Material Taken From:

Mathematicsfor the international student

Mathematical Studies SL

Mal Coad, Glen Whiffen, John Owen, Robert Haese, Sandra Haese and Mark Bruce

Haese and Haese Publications, 2004

Practice

• Worksheet S-47 #3

• y = x3 + 1.5x2 – 6x – 3

Find where the gradient is equal to zero.

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.

positivenegative

positivenegative

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

• 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

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

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

A rectangle has width x cm and length y cm. It has a constant area 20 cm2.– 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

An open rectangular box is made from thin cardboard. The base is 2x cm long and x cm wide and the volume is 50 cm3. 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 cm2, of cardboard used is given

by y = 2x2 + 150x – 1 c) Find the value of x that makes the area a minimum

and find the minimum area of cardboard used.

Example 4

Homework

• Worksheet S-47 #6, 7• Pg 629 #5,6,7• Worksheet, Optimization