Chapter 2 Differentiation Course Number Section 2.1 The Derivative an the Tangent Line Problem Objective: In this lesson you learne how to fin the erivative of a function using the limit efinition an unerstan the relationship between ifferentiability an continuity. Instructor Date Important Vocabulary Define each term or concept. Differentiation aaa aaaaaaa aa aaaaaaa aaa aaaaaaaaaa aa a aaaaaaaaa Differentiable a aaaaaaaa aa aaaaaaaaaaaaaa aa a aa aaa aaaaaaaaaa aaaaaa aa aa I. The Tangent Line Problem (Pages 96 99) Essentially, the problem of fining the tangent line at a point P boils own to aaa aaaaaaa aa aaaaaaa aaa aaaaa aa aaa aaaaaaa aaaa aa aaaaa a. You can approimate this slope using a aaaaaa aaaa through the point of tangency (c, f(c)) an a secon point on the curve (c +, f(c + )). The slope of the secant line through these two aaa a aaa a points is m sec aaaaa. aaa How to fin the slope of the tangent line to a curve at a point The right sie of this equation for the slope of a secant line is calle a aaaaaaaaaa aaaaaaaa. The enominator is the aaaaaa aa a, an the numerator y f ( c ) f ( c ) is the aaaaaa aa a. The beauty of this proceure is that you can obtain more an more accurate approimations of the slope of the tangent line by aaaaaaaa aaaaaa aaaaaa aaa aaaaaa aa aaa aaaaa aa aaaaaaaa. If f is efine on an open interval containing c, an if the limit lim y f ( c ) f ( c) lim 0 0 m eists, then the line passing 31
32 Chapter 2 Differentiation through (c, f(c)) with slope m is aaaaaa aaaa aa aaa aaaaa aa a aa aaa aaaaa aaa aaaaa. The slope of the tangent line to the graph of f at the point (c, f(c)) is also calle aa a a a. aaa aaaaa aa aaa aaaaa aa a Eample 1: Fin the slope of the graph of ( ) 9 2 f at the point (4, 7). a aaa Eample 2: Fin the slope of the graph of the point ( 1, 1). a f ( ) 2 3 2 at The efinition of a tangent line to a curve oes not cover the possibility of a vertical tangent line. If f is continuous at c an lim ( ) ( ) f c f c 0 or lim ( ) ( ) f c f c 0 vertical line = c passing through (c, f(c)) is aaaaaaa aaaa to the graph of f., the a aaaaaaaa II. The Derivative of a Function (Pages 99 101) The aaaaaaaaaa aa a aa a is given by f '( ) lim ( ) ( ), provie the limit eists. For all f f 0 for which this limit eists, f is a aaaaaaa aa a. How to use the limit efinition to fin the erivative of a function The erivative of a function of gives the aaaaa aa aaa aaaaaaa aaaa to the graph of f at the point (, f()), provie that the graph has a tangent line at this point. A function is ifferentiable on an open interval (a, b) if a aa aaaaaaaaaaaaaa aa aaaaaa aa aaa aaaaaaaa.
Section 2.1 The Derivative an the Tangent Line Problem 33 Eample 3: Fin the erivative of a aaaa a aa 2 f ( t) 4t 5. III. Differentiability an Continuity (Pages 101 103) Name some situations in which a function will not be ifferentiable at a point. a aaaaa aaaaaa a aaaaaaaa aaaaaaa aaaa aa a aaaaa aaaa a aaaaa aaaaa How to unerstan the relationship between ifferentiability an continuity If a function f is ifferentiable at = c, then aaaaaaaaaa aa a a a. a aa Complete the following statements. 1. If a function is ifferentiable at = c, then it is continuous at = c. So, ifferentiability aaaaaa continuity. 2. It is possible for a function to be continuous at = c an not be ifferentiable at = c. So, continuity aaaa aaa aaaaa ifferentiability.
