TOTAL NAME DATE PERIOD AP CALCULUS AB UNIT 4 ADVANCED DIFFERENTIATION TECHNIQUES DATE TOPIC ASSIGNMENT /6 10/8 10/9 10/10 X X X X 10/11 10/12

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1 NAME DATE PERIOD AP CALCULUS AB UNIT ADVANCED DIFFERENTIATION TECHNIQUES DATE TOPIC ASSIGNMENT 0 0 0/6 0/8 0/9 0/0 X X X X 0/ 0/ 0/5 0/6 QUIZ X X X 0/7 0/8 0/9 0/ 0/ 0/ 0/5 UNIT EXAM X X X TOTAL

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3 AP Calculus AB - Worksheet 0 Linear Approimations, Calculator Derivatives, Higher Orer Derivatives. If f, approimate f.0 using linearization centere at.. For the function f, f ' an f. What is the approimation for. approimation centere at?. Approimate.9.9 using linearization. f using the tangent line Fin an approimate value for f.9 on f Approimate using tangent line approimation: 7 Approimate using a tangent line approimation using linearization. 7 Fin y for y. Simplify. 8. Evaluate f ' for f using a calculator Evaluate ' for f f sin Evaluate f ' for f using a calculator. e using a calculator. Answers: y

4 AP Calculus AB - Worksheet Derivatives of Sine an Cosine functions Know the following theorems: sin cos an cos sin Eamples. y sin 5 y sin 5 y ' cos 5 5 y' 5cos 5. y cos9 y cos 9 y' sin 9 9 y' 9sin 9. y sin y ' sin cos y sin y ' sin sin y ' sin cos Fin the erivative of each function. Simplify, if necessary. y sin y 5cos f cos y sin cos 5 y cos 6 f 5sin 7 y cos 8 cos f sin 9 y cos 0 Fin the equations for the lines that are tangent an normal to the graph of f sin Fin the equation of the normal line to f sin cos at. Determine all values of in the interval 0, for which f cos at. has horizontal tangents. Answers ) y' 6 cos y ) ' ) 5 sin ) y' cos (common ientity) 5) y' cos sin 7) y' sec tan ) y f cos sin f ' 5cos 6) 8) f ' 0) T : y sin N : y

5 AP Calculus AB - Worksheet Derivatives of Trigonometric Functions Know the following Theorems tan sec cot csc sec sec tan csc csc cot Eamples. y tan 5 y tan 5 y y ' sec 5 5 ' 5sec 5. y sec 5 y sec 5 y ' sec 5 tan 5 5 y ' 5sec 5 tan 5. y cot y cot ' cot csc y ' cot csc y. y csc y csc y ' csc csc cot y ' 6csc cot Use the quotient rule to prove the erivative of: [Hint: change into sin an cos an then take erivative]. tan. cot. sec. csc

6 AP Calculus AB - Worksheet Know the following theorems: ln e e Fin y.. y e. y e. 5. y e 6. y e y e 0. y e. y ln y e. y e e 8. Derivatives of ln an e y e 5 y e e y ln. 0. y ln. y ln 5. y ln ln 6. y ln 7. A line with slope m passes through the origin an is tangent to y ln. What is the value of m? 8. Evaluate 9. Fin an equation of the tangent line to the graph of the given function at the inicate -value. a) f ln, b) f ln sin, 0. Fin ' f tan. Answers f for

7 AP Calculus AB Worksheet 5 L Hopital s Rule an Review Evaluate each Limit. Use L Hopital s Rule where appropriate. Fin each erivative. 9 f e 0 sin y e y ln sin y g cos 5 y ln e 5 Use a tangent line approimation to estimate the value of 5. 6 For Answers: f e, use a tangent line approimation centere at 0 to estimate 0. f. 9) f ' e 0) y' cos e sin ) y' cot ) y' g ' 5cos 5sin5 )

8 AP Calculus AB Worksheet 7 Implicit Differentiation Fin y. y y 6 y tan y sin y y 5 y 5 6 y 9 7 y 8 5 y 9 For y 8y, show that y 6y y 6 0 For y, fin the slope of the tangent line at the point,. For y y, fin the equations of the tangent lines at the point where. If y', fin three possible equations for y. Fin the secon erivative of f e Answers: ) y y y y ) y y cos y ) y y ) y cos y 5) y y 6) y 9 5 7) y 8) y 5 0) y ) Tangents : y y,y,

9 AP Calculus AB - Worksheet 8 Know the following Theorems. arcsin arctan Derivatives of Inverse Trigonometric Functions sec arccos cot csc Fin the erivative of y with respect to the appropriate variable.. y cos. y sin t. y sin t. y arcsin 5. y sec 5s 6. y t tan 7. Which of the following is sin? A) B) C) D) E) Fin the erivative of the function. 8. y y y. r tan 0. y cot t. y ln. y e 5. y ln sin 6. If f ln, use tangent line approimations to estimate the value of f. Answers 6. y'. y'. y ' t t t t 9. y' sin 5. y' 6. y' s 5s t t 7. E 8. y' 6 9. y' csc t 0. y'. y ' t. y '. y ' e. y ' tan sec 5. y' cot

10 AP Calculus AB Worksheet 9 Derivatives an Linear Approimation Review Fin each erivative f e y ln csc 5 f ln e f sec 5 y ln sin 6 cos y 7 y e 8 y 9 cot lim h0 h cot h 0 What is the slope of the line tangent to the graph of y e sin when? Approimate 7.6 using linearization. For 8 f.9. f, approimate The line y k is tangent to the graph of y. What is the value of k? Answers: ) f ' e ) y' cot 5) y' cos sin sin 9) csc 0) 0 5e e y' e ) f ' 5 5 cos 6) y' ln sin 7) ) 59 0 ) f ' 6sec tan 8) y' )