A4 – The Derivative A4 – The Derivative as a Functionas a Function

MCB4U - SantowskiMCB4U - Santowski

(A) Review – The (A) Review – The Derivative at a PointDerivative at a Point In a previous lesson on tangents and limits In a previous lesson on tangents and limits

and derivatives, we considered the derivative and derivatives, we considered the derivative of a function of a function ff at a fixed point ( at a fixed point (xx = = aa) and we ) and we developed a formula for finding the derivative developed a formula for finding the derivative at that point (tangent slope or instantaneous at that point (tangent slope or instantaneous rate of change) as:rate of change) as:

h

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h

)()(lim)(

0

(B) The Derivative as a (B) The Derivative as a FunctionFunction

Our change in today's lesson will be to Our change in today's lesson will be to change our point of view and let the value of change our point of view and let the value of aa vary (in other words, it will be a variable) vary (in other words, it will be a variable)

Consequently, we develop a new function - Consequently, we develop a new function - which we will now call the derived function which we will now call the derived function (AKA the derivative)(AKA the derivative)

We will do this as an investigation using two We will do this as an investigation using two different methods: a graphic/numeric different methods: a graphic/numeric approach and a more algebraic approachapproach and a more algebraic approach

(C) Deriving the Derivative (C) Deriving the Derivative Function Graphically & Function Graphically & NumericallyNumerically

We will work with the tangent concept, We will work with the tangent concept, draw tangents to given functions at draw tangents to given functions at various points, tabulate results, create various points, tabulate results, create scatter-plots and do a regression scatter-plots and do a regression analysis to determine the equation of analysis to determine the equation of the curve of best fit.the curve of best fit.

Example: f(Example: f(xx) = ) = xx² - 4² - 4xx - 8 for the - 8 for the interval [-3,8]interval [-3,8]

(C) Deriving the Derivative (C) Deriving the Derivative Function Graphically & Function Graphically & NumericallyNumerically Example: Example: yy = = xx² - 4² - 4xx - 8. for the interval [-3,8] - 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 2. Find the tangent slope at x = -3. Slope of tangents can be

determined by DRAW, TANGENT commands on a TI-83 graphing determined by DRAW, TANGENT commands on a TI-83 graphing calculator. Equations of tangent lines are given. Alternate use of calculator. Equations of tangent lines are given. Alternate use of graphing calculator is to use CALC, dy/dx option.graphing calculator is to use CALC, dy/dx option.

3. Repeat for x = -2,-1,….,7,8 and tabulate3. Repeat for x = -2,-1,….,7,8 and tabulate

X -3 -2 -1 0 1 2 3 4 5 6 7 8X -3 -2 -1 0 1 2 3 4 5 6 7 8 Slope -10 -8 -6 -4 -2 0 2 4 6 8 10 12Slope -10 -8 -6 -4 -2 0 2 4 6 8 10 12

4. Tabulate data: STAT, EDIT4. Tabulate data: STAT, EDIT 5. Create scatter-plot: STAT PLOT, 1, ON, GRAPH5. Create scatter-plot: STAT PLOT, 1, ON, GRAPH 6. Do a regression analysis: STAT, CALC, pick the most appropriate 6. Do a regression analysis: STAT, CALC, pick the most appropriate

regression eqn (linreg), L1,L2, Y2. regression eqn (linreg), L1,L2, Y2. 7. Equation generated is 7. Equation generated is yy = 2 = 2xx - 4 - 4

(C) Deriving the Derivative (C) Deriving the Derivative Function Graphically & Function Graphically & NumericallyNumerically

(C) Deriving the Derivative (C) Deriving the Derivative Function Graphically & Function Graphically & NumericallyNumerically

(C) Deriving the Derivative (C) Deriving the Derivative Function Graphically & Function Graphically & NumericallyNumerically

(C) Deriving the Derivative (C) Deriving the Derivative Function Graphically & Function Graphically & NumericallyNumerically Our function equation was f(Our function equation was f(xx) = ) = xx² - ² -

44xx - 8 - 8 Equation generated is f`(Equation generated is f`(xx) = 2) = 2xx - 4 - 4 The interpretation of the derived The interpretation of the derived

equation is that this "formula" (or equation is that this "formula" (or equation) will give you the slope of the equation) will give you the slope of the tangent (or instantaneous rate of tangent (or instantaneous rate of change) at every single point change) at every single point xx. .

The equation f`(The equation f`(xx) = 2) = 2xx - 4 is called - 4 is called the derived function, or the derivative the derived function, or the derivative of f(of f(xx) = ) = xx² - 4² - 4xx - 8 - 8

(D) Deriving the Derivative (D) Deriving the Derivative Function AlgebraicallyFunction Algebraically

Given f(x) = xGiven f(x) = x22 – 4x – 8, we will find the – 4x – 8, we will find the derivative at x = a using our “derivative derivative at x = a using our “derivative formula” of formula” of

Our one change will be to keep the variable Our one change will be to keep the variable xx in the “derivative formula”, since we do not in the “derivative formula”, since we do not wish to substitute in a specific value like wish to substitute in a specific value like aa

h

afhafaf

h

)()(lim)(

0

(D) Deriving the Derivative (D) Deriving the Derivative Function AlgebraicallyFunction Algebraically

f xf x h f x

h

f xx h x h x x

h

f xx xh h x h x x

h

f xxh h h

hf x x h

f x x

h

h

h

h

h

( ) lim( ) ( )

( ) lim[( ) ( ) ] [ ]

( ) lim[( ) ]

( ) lim

( ) lim

( )

0

0

2 2

0

2 2 2

0

2

0

4 8 4 8

2 4 4 8 4 8

2 4

2 4

2 4

(E) In-Class Examples(E) In-Class Examples

Use the algebraic method to determine the Use the algebraic method to determine the equations of the derivative functions of the equations of the derivative functions of the following and then state the domain of both following and then state the domain of both the function and the derivative functionthe function and the derivative function

(i) f(x) = (i) f(x) = (x+5)(x+5) (ii) g(x) = (x + 1)/(3x – 2)(ii) g(x) = (x + 1)/(3x – 2)

(F) Internet Links(F) Internet Links

From Paul Dawkins - Calculus I (Math 2From Paul Dawkins - Calculus I (Math 2413) - Derivatives - The Definition of th413) - Derivatives - The Definition of the Derivativee Derivative

From UTK - Visual Calculus - Definition From UTK - Visual Calculus - Definition of derivativeof derivative

Tutorial for Derivatives From Stefan Tutorial for Derivatives From Stefan WanerWaner @ @ HofstraHofstra U U

(G) Homework(G) Homework

Photocopy from Stewart text – Photocopy from Stewart text – “Calculus – A First Course”; Chap “Calculus – A First Course”; Chap 2.1, Q10,11,122.1, Q10,11,12