34 Chapter 2 Differentiation Aitional notes y y y y y y Homework Assignment Page(s) Eercises
Section 2.2 Basic Differentiation an Rates of Change 35 Section 2.2 Basic Differentiation an Rates of Change Objective: In this lesson you learne how to fin the erivative of a function using basic ifferentiation rules. Course Number Instructor Date I. The Constant Rule (Page 107) The erivative of a constant function is s. If c is a real number, then [] c II. The Power Rule (Pages 108 109) s. The Power Rule states that if n is a rational number, then the n function f ( ) is ifferentiable an n ss = 0, n must be a number such that. For f to be ifferentiable at ss s s. n 1 is s ss ss How to fin the erivative of a function using the Constant Rule How to fin the erivative of a function using the Power Rule Also, s. 1 Eample 1: Fin the erivative of the function f( ) 3. s s Eample 2: Fin the slope of the graph of ss 5 f ( ) at = 2. III. The Constant Multiple Rule (Pages 110 111) The Constant Multiple Rule states that if f is a ifferentiable function an c is a real number then cf is also ifferentiable an cf ( ) ss s. How to fin the erivative of a function using the Constant Multiple Rule Informally, the Constant Multiple Rule states that ss ss ss ss s ss ss ss.
36 Chapter 2 Differentiation Eample 3: Fin the erivative of f( ) 2 5 The Constant Multiple Rule an the Power Rule can be combine into one rule. The combination rule is c n. 2 Eample 4: Fin the erivative of y 5 5 s s IV. The Sum an Difference Rules (Page 111) The Sum an Difference Rules of Differentiation state that the sum (or ifference) of two ifferentiable functions f an g is itself ifferentiable. Moreover, the erivative of f g (or How to fin the erivative of a function using the Sum an Difference Rules f g) is the sum (or ifference) of the erivatives of f an g. That is, f ( ) g ( ) s s s ss s an f ( ) g ( ) s s s ss s Eample 5: Fin the erivative of ss s s ss ss 3 2 f ( ) 2 4 3 1 V. Derivatives of Sine an Cosine Functions (Page 112) sin s s cos s s s How to fin the erivative of the sine function an of the cosine function Eample 6: Differentiate the function ss s ss s s s 2 y 2cos.
Section 2.2 Basic Differentiation an Rates of Change 37 VI. Rates of Change (Pages 113 114) The erivative can also be use to etermine s ss ss ss s s ss s. How to use erivatives to fin rates of change Give some eamples of real-life applications of rates of change. s s ss s s The function s that gives the position (relative to the origin) of an object as a function of time t is calle a ss ss. The average velocity of an object that is moving in a straight line is foun as follows. Average velocity = ss = ss ss Eample 7: If a ball is roppe from the top of a builing that is 200 feet tall, an air resistance is neglecte, the height s (in feet) of the ball at time t (in secons) is 2 given by s 16t 200. Fin the average velocity of the object over the interval [1, 3]. s ss s If s s() t is the position function for an object moving along a straight line, the (instantaneous) velocity of the object at time t is s s ss vt () = ss. ss In other wors, the velocity function is the s ss the position function. Velocity can be ss ss ss. The ss of an object is the absolute value of its velocity. Spee cannot be ss.
