Math Review. Daniel B. Rowe, Ph.D. Professor Department of Mathematics, Statistics, and Computer Science. Copyright 2018 by D.B.

Transcription

1 Math Review Daniel B. Rowe, Ph.D. Professor Department of Mathematics, Statistics, and Computer Science Copyright 2018 by 1

2 Outline Differentiation Definition Analytic Approach Numerical Approach Integration Definition Analytic Approach Numerical Approach Summary 2

3 Differentiation - Definition A function f is a rule that assigns to each element in a set A eactly one element, called f(), in a set B. f f() The set A is the domain of f and the set B is the range of f. 3

4 Differentiation - Definition f() 4

5 Differentiation - Definition f() Slope of Line: h=δ m l f ( a h) f ( a) h a a+h 5

6 Differentiation - Definition f() Slope of Line: h=δ m l f ( a h) f ( a) h a a+h a+h 6

7 Differentiation - Definition f() Slope of Line: h=δ m l f ( a h) f ( a) h a a+h a+h a+h 7

8 Differentiation - Definition f() Slope of Line: h=δ m l f ( a h) f ( a) h Slope of Tangent Line: m t lim h0 f ( a h) f ( a) h a a+h a+h a+h 8

9 Differentiation - Analytic Approach m t lim h0 f ( a h) f ( a) h df ( ) d f '( ) f() -2 f'() f '( ) 2 2 f ( ) ( 1)

10 Differentiation - Analytic Approach f() -2 f'() f '( ) 2 2 f ( ) ( 1)

11 Differentiation - Analytic Approach A function f () is differentiable at =a if there eists only one unique tangent line to the graph of f () at =a. Differentiable Not Differentiable f() f() a a 11

12 Differentiation - Analytic Approach d c 0 d d n n n1 d d sin( ) cos( ) d d d cos( ) sin( ) d e d e 12

13 Differentiation - Analytic Approach Let f '() and g'() eist. Linearity Rule: d c f 1 ( ) c g 2 ( ) c f 1 '( ) c g 2 '( ) d Product Rule: Quotient Rule: Chain Rule: d cf g c f g f g d ( ) ( ) '( ) ( ) ( ) '( ) d f ( ) f '( ) g ( ) ( ) '( ) c c f g d g( ) g ( ) 2 d cf ( g ( )) cf '( g ( )) g '( ) d f g ( ) 0 '( g( )) must eist 13

14 Differentiation - Analytic Approach Eamples: Linearity Rule: Product Rule: Quotient Rule: Chain Rule: d c c 2 c c 22 d d c c d sin( ) 1sin( ) cos( ) d sin( ) cos( )cos( ) sin( )[ sin( )] c c d cos( ) cos( ) 2 2 csec ( ) d 2 1/2 1 c 2 1/2 ( 1) c ( 1) 2 d 2 14

15 Differentiation - Numerical Approach f() Slope of Line: h small d f ( a h) f ( a) f( ) d h a a a+h 15

16 Differentiation - Numerical Approach f() Slope of Line: h small d f ( a 2 h) f ( a h) f( ) d h ah a+h a+2h 16

17 Differentiation - Numerical Approach f() Slope of Line: h small d f ( a 3 h) f ( a 2 h) f( ) d h a2h a+2h a+3h 17

18 Differentiation - Numerical Approach Analytic Numerical h= Analytic Numerical f() f ( ) ( 1) 2 f'() >>polyder([-1 2-1]) ans= f '( ) If h=0.01, then lines look same! 18

19 Differentiation - Numerical Approach 19

20 Maimization - Analytic Approach f() Given a function f(), if it has a maima a 0 b in the interval [a,b] at 0, then the slope of f() is zero at 0. This means that f '()=0 at = 0. 0 is a global maima if it is unique. 20

21 Maimization - Numerical Approach Define values of want to find ma of f() for, a to b. f() Assume complicated function we can t analytically differentiate. a b 21

22 Maimization - Numerical Approach Define values of want to find ma of f() for, a to b. Set Δ to evaluate f() at increments. f() Assume complicated function we can t analytically differentiate. 0 is value that makes f() largest. a b a+δ 0 better results for Δ smaller 22

23 Integration - Area Under Curve f() 23

24 Integration - Area Under Curve f() Area under curve between a and b. a b 24

25 Integration - Area Under Curve f() Divide into intervals: Δ small Δ =(b-a)/n i i 1 a= 0 Δ 1 2 b= n n 5 25

26 Integration - Area Under Curve f() Find midpoints: Δ small Δ =(b-a)/n i i 1 a / 2 ( i 1) i i 1,..., n a= b= n n 5 26

27 Integration - Area Under Curve f() Area by rectangles: Δ small Δ =(b-a)/n i i 1 a / 2 ( i 1) i n A f ( ) i1 i i1,..., n a= b= n n 5 27

28 Integration - Area Under Curve f() A lim n 0 i1 f ( ) i b a f ( ) d ( b a) / n a / 2 ( i 1) i a b 28

29 Integration - Analytic Approach >> polyint([-1 2 3]) ans = -1/ f() 2 f() d f ( ) 4 ( 1) f ( ) d 4 ( 1) /

30 Integration - Analytic Approach cd c C 1 n 1 n n1 d C sin( ) d cos( ) C cos( ) d sin( ) C e d e C 30

31 Integration - Analytic Approach Linearity: c f ( ) c g( ) d c f ( ) d c g( ) d By Parts: f ( ) g '( ) d f ( ) g( ) f '( ) g( ) d Assuming f '() and g'() eist. Other methods: Trigonometric Substitution Partial Fractions 31

32 Integration - Numerical Approach numerical n i1 f ( ) n=10, Δ=0.4 i n=50, Δ= [4 ( 1) ] d analytic

33 Integration - Numerical Approach analytic numerical numerical n i1 f ( ) n=10, Δ=0.4 numint= i n=50, Δ=0.08, numint= analytic numerical [4 ( 1) ] d analytic

34 Integration - Area Under Curve Trapezoidal Rule: f() Area by trapezoids: Δ =(b-a)/n a= 0 Δ 1 2 b= n 34

35 Integration - Area Under Curve Simpson s Rule: f() Area by parabola arcs: Δ =(b-a)/n a= 0 Δ 1 2 b= n 35

36 Summary Differentiation Definition Analytic Approach Numerical Approach Integration Definition Analytic Approach Numerical Approach 36

37 Homework 1: f 3 ( ), 1) Differentiate analytically and evaluate at a=-1 and b=1. 2) Differentiate by hand numerically with Δ=0.5. 3) Write a Matlab program for 2) then let Δ=1/100. 4) Integrate analytically and evaluate from a=-1 to b=1. 5) Integrate by hand numerically using n=4. 0.5,,, 0.75, 0.25,0.25, ) Write a Matlab program for 5) then let n=