AP CALCULUS AB UNIT 3 BASIC DIFFERENTIATION RULES TOTAL NAME DATE PERIOD DATE TOPIC ASSIGNMENT /18 9/19 9/24 9/25 9/26 9/27 9/28 10/1 10/2 10/3

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1 NAME DATE PERIOD AP CALCULUS AB UNIT BASIC DIFFERENTIATION RULES DATE TOPIC ASSIGNMENT 0 0 9/8 9/9 9/ 9/5 9/6 9/7 9/8 0/ 0/ 0/ 0/ 0/5 TOTAL

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3 AP Calculus AB Worksheet 9 Average Rates of Change Find the average rate of change of f ( ) = + over the interval,. Find the average rate of change of f ( ) = + over the interval Find the average rate of change of f ( ) = + cos over the interval 0,. *** Calculator *** 0,. The amount of money in a bank account, called the balance, after t years is f ( t ) = 00(.08) t dollars. a) What are the units of the rate of change of f ( t ). b) Find the average rate of change of the balance over the time interval 0,0.5. Eplain the meaning of this value in the contet of the bank account. 5 *** Calculator *** The amount of money in a bank account, called the balance, after t years is f ( t ) = 00( 0.9) t dollars. Find the average rate of change of the balance over the time interval 0,0.5. Eplain the meaning of this value in the contet of the bank account. 6 The table below shows the weight, in lbs., of a grown man over a certain time period, t, in years. Consider the first year of the data collection period as t = 0. Year Weight a) Estimate the rate of change of the man s weight over the time period 0,5. Using correct units, eplain the meaning of this value in the contet of the man s weight. b) Estimate the rate of change of the man s weight over the time period eplain the meaning of this value in the contet of the man s weight.,6. Using correct units, 7 f + h f Evalauate lim h 0 h f 5 for a) = + b) f ( ) = 7 8 Find the equation of the line that passes through, and is perpendicular to y= 7. 9 Verify that f ( ) +, = +, is either continuous or discontinuous at =.

4 AP Calculus AB Worksheet 0 Limit Definition of the Derivative and the Tangent Line Problem Find the derivative of the function using the limit process. Remember: The derivative is a new function so it requires new notation; Eample: g( ) = 5 h s = + s f = Find the slope of the tangent line to the graph of the function at the given point. f ( ) = + ; (,) 5 f ( t) = t t ;, Find the equation of the tangent line to the graph of the function at the given value of. 6 f ( ) = + +, = 7 g ( ) = ( ), = dy f ' or or y ' d 8 What is the value of k if f ( )= k, +, is everywhere continuous? 9 5+ =, 6+ 8 a) Find the discontinuities and identify them as removable or non-removable. b) Identify the vertical asymptotes and describe the behavior to the left and right of the vertical asymptote. f. Write as a piecewise For the function f ( ) c) Write a new function, g( ), that removes the removable discontinuity from function. d) Identify any horizontal asymptotes. The table below shows selected values for a function, f ( ), at various values of f ( ) 8 0) Find the average rate of change of the function over the interval,. ) Find the average rate of change of the function over the interval 5,9. ) Estimate the slope of the function when =.

5 AP Calculus AB - Worksheet Power Rule Nearly all men can stand adversity, but if you want to test a man's character, give him power. - Abraham Lincoln Learn: The Power Rule: ' f = a n ( n ) f = n a d f = f ' = 8 = d Eample: Use the Power Rule to find the derivative of each function.. y f 5 =. =. f ( ) = +. y= 5, 0 5. f ( t) = t + t 6 6. f ( ) = 7 7. y = ( + ) 8. f ( ) =, 0 9. f ( ) = ( ) 0. f =, 0. y f = a + b + c =. Find the instantaneous rate of change of each function when =.. y =, 0. y = m + b 5. f ( ) = f ( ) = y =, 0 9. y = f t = t + t Find the slope of the curves at the indicated values of. 0. = +, =. f ( ) = +, = 0, y. f ( ) = =., y = + +, = 0. Evaluate ( + ) 5( + ) + 6 ( 5 + 6) 0 lim using the power rule. 5. Find two unique functions, f ( ), such that f ' = (Answers will vary)

6 AP Calculus AB - Worksheet Tangent and Normal Lines (Power Rule) Learn: Tangent and Normal Lines to a Curve Recall: Derivative = slope of the Tangent line at that point s -coordinate Eample: f = +, = = y = ( ) f ' f ' slope of the tangent line Tangent Line: Normal Line: y = ( ) For each of the following: a) Sketch a graph - USE GRAPH PAPER!! b) Find the slope of the tangent line at the given point. c) Find the equations of the tangent line at the given point. Sketch the line. d) Find the equation of the normal to the curve at the given point. Sketch the line.. y=, (,). f ( ) = 6 (,). f ( ) =, (, ). y=, (,) Find the equations of the tangent and normal lines to the curve at the given -value. 5. y = + 8 +, = 6. y = =, 7. Find the points on the curve y = + + where the tangent is horizontal. 8. Show that the curve y = has no tangent line with slope. 9. Find an equation of the tangent line to the curve y = that is parallel to the line y= Find a parabola with equation y = a + b + c that has slope at =, slope -8 at =, and passes through the point (,5 ). Hint: You will create different equations in order to solve for a, b, and c.. Evaluate lim 0 ( + ) ( + ) ( ) using the Power Rule.

