The derivative of a function f is a new function defined by. f f (x + h) f (x)

Transcription

1 Derivatives

2 Definition Te erivative of a function f is a new function efine by f f (x + ) f (x) (x). 0 We will say tat a function f is ifferentiable over an interval (a, b) if if te erivative function f (x) at every point in (a, b).

3 Definition Te erivative of a function f is a new function efine by f f (x + ) f (x) (x). 0 We will say tat a function f is ifferentiable over an interval (a, b) if if te erivative function f (x) at every point in (a, b). Q. How is tis te same or ifferent from wat we were oing yesteray wit tangent lines?

4 Definition Te erivative of a function f is a new function efine by f f (x + ) f (x) (x). 0 We will say tat a function f is ifferentiable over an interval (a, b) if if te erivative function f (x) at every point in (a, b). Q. How is tis te same or ifferent from wat we were oing yesteray wit tangent lines? A. Yesteray, we were calculating erivatives at iniviual points, an getting numbers for answers. Toay, we ll calculate te erivative function, an get out answers wit variables in tem (o all te points at once).

5 Example: let f (x) = x 2 Derivatives at a point: If I first ask wat is f (2)?, I coul calculate f (2) 0 (2 + ) = 4.

6 Example: let f (x) = x 2 Derivatives at a point: If I first ask wat is f (2)?, I coul calculate f (2) 0 (2 + ) = 4. But ten, if I ask wat is f (3)? we ave to o it all over again.

7 Example: let f (x) = x 2 Derivatives at a point: If I first ask wat is f (2)?, I coul calculate f (2) 0 (2 + ) = 4. But ten, if I ask wat is f (3)? we ave to o it all over again. Toay s goal: Write own a function f (x) wic as all te erivatives-at-a-point collecte togeter.

8 If a is a number, (like 2 or 3) ten f (a) = f (a + ) f (a) lim }{{} 0 }{{ } gets ri a function of of te s a s an s

9 If a is a number, (like 2 or 3) ten f (a) = f (a + ) f (a) lim = number }{{} 0 }{{ } gets ri of te s a function of a s an s

10 If a is a number, (like 2 or 3) ten f (a) = But x is a variable, so f (x) = f (a + ) f (a) lim }{{} 0 }{{ } gets ri a function of of te s a s an s f (x + ) f (x) lim }{{} 0 }{{ } gets ri a function of of te s x s an s = number

11 If a is a number, (like 2 or 3) ten f (a) = But x is a variable, so f (a + ) f (a) lim }{{} 0 }{{ } gets ri a function of of te s a s an s = number f (x) = f (x + ) f (x) lim = function of x s }{{} 0 }{{ } gets ri of te s a function of x s an s

12 Starting simple Suppose we consier te piecewise linear function x x 1 f (x) = 1 1 < x < 3 x x

13 Starting simple Suppose we consier te piecewise linear function x x 1 f (x) = 1 1 < x < 3 x x

14 Starting simple Suppose we consier te piecewise linear function x x 1 f (x) = 1 1 < x < 3 x x

15 Starting simple Suppose we consier te piecewise linear function x x 1 f (x) = 1 1 < x < 3 x x

16 Starting simple Suppose we consier te piecewise linear function x x 1 f (x) = 1 1 < x < 3 x x

17 Starting simple Suppose we consier te piecewise linear function x x 1 f (x) = 1 1 < x < 3 x x Te erivative is: 1 x < 1 f (x) = 0 1 < x < < x

18 Anoter example Wat is te erivative of f (x) = x? Write own te piecewise function an sketc it on te grap

19 Anoter example Wat is te erivative of f (x) = x? Write own te piecewise function an sketc it on te grap f (x) = { 1 x < 0 1 x > 0

20 Lines In general, if m an b are constants, an f (x) = mx + b b m=slope f (x) = m te slope of te tangent line = slope of te line

