Differentiation Techniques and Slope Functions Analysis" (Limit: 61 characters)

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Alternative methods for finding derivatives and analyze the slope functions of various equations. Examine the behavior of functions and determine slopes at different points. Includes visual representations and practical examples. Discover how to calculate slopes, interpret results, and understand the equations involved in the process. Utilize GeoGebra for exploring quadratic equations and enhancing understanding of slope functions. Answer questions related to finding equations of slope functions and tangents. Enhance your mathematical skills through visual aids and applications.

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Purpose Is there another way to work out the derivative?

Slope Functions- What is the Slope of the Functions below? 4 y ( ) f x = 3 3 2 ( ) 1 g x = 1 x -6 -4 -2 2 4 6 ( ) h x = 1 -1 -2 y = 3 -3 -4

Slope Functions 3 2 3 2 = + 3 y x 2 = x ( ) h x x = ( ) f x x = + ( ) g x = + = ( ) k x 3 x ( ) p x x 4 y 3 2 1 x -6 -4 -2 2 4 6 -1 -2 -3 -4

Slope Functions 3 2 3 2 = + 3 y x 2 = x ( ) h x x = ( ) f x x = + ( ) g x = + = ( ) k x 3 x ( ) p x x 4 y 3 2 1 x -6 -4 -2 2 4 6 -1 -2 -3 -4

Slope Functions 3 2 3 2 = + 3 y x 2 = x ( ) h x x = ( ) f x x = + ( ) g x = + = ( ) k x 3 x ( ) p x x 4 y 3 2 1 x -6 -4 -2 2 4 6 -1 -2 -3 -4

Slope Functions 3 2 3 2 = + 3 y x 2 = x ( ) h x x = ( ) f x x = + ( ) g x = + = ( ) k x 3 x ( ) p x x 4 y 3 2 1 x -6 -4 -2 2 4 6 -1 -2 -3 -4

Slope Functions 12 y 11 10 (3, 9) 1 (-3, 9) -3 9 8 7 -6 -6 -6 6 5 x f (x) (2, 4) 1 (-2, 4) -2 4 3 -4 -4 2 (1, 1) 1 (-1, 1) -1 1 -2 -2 x 0 0 -5 -4 -3 -2 -1 1 2 3 4 5 (0, 0) -1 1 2 2 4 -2 3 6

Slope Functions 12 y 11 10 (3, 9) (-3, 9) 9 8 7 4 4 6 5 x f (x) (2, 4) 2 (-2, 4) 4 -3 -6 1 3 2 -2 -4 2 2 (1, 1) 1 (-1, 1) 1 -1 -2 1 x 0 -5 -4 -3 -2 -1 1 2 3 4 5 (0, 0) 0 -1 -2

Slope Functions 12 y 11 10 (3, 9) 3 (-3, 9) 9 8 7 6 6 5 x f (x) (2, 4) (-2, 4) 4 -3 -6 3 1 2 -2 -4 (1, 1) (-1, 1) 1 -1 -2 x 0 0 -5 -4 -3 -2 -1 1 2 3 4 5 (0, 0) -1 1 2 -2 2 4

Slope Functions of other Quadratic Equations using GeoGebra http://www.soft32.com/blog/wp-content/uploads/2012/04/geogebra.png 10

Question 7

8. What is the equation of the slope function of f(x) = x2 x 6? 7 y 6 5 4 3 2 1 x -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 -1 -2 -3 -4 -5 -6 -7 f (x) = 2x 1 -8 -9 -10 The slope of the tangent at x = 3 is 7 What is the slope of f(x) for x=1.5? The slope of the tangent at x = 2 is 5 The slope of the tangent at x = 1 is 3 The slope of the tangent at x = 0 is 1 The slope of the tangent at x = 1 is +1 The slope of the tangent at x = 2 is +3 The slope of the tangent at x = 3 is +5 We can gain information about the slope of f(x) for x=1.5 from (i) the graph of f(x) (ii) the list of slopes on the left (iii) the graph of f (x) (iv) the equation of the slope function f (x)

Slope Functions of Cubic Equations using GeoGebra http://www.soft32.com/blog/wp-content/uploads/2012/04/geogebra.png 13

Question 13

Summary and Prediction What are the slope functions of: (i) h(x)=x8 (ii) g(x)=3x10 = = n y x (iii) f(x)=5x2-3x-6 dy dx 1 n nx

Consistency = = = = = = = 2 3 5 x 7 y x y y y 6 x 1 0 5 x 7 y y x y 6 x 6 1 = 1 y x = = = = = = = dy dx dy dx dy dx dy dx 0 1 2 6 x 5 (0)7 6 x x x dy dx dy dx dy dx 5(1) 0 6 2 x = dy dx 5

Slopes of Tangents of f(x)=x2 and g(x)=x2+3

Central Ideas Associating derivatives with slopes and tangent lines = n If ( ) then ( ) f x x f x = 1 n nx = = + n n If ( ) then ( ) and ( ) ( ) q x = p x x q x x c p x

Purpose To identify features of the graphs of functions in preparation for sketching their corresponding slope functions.

Describing Functions Pair up. One person describes a function. The other person draws the function on their whiteboard using only the descriptions their partner gives them.

Features of +x3 Graphs The function is f(x) is y is y 6 Positive and Positive and Increasing 5 Neither Increasing nor Decreasing i.e. Stationary Positive and Decreasing 4 Positive and Increasing 3 2 Neither Positive nor Negative and Decreasing Neither Positive nor Negative and Increasing Neither Positive nor Negative and Increasing 1 x -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 Negative -1 and Negative and Increasing Negative and Increasing -2 Decreasing -3 Negative and -4 Neither Increasing nor Decreasing i.e. Stationary -5 -6 21

Features of +x3 Graphs The function is f(x) is y is y 6 5 Neither Increasing nor Decreasing i.e. Stationary 4 3 Increasing Decreasing 2 1 x -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 -1 -2 Increasing -3 Neither Increasing nor Decreasing i.e. Stationary -4 -5 -6 23

Features of +x3 Graphs The slope of the function is y 6 5 f (x)=0 4 Positive f (x)>0 3 2 Negative f (x)<0 Positive f (x)>0 1 x -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 -1 -2 -3 f (x)=0 -4 -5 -6 24

http://www.soft32.com/blog/wp-content/uploads/2012/04/geogebra.pnghttp://www.soft32.com/blog/wp-content/uploads/2012/04/geogebra.png

Features of +x3 Graphs The slope function is f (x) is dy/dx is y 6 5 4 3 f (x) is decreasing 2 f(x) is increasing 1 f (x) is neither increasing nor decreasing x -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6 26

+x2 Graphs: Relating the Function to the Slope Function y 6 The slope function will be positive for these x-values 5 4 The slope of the function is negative The slope of the function is positive 3 2 1 x -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 The slope of the function is zero The slope function will be negative for these x-values -1 -2 The slope function will be 0 for this x-value. -3 -4 -5 -6 27

+x3 Graphs: Relating the Function to the Slope Function y 6 The slope function will be positive for these x-values The slope function will be positive for these x-values 5 4 The slope of the function is positive The slope of the function is negative 3 2 1 x -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 The slope function will be negative for these x- values -1 -2 The slope of the function is positive -3 -4 -5 -6 28