Objectives:

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. Objectives: • To find the second derivative of a function. • To identify where functions are increasing and decreasing. • To interpret when the f ’(x)=0, f ’(x)>0, and f ’(x) < 0.

3. IB Subject Guide • Gradients of curves for given values of x. • Values of x where f ′(x) is given. • Equation of the tangent at a given point. • Increasing and decreasing functions. • Graphical interpretation of f′(x)>0, f′(x)=0, f′(x)<0. • Values of x where the gradient of a curve is 0 (zero): solution of f′(x) =0. • Local maximum and minimum points.

4. GDC • Equations of tangentsin the calculator: • put function in [Y=] • graph • [2nd] [prgm] Draw 5: Tangent( • type the value where you want your tangent

5. Section 19G – The Second Derivative Find f ’’(x) given that Example 1

6. Increasing function An increase in x produces an increase in y Decreasing function An increase in x produces a decrease in y. Section 19HI – Curve Properties

7. Consider y = x3 – 3x + 4 • What is happening to the slopes of the tangent lines? • Where f(x) is increasing, f ’(x) is _____. • Where f(x) is decreasing f ’(x) is _____. • Where f(x) is at a maximum or minimum, f ’(x) is _____.

8. Understanding: • Derivative = slope of tangent line. • If the tangent line has a negative slope, then the derivative is negative. • This happens where the function, f(x), is decreasing. • If the tangent line has a positive slope, then the derivative is positive. • This happens where the function, f(x), is increasing. • If the tangent line is horizontal, then the derivative is zero. • This happens where the function, f(x), is at a maximum or minimum.

9. IB Example 1 Given the graph of f (x) state: • the intervals from A to L in which f (x) is increasing. • b) the intervals from A to L in which f (x) is decreasing.

10. IB Example 2 The function f(x) is given by the formula f(x) = 2x3 – 5x2 + 7x – l • Evaluate f (1). b) Calculate f '(x). c) Evaluate f '(2). d) State whether the function f (x) is increasing or decreasing at x = 2.

11. Consider y = x3 – 3x + 4 • What is the tangent line at the maximum and the minimum? In order to find the x-coordinate of any maximum or minimum points, solve the equation f’(x) = 0

12. Example 3 The function f(x) is defined as Determine the x-coordinates of the points where the graph has a gradient of zero.

13. Example 4 The function f(x) is defined as Determine the x-coordinates of the points where the graph has a gradient of zero.

14. IB Example 5 Consider the function f(x)=2x3 – 3x2 – 12x + 5 a) (i) Find f ‘(x). (ii) Find the gradient of the curve f(x) when x = 3. b) Find the x-coordinates of the points on the curve where the gradient is equal to –12. c) (i) Calculate the x-coordinates of the local maximum and minimum points. (ii) Hence find the coordinates of the local minimum. d) For what values of x is the value of f(x) increasing?

15. Homework • Worksheet S-46a • Worksheet S-46b • Pg 624 #1bce • Pg 626 #3ace • Worksheet S-47 #1-2