DEFINITION OF A DERIVATIVE Section 2.1 Calculus AP/Dual, Revised 2017 viet.dang@umbleisd.net 2.1: Definition of a Derivative 1
DEFINITION A. Te derivative of a function allows you to find te SLOPE OF THE TANGENT LINE at a point. 1. Te slope of a curve is DIFFERENT at every point wereas te slope of a line is te same. B. Different ways of writing te derivative, it is known as te derivative of y wit respects to x 1. f (x): 1 st Derivative of f 2. dy dx : Derivative of y wit respect to x 3. y : 1 st Derivative of y 4. d dx f x : 1st Derivative of f(x) wit respect to x 2.1: Definition of a Derivative 2
EQUATION Q x +, f x + P x, f x As Q approaces P, te secant line approaces to te tangent line f ' x lim f x f x 0 2.1: Definition of a Derivative 3
EQUATION f ' x lim f x f x 0 2.1: Definition of a Derivative 4
STEPS A. Purpose of a Tangent Line 1. Te line sares a point wit te curve in question. 2. At te sared point, te derivative of te curve is equal to te slope of te line. B. Finding te Derivative by te Limit Process 1. Apply te Difference Quotient te equation 2. Add te limit to eac step as te goal is to determine te limit wen 0; te difference is. It approaces at zero because te slope of te grap of f at x = c. 2.1: Definition of a Derivative 5
REVIEW 1) Solve for f 3 of f x = x 2 3 f 3 6 2) Solve for f of f x = x 2 3 2 f 3 3) Solve for f x + x of f x = x 2 3 2 2 f x x x 2xx x 3 2.1: Definition of a Derivative 6
EXAMPLE 1 Use te limit process to find te derivative of f x = 2x 3 f x f x f x lim 0 ' 2 x 3 2x 3 lim 0 2 2 3 2 3 2 lim 0 x x lim 0 f ' x 2 Wat is te slope of tis function? 2.1: Definition of a Derivative 7
EXAMPLE 1 Use te limit process to find te derivative of f x = 2x 3 y x 2.1: Definition of a Derivative 8
EXAMPLE 2 Use te limit process to find te derivative of f x = 2x 2 16x + 35 f ' x 4x 16 2.1: Definition of a Derivative 9
YOUR TURN Use te limit process to find te derivative of f x = x 2 3 f ' x 2x 2.1: Definition of a Derivative 10
EXAMPLE 3 Use te limit process to find te derivative of f x = 5x 8 lim f x f x 0 5 8 5 8 x x lim 0 5 5 8 5 8 x x lim 0 2.1: Definition of a Derivative 11
EXAMPLE 3 Use te limit process to find te derivative of f x = 5x 8 5 5 8 5 8 x x lim 0 5 5 8 5 8 5 5 8 5 8 5x 5 8 5x 8 x x x x lim 0 5 5 8 5 8 5 lim 5x 5 8 5x 8 0 5 x 5 8 5 x 8 x x lim 0 2.1: Definition of a Derivative 12
EXAMPLE 3 Use te limit process to find te derivative of f x = 5x 8 lim 5 5 8 5 8 0 x x lim 0 x x 5 5 0 8 5 8 5 5 5 5x8 5x8 2.1: Definition of a Derivative 13
EXAMPLE 3 Use te limit process to find te derivative of f x = 5x 8 5 5x8 5x8 5 2 5x 8 5 2 5x 8 2.1: Definition of a Derivative 14
YOUR TURN Use te limit process to find te derivative of f x = x 1 1 2 x 1 2.1: Definition of a Derivative 15
USING THE LIMIT PROCESS TO FIND TANGENT LINE A. Use te limit process to find te derivative B. Plug in x of te derivative to determine te slope C. Plug into Point-Slope form, y y 1 = m(x x 1 ) wit te given x and y coordinates, and slope D. AP Hint: For free-response AP Questions, do not need to simplify point-slope form 2.1: Definition of a Derivative 16
EXAMPLE 4 Find te slope of te grap of f x = x 2 8x + 9 at te point 3, 6 using te limit process. Ten, find te equation of te tangent line. f ' x f x f x lim 0 2 8 9 2 8 9 x x x x lim 0 2 2 2 x x x x x lim 0 2 8 8 9 8 9 2.1: Definition of a Derivative 17
