SECTION THE TANGENT AND VELOCITY PROBLEMS. 2 y= at the points with x-coordinates

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1 Tangent Problem SECTION.1 - THE TANGENT AND VELOCITY PROBLEMS Definition (Secant Line & Tangent Line) Eample 1 Find the equation of the tangent lines to the curve = 0 and = 1. y= at the points with -coordinates 1 3 1

2 Eample Susie and Johnny are running a lemonade stand. The following table gives a running total of money they made as added each hour. Estimate the slope of the tangent line to the graph at t= 3. # of hours total $ $0 $4.50 $10.5 $18.75 $6.50 $33.75

3 Velocity Problem Definition (Average Velocity & Instantaneous Velocity) Eample 3 A baseball is rolled down a ramp. The distance the ball has traveled, measured in feet, is t modeled by the function d( t) = + t+ 1 where time, t, is measured in seconds. Find the 4 average velocity of the ball over the following time periods. [ 1, 3] [ 1, ] [ 1, 1.5] [ 1, 1.1] Find the instantaneous velocity of the ball when t= 1. 3

4 SECTION. THE LIMIT OF A FUNCTION Definition (Limit) Eample 1 Find each limit or eplain why it does not eist. lim f( )= lim f 4 ( )= lim f( )= lim f 0 ( )= 4 Eample lim Estimate ( ) 6 4

5 Definition (One-Sided Limit) Eample 3 Let f( ) = + 1 if if 0 3 > 3. Find lim f( ) and lim f( ) Definition (Infinite Limit) 5

6 Eample 4 1 Find lim+ 0 and lim 0 1. Eample 5 Find the following limits. lim 0 csc = lim 1 ( ) = Definition (Vertical Asymptote) (See eamples 4 & 5.) 6

7 Eample 6 Find the vertical asymptotes of the function f( ) =. 3 7

8 SECTION.3 CALCULATING LIMITS USING LIMIT LAWS Limit Laws Suppose c is a constant and the limits lim f( a ) and lim g( a ) 1.) lim ( f( ) + g( ) ) = lim f( ) + lim g( ) a a a.) lim ( f( ) g( ) ) = lim f( ) lim g( ) a a 3.) lim cf( ) = c lim f( ) a a a 4.) lim ( f( ) g( ) ) = lim f( ) * lim g( ) 5.) a lim a f g a a ( ) lim f( ) a = if lim g( ) 0 ( ) lim g( ) a a ( ) n n 6.) lim ( f( ) ) lim f( ) a = where n is a positive integer a eist. 7.) lim c = c a 8.) lim = a a 9.) lim n = a a n where n is a positive integer n 10.) lim = a n a where n is a positive integer. If n is even, we assume that a> ) lim n f( ) = n lim f( ) where n is a positive integer. If n is even, we assume that ( ) 0 a a f >. 8

9 Eample 1 Calculate the following limits. lim ( ) lim lim lim

10 Direct Substitution Property Eample Find the following limits. lim ( )= 9 lim = lim h 0 ( 3+ h) h 18 = Theorem 1 10

11 Eample 3 Determine if the following limits eist if 4 lim f( 4 ) where f( ) = + 1 if > 4 lim Theorem Squeeze Theorem 11

12 Eample 4 Use the squeeze theorem to find the following limits. 4 + lim sin 0 3 lim 1 ( ) cos 1

13 SECTION.4 THE PRECISE DEFINITION OF A LIMIT Definition (Limit Precise Definition) Definition (One Sided Limit Precise Definition) Definition (Infinite Limits Precise Definition) 13

14 Eample 1 Use the precise definition to prove the following statements. lim ( 7 7) = 8 5 lim ( 3) = lim 1 3 ( ) = 14

15 Eample Prove the following statement by using the precise definition of an infinite limit. lim =

16 Definition (Continuous at a number) SECTION.5 CONTINUITY Eample 1 Find all of the discontinuities of f( ) = ( 3) 3 if if if < 0 0 <

17 Definition (Discontinuity) Definition (Continuous from the left/right) Eample For each of the numbers, 3, and 4, determine whether g is continuous from the left, if 0 if < 3 continuous from the right, or continuous at the number where g( ) =. 4 if 3< < 4 π if 4 17

18 Definition (Continuous On An Interval) Eample 3 Show that g( ) 9 = is continuous on the interval [ 3, 3]. 9 18

19 Theorem 4 Theorem 5 Eample 4 f = csc is continuous. Find where ( ) 19

20 Theorem 7 Theorem 8 Eample 5 Show that f ( ) = sec is continuous at =. π Theorem 9 0

21 Eample 6 Show that the funtions are continuous on their domain. sin h = e ( ) g( ) = 9 4 1

22 Theorem 10 (Intermediate Value Theorem) Eample 7 Use the intermediate value theorem to show that there is a root of the equation in the given interval. 3 + = 0, 1 + on ( ) e = on ( 0, 1)