38 Chapter 2 Differentiation Eample 8: If a ball is roppe from the top of a builing that is 200 feet tall, an air resistance is neglecte, the height s (in feet) of the ball at time t (in secons) is 2 given by s( t) 16t 200. Fin the velocity of the ball when t = 3. s ss s The position function for a free-falling object (neglecting air resistance) uner the influence of gravity can be represente by the equation s s ss s s s s s s s s, where s 0 is the initial height of the object, v 0 is the initial velocity of the object, an g is the acceleration ue to gravity. On Earth, the value of g is s s ss s. Homework Assignment Page(s) Eercises
Section 2.3 Prouct an Quotient Rules an Higher-Orer Derivatives 39 Section 2.3 Prouct an Quotient Rules an Higher-Orer Derivatives Course Number Instructor Objective: In this lesson you learne how to fin the erivative of a function using the Prouct Rule an Quotient Rule. Date I. The Prouct Rule (Pages 119 120) The prouct of two ifferentiable functions f an g is itself ifferentiable. The Prouct Rule states that the erivative of the fg is equal to ss ss ss s How to fin the erivative of a function using the Prouct Rule ss s s ss ss ss. That is, s ss f ( ) g ( ) f ( ) g ( ) g ( ) f ( ). Eample 1: Fin the erivative of ss s s s s s 2 y (4 1)(2 3). The Prouct Rule can be etene to cover proucts that have more than two factors. For eample, if f, g, an h are ifferentiable functions of, then f ( ) g ( ) h ( ) s s ss ss s s Eplain the ifference between the Constant Multiple Rule an the Prouct Rule. s s ss ss ss s ss ss s ss s s ss s ss s ss s s s ss s s ss ss ss
40 Chapter 2 Differentiation II. The Quotient Rule (Pages 121 123) The quotient f / g of two ifferentiable functions f an g is itself ifferentiable at all values of for which g ( ) 0. The How to fin the erivative of a function using the Quotient Rule erivative of f / g is given by ss ss s ss ss s ss ss by ss ss. ss, all ivie This is calle the ss s, an is given by f ( ) g( ) f ( ) f ( ) g ( ) 2, g ( ) 0. g( ) g ( ) Eample 2: Fin the erivative of s ss s s y 2 5. 3 With the Quotient Rule, it is a goo iea to enclose all factors an erivatives ss ss an to pay special attention to ss ss ss. III. Derivatives of Trigonometric Functions (Pages 123 124) tan s s s cot s s s s sec s s s csc s s s s How to fin the erivative of a trigonometric function Eample 3: Differentiate the function f ( ) sin sec. ss s s s s s s s
Section 2.3 Prouct an Quotient Rules an Higher-Orer Derivatives 41 IV. Higher-Orer Derivatives (Page 125) The erivative of f ( ) is the secon erivative of f() an is enote by s s. The erivative of f ( ) is the How to fin a higherorer erivative of a function ss s of f() an is enote by f. These are eamples of ss of f(). The following notation is use to enote the ss s s of the function y f () : 6 y 6 D 6 [ y ] (6) y 6 [ f ( )] 6 f (6) ( ) (5) Eample 4: Fin y for ss s s 7 5 y 2. Eample 5: On the moon, a ball is roppe from a height of 100 feet. Its height s (in feet) above the moon s 27 2 surface is given by s t 100. Fin the 10 height, the velocity, an the acceleration of the ball when t = 5 secons. s s s ss s s s ss s s
42 Chapter 2 Differentiation Eample 6: Fin y for s s s s y sin. Aitional notes Homework Assignment Page(s) Eercises
Section 2.4 The Chain Rule 43 Section 2.4 The Chain Rule Objective: In this lesson you learne how to fin the erivative of a function using the Chain Rule an General Power Rule. Course Number Instructor Date I. The Chain Rule (Pages 130 132) The Chain Rule, one of the most powerful ifferentiation rules, eals with functions. Basically, the Chain Rule states that if y changes y/u times as fast as u, an u changes u/ times as fast as, then y changes ss times as fast as. How to fin the erivative of a composite function using the Chain Rule The Chain Rule states that if y f () u is a ifferentiable function of u, an u g() is a ifferentiable function of, then y f ( g( )) is a ifferentiable function of, an y f ( g ( )) or, equivalently, s. When applying the Chain Rule, it is helpful to think of the composite function f g as having two parts, an inner part an an outer part. The Chain Rule tells you that the erivative of y f () u is the erivative of the ss ss (at the inner function u) times the erivative of the ss ss. That is, y s s s ss. Eample 1: Fin the erivative of s s ss s 2 5 y (3 2).
