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
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.
positive to the left of p
– p is a maximum if f ’(x) is ________
negative to the right of p.
and ________
negative to the left of p
– p is a minimum if f ’(x) is ________
positive to the right of p.
and ________
Remember:
increasing
if f ’(x) is positive then f(x) is ___________.
if f ’(x) is negative then f(x) is ___________.
decreasing
Worksheet S-47 #4, 5
Section 19JK - Optimization
• 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.
Example 1
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.
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.
See animation in HL book, page 653
Example 2
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 3
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 4
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.
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.
See animation in HL book, page 653
Homework
• Worksheet S-47 #6, 7
• Pg 629 #5,6,7
• Worksheet, Optimization