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Practice

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 = x 3 + 1.5 x 2 – 6 x – 3

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Practice

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  1. 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

  2. Practice • Worksheet S-47 #3 • y = x3 + 1.5x2 – 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 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

  4. 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.

  5. 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

  6. 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.

  7. 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.

  8. 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

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

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