sandra Haese and Mark Bruce Haese and Haese Publications, 2004

2. Practice Worksheet S-47 #3 y = x 3 + 1.5x 2 – 6x – 3 Find where the gradient is equal to zero.

3. Maximum vs. Minimum If f ’(p)=0, then p is a max or min. – p is a maximum if f ’(x) is ________ to the left of p and ________ to the right of p. – p is a minimum if f ’(x) is ________ to the left of p and ________ to the right of p. positive negative positive negative Remember: if f ’(x) is positive then f(x) is ___________. if f ’(x) is negative then f(x) is ___________. increasing decreasing Worksheet S-47 #4, 5

4. At a maximum or minimum  tangent line is horizontal  derivative is zero. We can use that information to find the maximum and minimum of a real-world situation. Section 19JK - Optimization

5. A sheet of thin card 50 cm by 100 cm has a square of side x cm cut away from each corner and the sides folded up to make a rectangular open box. See animation in HL book, page 653 a)Find the volume, V, of the box in terms of x. b)Using calculus, find the value of x which gives a maximum volume of the box. c)Find this maximum volume. Example 1

6. A rectangular piece of card measures 24 cm by 9 cm. Equal squares of length x cm are cut from each corner of the card as shown in the diagram below. What is left is then folded to make an open box, of length l cm and width w cm. (a)Write expressions, in terms of x, for (i)the length, l; (ii)the width, w. (b)Show that the volume (B m3) of the box is given by B = 4x3 – 66x2 + 216x. (c)Find. (d)(i)Find the value of x which gives the maximum volume of the box. (ii)Calculate the maximum volume of the box. Example 2

7. A rectangle has width x cm and length y cm. It has a constant area 20 cm 2. – Write down an equation involving x, y and 20. – Express the perimeter, P, in terms of x only. – Find the value of x which makes the perimeter a minimum and find this minimum perimeter. Example 3

8. An open rectangular box is made from thin cardboard. The base is 2x cm long and x cm wide and the volume is 50 cm 3. Let the height be h cm. See animation in HL book, page 653 a)Write down an equation involving 50, x, and h. b)Show that the area, y cm 2, of cardboard used is given by y = 2x 2 + 150x – 1 c)Find the value of x that makes the area a minimum and find the minimum area of cardboard used. Example 4

9. Homework Worksheet S-47 #6, 7 Pg 629 #5,6,7 Worksheet, Optimization