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Learn how to find second derivatives of functions, identify function trends, interpret derivative values, and solve tangent equations. Includes graphical interpretations and concept explanations. Ideal for IB students studying Mathematical Studies.
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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
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.
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.
GDC • Equations of tangentsin the calculator: • put function in [Y=] • graph • [2nd] [prgm] Draw 5: Tangent( • type the value where you want your tangent
Section 19G – The Second Derivative Find f ’’(x) given that Example 1
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
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 _____.
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.
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.
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.
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
Example 3 The function f(x) is defined as Determine the x-coordinates of the points where the graph has a gradient of zero.
Example 4 The function f(x) is defined as Determine the x-coordinates of the points where the graph has a gradient of zero.
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?
Homework • Worksheet S-46a • Worksheet S-46b • Pg 624 #1bce • Pg 626 #3ace • Worksheet S-47 #1-2