7 Worksheet Answers ) y' = ) f ' = ) = y ' = ( ) tangent: y = normal: y = ( ) ( ) ( ) f ' = ( ) tangent: y = normal: y = ( ) ) 5) 6) y' = 8 y ' = tangent: y = normal: y = y = 9 y = + + y' = + 8 y ' = ( ) tangent: y 9 = normal: y 9 = ( ) f = f '( ) = f ' = tangent: y = normal: y = y = y ( ) = 0 y ' = 6 y ' = ( ) ( ) ( ) tangent: y 0 = + normal: y 0 = + ( ) 7) y ' = = 0 8) + = 0 ( )( ) + = 0 = ; = y = ; y = 6 (, ); (, 6) y = + y 6 5 ' = = 8 = No Real Solution 9) 0) y = = y' = = = = ( ) y 8 = y = a + b + c y ' = a + b slope at = a + b = slope 8 at = a + b = 8 passes thru,5 a + b + c = 5 a+ b= a+ b= 8 b = a = + c = 5 b= c= 7 y = + 7

8 Worksheet f = a f ' = n a Learn: Package Rule n n d d The Package Rule Eample: f ( ) = ( + 7 ) f '( ) = ( 9)( )( + 7) ( ) ( 7) = 5 ( + 7) At the specified point, find the equation of the normal to the curve: f ( ) 00 at 6, f ( 6). For what value of k, is For problems -6, find f ' y k f = +, is a normal line to the graph of. Do not simplify your answer.. f ( ) = ( + ) 8. f ( ) ( ) 6 = + 7. f ( ) f 6. 5 f 9. =. = +. = 5. = ( + ) f = 8. f ( ) = + = + 0. f ( ) = +. f ( ) =. f =. f ( ) f ( ) ( )( ) = 6. 5 f =. f ( ) = + = 5 Write the equation of the normal line at the given -value. 7. f ( ) = 7 when = f ( ) = when = 5 0. f = + when = 0 f = when =

9 AP Calculus AB - Worksheet Product and Quotient Rules for Derivatives Differentiate the following functions. Do not simplify the answer... 5 g( t) = 6t. h( z) = 8z. f ( s) 5 s s 5s =. B = v Gv () = v 8 6+ t gt () = t f ( ) = = +. f( ) g( ) = 5. p = k( ) ( )( 6 5) = h( ) 5 F() t = t + = + 7. h( ) = ( 5 ) = + 8. S( ) = ( + ) 8. k( r) r ( r 7r r) t 9. H ( z) ( z 5 z )( 7z z ) = z N( z) = z f( ) = + 0. M( ) = 7 + +

10 AP Calculus AB Worksheet 5

11 AP Calculus AB Worksheet 6 Derivatives Using Tables a) Given h( ) = f ( ) + g ( ), find h' b) Given s( ) = f ( ) g ( ), find s' ( ) c) Given p( ) = f ( ) g ( ), find p' ( ) g( ) d) Given q( ) =, find q' ( 5) f ( ) e) Given c ( ) = ( f ( )) find c' ( ) f) Given m( ) = f ( g ( ) ), find h' ( 6)

12 AP Calculus AB Worksheet 7 Differentiability Describe the -values at which f is a) differentiable. b) Continuous but not differentiable. c) Neither continuous or differentiable ) ) ) ) 5) 6) Determine if the function is differentiable at the given value., f =, =, 7) +, f =, =, 8) 9) Sketch the graph of a continuous function such that: f 0 = ; ' f = for 0) Sketch the graph of a continuous function such that: f 0 = f ' 0 is undefined f ' 0 for 0 f ' 0 for 0

13 AP Calculus AB Worksheet 8 Derivatives (Power, Package, Product and Quotient Rules) Review Use the limit definition of the derivative to find f '( ) for f ( ) Find the derivative of each function below. Simplify your answer. y = + y = 5 + f ( ) = 5 f ( ) = ( )( + ) 6 y = + 5 = + Answer each question about tangents and normals. 7 Find the points on the curve y = + 0 where the tangent line is parallel to the -ais Find the slope of the normal to f = + at the point where = Find the equation of the tangent to y = + at the point where =. Find dy d for y ( )( ) = + +, then find the slope of the normal when =. If the line y = is tangent to the function f = +, what is the point of tangency? If the slope of a tangent line is 5 then what is slope of the normal line to the same curve at the same point? At what -value is y = tangent to f ( ) = +. For #-9, use the table below to find the indicated value. h( ) = f ( ) + g ( ) h 5. Given d ( ) f ( ) g ( ), find d '( ) 6. Given p( ) = f ( ) g ( ), find p' ( ). g( ) 7. Given q( ) =, find q' ( ). f ( ) 8. Given k ( ) = ( f ( ) ), find k. 9. Given c ( ) = f ( g ( ) ), find c Given, find '. =.

14 Answers: ) f ' = 5) f '( )= + + 9) y = ) dy d = 0) dy 7 ( +) ) y' = ) y' = ) 6 d = (, 7) (, 0) ) 0, ) f ' 8) 5 ) = = ( )