23 SECTION.6 LIMITS AT INFINITY: HORIZONTAL LIMITS Definition (Limit of a function as approaches positive/negative infinity) Definition (Horizontal Asymptote) 3

24 Eample 1 Find the horizontal asymptotes of the following functions. cos y= y= Theorem 5 Notation 4

25 Eample Find each of the following limits. 1+ lim lim + 3 lim 9 6 lim e 3 5

26 Definition (Precise Definitions of Limits at Infinity) 6

27 Eample For the limit lim 0 =, illustrate the precise definition by finding values of N that + 5 correspond to ε = Eample 4 lim + 1 = Prove that ( ) by using the precise definition. 7

28 Definition (Tangent Line) SECTION.7 DERIVATIVES AND RATES OF CHANGE Eample 1 Find the equation of the tangent line to the curve y = 9 at the point (, 1). Eample Find the equation of the tangent line to the curve ( ) 3 f = that goes through the point (, 0) 6. 8

29 Definition (Average vs. Instantaneous Velocity) Eample 3 If a ball is thrown into the air with a velocity of 40ft/s, its height in feet after t seconds is given by y= 40t 16t. Find the velocity of the ball after seconds. Definition (Instantaneous Rates of Change) Eample 4 According to Boyle s Law, if the temperature of a confined gas is held fied, then the product of the pressure, P, and the volume, V, is a constant. Suppose that, for a certain gas, PV = 800, where P is measured in pounds per square inch and V is measured in cubic inches. Find the instantaneous rate of change of P when V is 00in 3. 9

30 Definition (Derivative at a Number) Note: Eample 5 The graph of the function f is given below. Put the following numbers in order from smallest to largest. f ( ) f ( ) f ( 1) f ( 1) f ( 3) f ( 3) 30

31 Eample 6 Use the definition of derivative to find f ( ), where f( ) 3 =. Eample 7 Find a function and a number b such that f ( b) = lim h 0 ( + h) 6 h

32 SECTION.8 THE DERIVATIVE AS A FUNCTION Notation Eample 1 Find the derivative of f ( ) 1 = and compare the graphs of f and its derivative. Eample The cost of repaying a student loan at an interest rate of r% per year is f( r) What is the meaning of f ( r)? C=. What are its units? mean? What does the statement f ( 10) = 100 Is f ( r) always positive or does it change sign? 3

33 Definition (Differentiable) Eample 3 Where is the function ( ) f = differentiable? 33

34 Theorem 4 Eample 4 Show that each of the following functions is continuous at a by showing the functions are differentiable at a. 1 f( ) =, a= g ( ) + 1 =, a= 4 34

35 Note: A function f is not differentiable at a point a if: We will come back to higher derivatives in chapter 3. 35

36 SECTION 3.1 DERIVATIVES OF POLYNOMIALS AND EXPONENTIAL FUNCTIONS Notation Derivatives Rules Definition (e) 36

37 Eample 1 Find the derivatives of the following functions. f = + + ( ) 5 g ( ) = ( ) h = + 4 k 4 ( ) = e s 1e 3 ( ) = + ( ) 5 37

38 SECTION 3. PRODUCT AND QUOTIENT RULES Product Rule Quotient Rule Table of Derivative Rules 38

39 Eample 1 Find the derivatives of the following functions. f = e + 1 ( ) ( ) g ( ) = e + 1 h ( ) = ( ) ( ) k = ( ) = ( e + 4+ )( e ) s + r e ( ) = 3 e

40 SECTION 3.3 DERIVATIVES OF TRIG FUNCTIONS Important Limits Eample 1 Find the following limits. cos( 4) 1 lim = 0 6 lim 0 ( + 4) csc = Derivatives of Sin & Cos 40

41 Eample Find the derivatives of the following functions. f = tan ( ) ( ) sec g = ( ) csc h = ( ) cot k = ( ) = 3cos r + s 4 ( ) = cos ( ) csc tan + e = sec p 41

42 Derivatives of Trig Functions 4

43 SECTION 3.4 CHAIN RULE Chain Rule Power Rule Combined With The Chain Rule Derivative Rule Eample 1 f Let f ( ) and g ( ) be differentiable functions and let ( ) ( ) h =. Find ( ) g( ) quotient rule. h without using the 43

44 Eample Find the derivatives of the following functions. f = sin cos ( ) ( ) sin ( ) e g = h sin ( ) ( 4 + ) = + e

45 k 1 ( ) = e ( ) r = sin sin( ) s ( ) = tan( sec( e ) 45

46 SECTION.8 HIGHER DERIVATIVES (CONTINUED) Notation Position, Velocity, and Acceleration Eample 1 3 A particle moves on a vertical line so that its coordinate at time t is y= t 1t+ 3, t 0. Find the velocity and acceleration functions. When is the particle moving upward and when is it moving downward? Find the distance that the particle travels in the first three seconds. 46