21 Roug sape of te erivative if f (x) is increasing f (x) is positive! if f (x) is ecreasing f (x) is negative!

22 Matc em up! Here are graps of two functions an teir erivatives. Wic are wic?

23 Matc em up! Here are graps of two functions an teir erivatives. Wic are wic?

24 Matc em up! Here are graps of two functions an teir erivatives. Wic are wic? f (x) g(x) g (x) f (x)

25 A little more notation Back to wat lim 0 f (x + ) f (x) means:

26 A little more notation Back to wat lim 0 f (x + ) f (x) means: Rename = x an f (x + ) f (x) = y ( means cange )

27 A little more notation Back to wat lim 0 f (x + ) f (x) means: Rename = x an f (x + ) f (x) = y ( means cange ) Δy Δx So m = f (x+) f (x) = y x.

28 A little more notation Back to wat lim 0 f (x + ) f (x) means: Rename = x an f (x + ) f (x) = y ( means cange ) Δy Δx So m = f (x+) f (x) = y x.

29 A little more notation Back to wat lim 0 f (x + ) f (x) means: Rename = x an f (x + ) f (x) = y ( means cange ) Δy Δx So m = f (x+) f (x) = y x.

30 A little more notation Back to wat lim 0 f (x + ) f (x) means: Rename = x an f (x + ) f (x) = y ( means cange ) Δx Δy So m = f (x+) f (x) = y x.

31 A little more notation Back to wat lim 0 f (x + ) f (x) means: Rename = x an f (x + ) f (x) = y ( means cange ) Δx Δy So m = f (x+) f (x) = y x.

32 A little more notation Back to wat lim 0 f (x + ) f (x) means: Rename = x an f (x + ) f (x) = y ( means cange ) Δy Δx So m = f (x+) f (x) = y x.

33 A little more notation Back to wat lim 0 f (x + ) f (x) means: Rename = x an f (x + ) f (x) = y ( means cange ) As 0, x an y get infinitely small. Δy Δx So m = f (x+) f (x) = y x.

34 A little more notation Back to wat lim 0 f (x + ) f (x) means: Rename = x an f (x + ) f (x) = y ( means cange ) As 0, x an y get infinitely small. x x y y x y So m = f (x+) f (x) = y x. y y x x infinitesimals

35 Leibniz notation One way to write te erivative of f (x) versus x is f (x). Anoter way to write it is f (x) = f x = x f (x).

36 Leibniz notation One way to write te erivative of f (x) versus x is f (x). Anoter way to write it is f (x) = f x = x f (x). Derivatives at a point: f (a) means te erivative of f (x) evaluate at a. Anoter way to write it is f (a) = f x = x=a x x=a f (x)

37 Leibniz notation One way to write te erivative of f (x) versus x is f (x). Anoter way to write it is f (x) = f x = x f (x). Derivatives at a point: f (a) means te erivative of f (x) evaluate at a. Anoter way to write it is f (a) = f x = x=a x x=a f (x) Example: We can write te erivative of x 2 as x x 2 an te erivative of x 2 at x = 5 as x x 2 x=5

38 Go to work: builing our first erivative rule. Example 1: Wat is te erivative of x 2?

39 Go to work: builing our first erivative rule. Example 1: Wat is te erivative of x 2? x x 2 f (x + ) f (x) (x + ) 2 (x) 2 0 0

40 Go to work: builing our first erivative rule. Example 1: Wat is te erivative of x 2? x x 2 f (x + ) f (x) (x + ) 2 (x) x 2 + 2x + 2 x 2 0

41 Go to work: builing our first erivative rule. Example 1: Wat is te erivative of x 2? x x 2 f (x + ) f (x) (x + ) 2 (x) x 2 + 2x + 2 x 2 0 2x + 2

42 Go to work: builing our first erivative rule. Example 1: Wat is te erivative of x 2? x x 2 f (x + ) f (x) (x + ) 2 (x) x 2 + 2x + 2 x 2 0 2x + 0 2x + 2