EXAMPLE 4 Find te slope of te grap of f x = x 2 8x + 9 at te point 3, 6 using te limit process. Ten, find te equation of te tangent line. 2 2 2 x 2x 8x 8 9 x 8x 9 lim 0 2 2x 8 lim 0 lim x 0 2 8 2.1: Definition of a Derivative 18
EXAMPLE 4 Find te slope of te grap of f x = x 2 8x + 9 at te point 3, 6 using te limit process. Ten, find te equation of te tangent line. 2x 8 lim 0 lim 2 x 8 0 lim 2 3 0 8 0 2x 8 y x 2.1: Definition of a Derivative 19 6 2 3
EXAMPLE 5 Find te slope of te grap of f x = 3 at te point (3, 1) using te x limit process. Ten, find te equation of te tangent line. lim f x f x 0 3 3 lim x x 0 3 3 1 x x x x lim 0 2.1: Definition of a Derivative 20
EXAMPLE 5 Find te slope of te grap of f x = 3 at te point (3, 1) using te x limit process. Ten, find te equation of te tangent line. lim 0 3 1 x x 3 lim 0 x x f ' x 3 x 2 2.1: Definition of a Derivative 21
EXAMPLE 5 Find te slope of te grap of f x = 3 at te point (3, 1) using te x limit process. Ten, find te equation of te tangent line. f f f ' x 2 '3 '3 3 3 2 1 3 y y m x x 1 1 3 x 2.1: Definition of a Derivative 22
EXAMPLE 5 Find te slope of te grap of f x = 3 at te point (3, 1) using te x limit process. Ten, find te equation of te tangent line. y y m x x 1 1 1 y y 1 x x1 3 1 y 1 x 3 3 y 1 1 x 3 3 2.1: Definition of a Derivative 23
EXAMPLE 5 Find te slope of te grap of f x = 3 at te point (3, 1) using te x limit process. Ten, find te equation of te tangent line. 2.1: Definition of a Derivative 24
YOUR TURN Find te slope of te grap of f x = x 2 + 2x + 1 at te point ( 3, 4) using te limit process. Ten, find te equation of te tangent line. y 4 4 x 3 2.1: Definition of a Derivative 25
EXAMPLE 6 Given te function y = 1 ancored at te point 2, 1, find te slope x 2 of te secant line drawn troug te point wit x-coordinate 2 +. Use te expression to find te slope of te tangent to te grap of y = 1 x at te point 2, 1 2. 2 Questions: Wat is a secant line? Wat is a tangent line? 2 points tat make a line 1 point tat make a line 2.1: Definition of a Derivative 26
EXAMPLE 6 Given te function y = 1 ancored at te point 2, 1, find te slope x 2 of te secant line drawn troug te point wit x-coordinate 2 +. Use te expression to find te slope of te tangent to te grap of y = 1 x at te point 2, 1 2. lim f x f x 0 f 2 f 2 lim 0 1 1 lim 2 2 0 2.1: Definition of a Derivative 27
EXAMPLE 6 Given te function y = 1 ancored at te point 2, 1, find te slope x 2 of te secant line drawn troug te point wit x-coordinate 2 +. Use te expression to find te slope of te tangent to te grap of y = 1 x at te point 2, 1 2. 1 1 lim 2 2 0 lim 0 lim 1 2 2 2 1 2 2 2 0 Slope of Tangent Line: 2.1: Definition of a Derivative 28 1 4
AP MULTIPLE CHOICE PRACTICE QUESTION 1 (NON-CALCULATOR) If f 2 = 3 and f 2 = 1, find an equation of te tangent line wen x = 2 (A) y + 1 = 2(x 2) (B) y 3 = 2(x + 1) (C) y 2 = 3(x + 1) (D) y 3 = 1(x 2) 2.1: Definition of a Derivative 29
AP MULTIPLE CHOICE PRACTICE QUESTION 1 (NON-CALCULATOR) If f 2 = 3 and f 2 = 1, find an equation of te tangent line wen x = 2 Vocabulary Connections and Process Answer and Justifications Definition of a Derivative Function Notation lim 0 f x f x y y m x x 1 1 m f ' x 1 ' 1 3 ' 2 3 1 x 2 y y f x x y f x y f ' x D At 2,3, te slope at te point is -1. Terefore, te equation of te tangent line is y 3 1 x 2 2.1: Definition of a Derivative 30
ASSIGNMENT Page 103 5-11 odd, 17, 21-29 odd 2.1: Definition of a Derivative 31