47 Eample Let f ( t) = 4t+ 1, g ( ) = sin, h ( ) = e, and k( ). Find f ( ) 1 =.. Find g ( ) Find ( h n ) ( ). Find ( k n ) ( ). 47

48 SECTION 3.5 IMPLICIT DIFFERENTIATION Method of Implicit Differentiation Eample 1 1 dy Let y= cos. Find. d Derivatives of Inverse Trig Functions 48

49 Eample dy For each of the following equations, find. d + 4y+ y = 13 ( y) y cos y+ sin = e y = y 1 sin ( y) = y 49

50 SECTION 3.6 DERIVATIVES OF LOGARITHMIC FUNCTIONS Derivatives Method of Logarithmic Differentiation Eample 1 Find the derivatives of the following functions. f = ln csc ( ) ( ) 4 ( t) tln( t ) g = ( ) = log ( 1 ) h

51 Eample dy For each of the following equations, find using logarithmic differentiation. d e y= e sin y= y= tan ( cos ) y=

52 SECTION 3.7 RATES OF CHANGE IN THE NATURAL & SOCIAL SCIENCES Definition (Instantaneous Rate of Change) Do #4, #18, & #30 from the book. 5

53 SECTION 3.9 RELATED RATES Strategy Do #6, #16, & #44 from the book. 53

54 SECTION 3.10 LINEAR APPROXIMATIONS AND DIFFERENTIALS Definition (Linearization & Linear Approimation of f at a) Eample 1 Find the linear approimation of f( ) = 5 near 3. For what values of is the linear approimation accurate to within.1? 54

55 Eample 3 Find the linearization of ( ) give an approimate value for f = 1+ 3 at a= 0. Use the corresponding linear approimation to Eample 3 Use a linear approimation to estimate

56 Definition (Differential) Definition (Ma Error, Relative Error, % Error) Eample 4 3 Let y= + 1, =, and d= 0. = d.. Compare y and dy for the given values of and 56

57 Eample 5 A window has the shape of a square surmounted by a semicircle. The base of the window is measured as having width 60cm with a possible error in measurement of 0.1cm. Use differentials to estimate the maimum error possible, relative error, and percentage error in computing the area of the window. 57

58 SECTION 4.1 MAXIMUM & MINIMUM VALUES Definition (Absolute Ma/Min & Local Ma/Min) Eample 1 Find the absolute ma, absolute min, local ma(s), local min(s), ma value, and min value of the following function. 58

59 The Etreme Value Theorem Fermat s Theorem Eample Find the absolute ma/min of the following functions. f = + ( ) ( ) 1 g ( ) = 1 59

60 Definition (Critical Number) The Closed Interval Method 60

61 Eample 3 Find the etreme values of the functions on the given intervals. f 0, 4 ( ) = on [ ] ( ) ( ) 3 g + = on [, 1] ln h ( ) = on [ 1, 3] 61

62 SECTION 4. THE MEAN VALUE THEOREM Rolle s Theorem The Mean Value Theorem Theorem 5 Corollary 7 6

63 Eample 1 Verify that the functions satisfy the hypotheses of Rolle s Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle s Theorem. 3 f 3 + 0, 3 ( ) 7 = on [ ] π 3π g ( ) = ln( sin) on, 4 4 Verify that the functions satisfy the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers c that satisfy the conclusion of the Mean Value Theorem. h 3+ 0, ( ) 1 = on [ ] k ( ) = on [ 1, 1]

64 Eample Show that the equation + + 1= 0 has eactly one real root. 3 Show that the equation = 0 has at most one real root. 64

65 SECTION 4.3 HOW DERIVATIVES AFFECT THE SHAPE OF THE GRAPH Increasing/Decreasing Test First Derivative Test 65

66 Eample 1 Find where each of the following functions is increasing/decreasing and the local maima/minima. 3 f = ( ) 4 g ( ) = + 1 h ( ) = e 66

67 Definition (Concave Up & Concave Down) Concavity Test Definition (Inflection Point) Second Derivative Test 67

68 Eample Find the intervals of concavity and inflection points of the following functions. 3 f = ( ) 4 g ( ) = + 1 h ( ) = e 68

