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Practice

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Material Taken From:Mathematicsfor the international student Mathematical Studies SLMal Coad, Glen Whiffen, John Owen, Robert Haese, Sandra Haese and Mark BruceHaese and Haese Publications, 2004

- Worksheet S-47 #3
- y = x3 + 1.5x2 – 6x – 3
Find where the gradient is equal to zero.

- 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

negative

positive

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

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.

Find the volume, V, of the box in terms of x.

Using calculus, find the value of x which gives a maximum volume of the box.

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.

Write down an equation involving 50, x, and h.

Show that the area, y cm2, of cardboard used is given by y = 2x2 + 150x – 1

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

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