1. A4 – The Derivative as a Function MCB4U - Santowski

2. (A) Review – The Derivative at a Point In a previous lesson on tangents and limits and derivatives, we considered the derivative of a function f at a fixed point (x = a) and we developed a formula for finding the derivative at that point (tangent slope or instantaneous rate of change) as: In a previous lesson on tangents and limits and derivatives, we considered the derivative of a function f at a fixed point (x = a) and we developed a formula for finding the derivative at that point (tangent slope or instantaneous rate of change) as:

3. (B) The Derivative as a Function Our change in today's lesson will be to change our point of view and let the value of a vary (in other words, it will be a variable) Our change in today's lesson will be to change our point of view and let the value of a vary (in other words, it will be a variable) Consequently, we develop a new function - which we will now call the derived function (AKA the derivative) Consequently, we develop a new function - which we will now call the derived function (AKA the derivative) We will do this as an investigation using two different methods: a graphic/numeric approach and a more algebraic approach We will do this as an investigation using two different methods: a graphic/numeric approach and a more algebraic approach

4. (C) Deriving the Derivative Function Graphically & Numerically We will work with the tangent concept, draw tangents to given functions at various points, tabulate results, create scatter-plots and do a regression analysis to determine the equation of the curve of best fit. We will work with the tangent concept, draw tangents to given functions at various points, tabulate results, create scatter-plots and do a regression analysis to determine the equation of the curve of best fit. Example: f(x) = x² - 4x - 8 for the interval [-3,8] Example: f(x) = x² - 4x - 8 for the interval [-3,8]

5. (C) Deriving the Derivative Function Graphically & Numerically Example: y = x² - 4x - 8. for the interval [-3,8] Example: y = x² - 4x - 8. for the interval [-3,8] 1. Draw graph. 1. Draw graph. 2. Find the tangent slope at x = -3. Slope of tangents can be determined by DRAW, TANGENT commands on a TI-83 graphing calculator. Equations of tangent lines are given. Alternate use of graphing calculator is to use CALC, dy/dx option. 2. Find the tangent slope at x = -3. Slope of tangents can be determined by DRAW, TANGENT commands on a TI-83 graphing calculator. Equations of tangent lines are given. Alternate use of graphing calculator is to use CALC, dy/dx option. 3. Repeat for x = -2,-1,….,7,8 and tabulate 3. Repeat for x = -2,-1,….,7,8 and tabulate X -3 -2 -1 0 1 2 3 4 5 6 7 8 X -3 -2 -1 0 1 2 3 4 5 6 7 8 Slope -10 -8 -6 -4 -2 0 2 4 6 8 10 12 Slope -10 -8 -6 -4 -2 0 2 4 6 8 10 12 4. Tabulate data: STAT, EDIT 4. Tabulate data: STAT, EDIT 5. Create scatter-plot: STAT PLOT, 1, ON, GRAPH 5. Create scatter-plot: STAT PLOT, 1, ON, GRAPH 6. Do a regression analysis: STAT, CALC, pick the most appropriate regression eqn (linreg), L1,L2, Y2. 6. Do a regression analysis: STAT, CALC, pick the most appropriate regression eqn (linreg), L1,L2, Y2. 7. Equation generated is y = 2x - 4 7. Equation generated is y = 2x - 4

6. (C) Deriving the Derivative Function Graphically & Numerically

9. Our function equation was f(x) = x² - 4x - 8 Our function equation was f(x) = x² - 4x - 8 Equation generated is f`(x) = 2x - 4 Equation generated is f`(x) = 2x - 4 The interpretation of the derived equation is that this "formula" (or equation) will give you the slope of the tangent (or instantaneous rate of change) at every single point x. The interpretation of the derived equation is that this "formula" (or equation) will give you the slope of the tangent (or instantaneous rate of change) at every single point x. The equation f`(x) = 2x - 4 is called the derived function, or the derivative of f(x) = x² - 4x - 8 The equation f`(x) = 2x - 4 is called the derived function, or the derivative of f(x) = x² - 4x - 8

10. (D) Deriving the Derivative Function Algebraically Given f(x) = x 2 – 4x – 8, we will find the derivative at x = a using our “derivative formula” of Given f(x) = x 2 – 4x – 8, we will find the derivative at x = a using our “derivative formula” of Our one change will be to keep the variable x in the “derivative formula”, since we do not wish to substitute in a specific value like a Our one change will be to keep the variable x in the “derivative formula”, since we do not wish to substitute in a specific value like a

11. (D) Deriving the Derivative Function Algebraically

12. (E) In-Class Examples Use the algebraic method to determine the equations of the derivative functions of the following and then state the domain of both the function and the derivative function Use the algebraic method to determine the equations of the derivative functions of the following and then state the domain of both the function and the derivative function (i) f(x) =  (x+5) (i) f(x) =  (x+5) (ii) g(x) = (x + 1)/(3x – 2) (ii) g(x) = (x + 1)/(3x – 2)

13. (F) Internet Links From Paul Dawkins - Calculus I (Math 2413) - Derivatives - The Definition of the Derivative From Paul Dawkins - Calculus I (Math 2413) - Derivatives - The Definition of the Derivative From Paul Dawkins - Calculus I (Math 2413) - Derivatives - The Definition of the Derivative From Paul Dawkins - Calculus I (Math 2413) - Derivatives - The Definition of the Derivative From UTK - Visual Calculus - Definition of derivative From UTK - Visual Calculus - Definition of derivative From UTK - Visual Calculus - Definition of derivative From UTK - Visual Calculus - Definition of derivative Tutorial for Derivatives From Stefan Waner @ Hofstra U Tutorial for Derivatives From Stefan Waner @ Hofstra U Tutorial for Derivatives From Stefan Waner @ Hofstra U Tutorial for Derivatives From Stefan Waner @ Hofstra U

14. (G) Homework Photocopy from Stewart text – “Calculus – A First Course”; Chap 2.1, Q10,11,12 Photocopy from Stewart text – “Calculus – A First Course”; Chap 2.1, Q10,11,12