44 Chapter 2 Differentiation II. The General Power Rule (Pages 132 133) The General Power Rule is a special case of the s. ss How to fin the erivative of a function using the General Power Rule The General Power Rule states that if y u() n, where u is a ifferentiable function of an n is a rational number, then y s ss or, equivalently, u n ss ss s 4 Eample 2: Fin the erivative of y 3 (2 1) ss s s s ss s. III. Simplifying Derivatives (Page 134) Eample 3: Fin the erivative of y ss s ss s s ss s ss s s s s s 2 3 (1 ) 3 2 an simplify. How to simplify the erivative of a function using algebra
Section 2.4 The Chain Rule 45 IV. Trigonometric Functions an the Chain Rule (Pages 135 136) Complete each of the following Chain Rule versions of the erivatives of the si trigonometric functions. How to fin the erivative of a trigonometric function using the Chain Rule sinu s ss ss s cosu s s ss ss s tan u s s ss ss s cot u s s s ss ss s secu s s ss ss s cscu s s s ss ss s Eample 4: Differentiate the function ss s s ss ss y sec4. Eample 5: Differentiate the function ss s ss s s s ss 2 y cos(2 1).
46 Chapter 2 Differentiation Aitional notes Homework Assignment Page(s) Eercises
Section 2.5 Implicit Differentiation 47 Section 2.5 Implicit Differentiation Objective: In this lesson you learne how to fin the erivative of a function using implicit ifferentiation. Course Number Instructor Date I. Implicit an Eplicit Functions (Page 141) Up to this point in the tet, most functions have been epresse in eplicit form y f (), meaning that ss s ss s s ss s ss ss s. However, some functions are only s by an equation. How to istinguish between functions written in implicit form an eplicit form Give an eample of a function in which y is implicitly efine as a function of. s s ss ss s s s s s ss ss ss ss Implicit ifferentiation is a proceure for taking the erivative of an implicit function when you are unable to ss s ss s ss ss s. To unerstan how to fin y implicitly, realize that the ifferentiation is taking place s s ss s. This means that when you ifferentiate terms involving alone, s ss ss. However, when you ifferentiate terms involving y, you must apply s s because you are assuming that y is efine s as a ifferentiable function of. ss Eample 1: Differentiate the epression with respect to : 2 4 y s ss ss s ss ss s ss
48 Chapter 2 Differentiation II. Implicit Differentiation (Pages 142 145) Consier an equation involving an y in which y is a ifferentiable function of. List the four guielines for applying implicit ifferentiation to fin y/. How to use implicit ifferentiation to fin the erivative of a function 1. s s ss ss ss s s ss ss 2. s ss ss ss s s ss ss s ss ss ss ss s ss 3. ss ss s s ss 4. ss Eample 2: Fin y/ for the equation ss s 2 2 4y 1. Homework Assignment Page(s) Eercises
Section 2.6 Relate Rates 49 Section 2.6 Relate Rates Objective: In this lesson you learne how to fin a relate rate. I. Fining Relate Variables (Page 149) Another important use of the Chain Rule is to fin the rates of change of two or more relate variables that are changing with respect to s. Course Number Instructor Date How to fin a relate rate Eample 1: The variables an y are ifferentiable functions of 3 t an are relate by the equation y 2 4. When = 2, /t = 1. Fin y/t when = 2. II. Problem Solving with Relate Rates (Pages 150 153) List the guielines for solving a relate-rate problems. 1. ss ss s s ss ss ss s s ss ss How to use relate rates to solve real-life problems 2. ss ss ss ss ss ss ss ss ss ss ss 3. ss ss ss s s s ss ss ss s s ss s ss 4. ss s s ss s s ss ss ss ss ss ss s s ss ss s ss s Eample 2: Write a mathematical moel for the following relate-rate problem situation: The population of a city is ecreasing at the rate of 100 people per month. s s ss ss ss s s ss
50 Chapter 2 Differentiation Aitional notes Homework Assignment Page(s) Eercises