Section 2.1 The Definition of the Derivative. We are interested in finding the slope of the tangent line at a specific point.

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1 Popper 6: Review of skills: Find tis difference quotient. f ( x ) f ( x) if f ( x) x Answer coices given in audio on te video. Section.1 Te Definition of te Derivative We are interested in finding te slope of te tangent line at a specific point. We need a way to find te slope of te tangent line analytically for every problem tat will be exact every time. We can draw a secant line across te curve, ten take te coordinates of te two points on te curve, P and Q, and use te slope formula to approximate te slope of te tangent line. Now suppose we move point Q closer to point P. Wen we do tis, we ll get a better approximation of te slope of te tangent line. Section.1 Te Definition of a Derivative 1

2 Wen we continue to move point Q even closer to point P, we get an even better approximation. We are letting te distance between P and Q get smaller and smaller. Now let s give tese two points names. We ll express tem as ordered pairs. Now we ll apply te slope formula to tese two points. m f ( x ) f ( x) f ( x ) ( x ) x f ( x) Tis expression is called a difference quotient also called te average rate of cange. Popper 7: Te difference quotient gives us te slope of te : A. Tangent Line B. Secant Line C. Neiter D. Bot Popper 8: Te represents a cange in te: A. X values B. Y values Section.1 Te Definition of a Derivative

3 Te last ting tat we want to do is to let te distance between P and Q get arbitrarily small, so we ll take a limit. Tis gives us te definition of te slope of te tangent line. Te slope of te tangent line to te grap of f at te point P( x, f ( x)) is given by f ( x ) f ( x) lim 0 provided te limit exists. We find te instantaneous rate of cange wen we take te limit of te difference quotient. Te derivative of f wit respect to x is te function f ' (read f prime ) defined by f ( x) f( x) f '( x) lim. Te domain of f '( x) is te set of all x for wic te limit exists. 0 d dy Note tat: f( x) y' dx dx Section.1 Te Definition of a Derivative 3

4 Example 1: Use te limit definition of te derivative to find f '( x ) for f ( x) 3x x. Recall: f '( x) lim 0 f ( x) f( x) Ten find f '( c ) wen c = 1. Example : Use te limit definition of te derivative to find f '( x ) for f( x). x 1 Recall: f '( x) f x f x lim 0 ( ) ( ) Section.1 Te Definition of a Derivative 4

5 Try tis one: Find te derivative of f( x) x f ( x) f( x) Recall: f '( x) lim 0 Section.1 Te Definition of a Derivative 5

6 Since te derivative is a formula for finding te slope of a tangent line, ten given a certain x- value, we can find its slope AND its equation. We ll may use te point-slope equation of a line: y y1 m( x x1) Example 3: Find te slope of te line tangent to te function f ( x) x x at te point (, 4). Recall: f '( x) lim 0 f ( x) f( x) Section.1 Te Definition of a Derivative 6

7 Example 4: Given [ 5 (5 )] (5 5) lim, give te function f and te value c 0 3 tan 6 3 Try tis one: Given lim, give te function f and te value c. 0 Example 5: If f(1) = 5 and f '(1) 6, give te equation of te tangent line at x = 1. Differentiability A function f is differentiable at c if lim 0 f ( c ) f( c) exists. For example, given te grap of a function f below, is it differentiable at x = 0? Wy? Section.1 Te Definition of a Derivative 7

8 If te limit fails to exist, we say tat te function is not differentiable at c. Fact: If f is differentiable at c, ten it is continuous at c. However, te converse of tis statement is not always true. A function as to be continuous at a point c to POSSIBLY be differentiable. For example, take f ( x) x. Tis function is continuous everywere, but it s not differentiable at x 0, since te one-sided limits do not agree tere. A function is also not differentiable at cusps nor vertical tangents. We ll study tese later. Example 6: Is tis function differentiable? x, if x > 1 f( x) 3 x 1, if x 1 To begin we must ceck to see if it is continuous at x = 1. After cecking wit te 3-step metod, yes, it s continuous at x = 1. f (1 ) f(1) We know tat f '(1) lim. Since f is defined by different formulas on different 0 sides of 1, we need to ceck te one-sided limits ere. 3 3 f(1 ) f(1) (1 ) 1 (1 1) lim lim f(1 ) f(1) (1 ) (1) lim lim 0 0 Section.1 Te Definition of a Derivative 8

9 Bx C, if x > 1 Try tis one: Given f( x). Determine te values of te constants B and C 3 8 x, if x 1 so tat f is differentiable. Hint: You must first ceck to see if it s continuous at x = 1. Section.1 Te Definition of a Derivative 9