43 Go to work: builing our first erivative rule. Example 1: Wat is te erivative of x 2? x x 2 f (x + ) f (x) (x + ) 2 (x) x 2 + 2x + 2 x 2 0 2x + = 2x 0 2x + 2

44 Go to work: builing our first erivative rule. Example 1: Wat is te erivative of x 2? x x 2 f (x + ) f (x) (x + ) 2 (x) x 2 + 2x + 2 x 2 0 2x + 2 2x + = 2x (so 0 x x 2 = 2 5) x=5

45 Go to work: builing our first erivative rule. Example 1: Wat is te erivative of x 2? x x 2 f (x + ) f (x) (x + ) 2 (x) x 2 + 2x + 2 x 2 0 2x + 2 2x + = 2x (so 0 x x 2 = 2 5) x=5 By taking limits, fill in te rest of te table: f (x) 1 x x 2 x 3 1 x 1 x 2 x 3 x f (x) 2x Hints: For 1 x 2, fin a common enominator, an ten expan. For 3 x, try multiplying an iviing by ( 3 x + ) 2 + ( 3 x + )( 3 x) + ( 3 x) 2.

46 Go to work: builing our first erivative rule. Example 1: Wat is te erivative of x 2? x x 2 f (x + ) f (x) (x + ) 2 (x) x 2 + 2x + 2 x 2 0 2x + 2 2x + = 2x (so 0 x x 2 = 2 5) x=5 By taking limits, fill in te rest of te table: f (x) 1 x x 2 x 3 1 x 1 x 2 x 3 x f (x) 0 1 2x 3x 2 1 x 2 2 x x 1 3( 3 x) 2 Hints: For 1 x 2, fin a common enominator, an ten expan. For 3 x, try multiplying an iviing by ( 3 x + ) 2 + ( 3 x + )( 3 x) + ( 3 x) 2.

47 f (x) f (x) x 0 = 1 0 x 1 = x 1 x 2 2x x 3 3x 2 1 x = x 1 x 2 1 = x 2 2x 3 x 2 x = x 1/2 (1/2)x 1/2 3 x = x 1/3 (1/3)x 2/3

48 f (x) f (x) x 0 = 1 0 x 1 = x 1 x 2 2x x 3 3x 2 1 x = x 1 x 2 1 = x 2 2x 3 x 2 x = x 1/2 (1/2)x 1/2 3 x = x 1/3 (1/3)x 2/3 Power rule: x x a = ax a 1

49 Use te power rule to take consecutive erivatives of x 5/2 : x 5/2 x x x x

50 Use te power rule to take consecutive erivatives of x 5/2 : x 5/2 x 5 2 x 3/2 x x x

51 Use te power rule to take consecutive erivatives of x 5/2 : x 5/2 x 5 2 x 3/2 x x 1/2 x x 1/2 x ( 1 2) x 3/2

52 Use te power rule to take consecutive erivatives of x 5/2 : x 5/2 x 5 2 x 3/2 1 st erivative x x 1/2 2 n erivative x x 1/2 3 r erivative x ( 1 2) x 3/2 4 t erivative

53 Use te power rule to take consecutive erivatives of x 5/2 : x 5/2 x 5 2 x 3/2 1 st erivative = f (x) = x x 2 x x 1/2 2 n erivative = f (x) = 2 x 2 x 2 x x 1/2 3 r erivative = f (3) (x) = 3 x 3 x 2 x ( 1 2) x 3/2 4 t erivative = f (4) (x) = 4 x 4 x 2

54 Use te power rule to take consecutive erivatives of x 5/2 : x 5/2 x 5 2 x 3/2 1 st erivative = f (x) = x x 2 x x 1/2 2 n erivative = f (x) = 2 x 2 x 2 x x 1/2 3 r erivative = f (3) (x) = 3 x 3 x 2 x ( 1 2) x 3/2 4 t erivative = f (4) (x) = 4 x 4 x 2 Definition: Te n t erivative of f (x) is x x... f (x) = n }{{ x} x n f (x) = f (n) (x). n