69 SECTION 4.4 INDETERMINATE FORMS AND L HOSPITAL S RULE Definition (Indeterminate Forms) L Hospital s Rule 69

70 Eample 1 Find the limits. 1 cos lim = 0 + lim e = lim e 1 4 = lim 3 e = 70

71 Definition (Indeterminate Products & Differences) Strategy Definition (Indeterminate Powers) Strategy 71

72 Eample Find the limits. 1 lim ln = lim ln = + 0 lim = lim π cos ( tan ) = 7

73 SECTION 4.5 SUMMARY OF CURVE SKETCHING Checklist Of Important Info Needed To Sketch A Curve 73

74 Eample 1 Sketch the following curve. 4 3 y =

75 Eample Sketch the following curve. y =

76 Eample 3 Sketch the following curve. π π y = 4 tan, < < 76

77 SECTION 4.7 OPTIMIZATION PROBLEMS Strategy First Derivative Test For Absolute Etreme Values Do Problems 4, 1, 18, & 6 in class. 77

78 What is Newton s Method used for? SECTION 4.8 NEWTON S METHOD How does Newton s Method work? 78

79 Eample Use Newton s Method to find the root of the equation = 0 in the interval 1, correct to si decimal places. [ ] Eample Use Newton s Method to find the absolute maimum value of the function f( t) = cos t+ t t correct to eight decimal places. 79

80 SECTION 4.9 ANTIDERIVATIVES Definition (Antiderivative) Theorem 1. Eample 1 Find the general antiderivative for the following functions. f = ( ) 3 g ( ) = 3e csc cot h 3 ( ) = 5 80

81 Eample For each of the following, find the particular antiderivitive. f t = t 3 sin with f ( 0) = 5 ( ) t g 3 ( ) = with g ( 0) = and g ( 1) = 0 81

82 TABLE OF ANTIDERIVATIVES 8

83 SECTION 5.1 AREAS AND DISTANCES The Area Problem Eample 1 Estimate the area under the curve y = from 0 to. 83

84 Definition (Area) Notation Eample Find the area under the curve y = from 0 to. Find the area under the curve y = from 1 to 3. 84

85 The Distance Problem Eample 3 A radar gun was used to record the speed of a runner. Use the data to estimate the distance the runner covered during those 5 seconds. time (s) velocity (ft/s)

86 Definition (Definite Integral) SECTION 5. THE DEFINITE INTEGRAL Eample 1 Epress lim n sin as a definite integral on the interval [ 0, π]. n i= 1 i Theorem 3 86

87 Theorem 4 Eample 3 Evaluate the integral by interpreting it in terms of area; ( ) 3 9 d. Equations Involving Sums of Powers of Positive Integers (See Appendi E) 87

88 Eample 3 Evaluate the Riemann sum for f( ) a= 0, b =, and 8 3 n=, then evaluate ( 1) = by taking sample points to be right endpoints and + d

89 Eample 4 Evaluate the Riemann sum for f( ) a= 1, b= 3, and 6 3 n=, then evaluate ( ) = by taking sample points to be right endpoints and d. 1 89

90 The Midpoint Rule Eample 5 1 Use the Midpoint Rule with n= 5 to approimate d. 0 Properties of the Definite Integral 90

91 Eample 6 3 Evaluate the following integral ( + 9 ) 3 4 d. 91

92 Eample 1 SECTION 5.3 THE FUNDAMENTAL THEOREM OF CALCULUS Let g ( X) = f( t) dt where f( ) 0 Find the values of g ( 0), ( 1) graph of g ( ). 3 ( 3) = if if if if 0 < < < 8 g, g ( ), g ( ), g ( ), g ( 5), g ( 6), g ( 7), and ( 8) g then sketch a rough 9

93 The Fundamental Theorem of Calculus (Part 1) Eample Find the derivative of each of the following functions. g ( ) = 1+ 1 t 4 dt h π ( ) = tan( t ) dt k ( ) = 0 t dt 3 1+ t The Fundamental Theorem of Calculus (Part ) 93

94 Eample 3 Evaluate the following integrals. 4 ( ) + d 9 0 e d 3π ( t 3sint ) π dt The Fundamental Theorem of Calculus 94

95 SECTION 5.4 INDEFINITE INTEGRALS AND THE NET CHANGE THEOREM Definition (Indefinite Integral) Table of Indefinite Integrals 95

96 Eample 1 Evaluate the following integrals d ( cost sec t) + dt d The Net Change Theorem Eample = t t, where v is measured in meters per second. Find the displacement and the distance traveled by the particle during the 0, 5. A particle moves along a line with a velocity function of v( t) time interval [ ] 96

97 SECTION 5.5 THE SUBSTITUTION RULE The Substitution Rule (U-Substitution) Eample 1 Evaluate the following integrals. + d + 4 sin cos ( cos ) d ( ) tan ln cos d 97

98 Substitution Rule for Definite Integrals Eample Evaluate the following integrals. 1 ( y 1) y dy 1 0 v cos 3 ( v ) dv π 4 3 ( 1+ tant) sec t dt 0 98

99 Integrals of Symmetric Functions Eample 3 Evaluate the following integrals. ( 14 ) d 7 sin d 99