AP Calculus BC Final Exam Preparatory Materials December 2016

Transcription

1 AP Calculus BC Final Eam Preparatory Materials December 06 Your first semester final eam will consist of both multiple choice and free response questions, similar to the AP Eam The following practice problems are a comprehensive review of first semester topics I strongly recommend you work through all of the problems in this review and ask me questions prior to the day of the eam Your final eam will take place over the course of two class periods: The multiple choice portion of the eam will be on Wednesday, December 7 The free response portion of the eam will be on Thursday, December 8 Class time will be dedicated to reviewing for the final on Monday, December 5 and Tuesday, December 6 I will also be available for stuing during lunch that week (Dec 5-8) and after school every day between now and the final

2 MULTIPLE CHOICE SECTION Graphing calculators are allowed The following is a comprehensive collection of questions from first semester calculus material On the final eam, you will be given 0 of these problems in the form of multiple choice questions You will be epected to show work to support your answer when requested Such problems will be noted with SW on this list A particle travels along the -ais The velocity of the particle is shown in the graph below At what value(s) of t does the particle change direction? Write the equation of the line tangent to the graph of 3 y at the point (,4) SW Consider the piece-wise defined function shown below Find the value of the right-hand derivative at 4 Determine the continuity and differentiability of the function 8 6, 4 at 4 SW, If 5 4y y, find SW d 6 If 4 (4) ( ), find f (0) SW

3 7 Find d at 4 given the functions defined below SW g( ) g() f() 8 Determine where the graph of y is concave up SW 9 If y cos(), find d 0 Find e lim SW ln Find the average rate of change of y 5 4 on [,] SW If the radius of a sphere is increasing at in/sec, how fast, in cubic inches per second, is the volume increasing when 4 the radius is 0 inches? V r 3 SW 3 3 Write the limit statement(s) to support that a function has a vertical asymptote at 3 4 If f ( ) ( ) ( ), then determine where the graph of f () has points of inflection 5 Find the equation of the line normal to the graph of y when SW

4 6 The graphs of the differentiable functions f () and g() are shown below If P( ) g( ) then determine the following: Is P (0) positive, negative, or zero? Is P () positive, negative, or zero? SW Graph of f () Graph of g () 7 If the graph of f () is shown below, sketch the graph of f () 8 If lim L is a real number, determine if each of the following statements must be true or could be false: a I f (a) eists II f () is continuous at a III f () is defined at a IV f ( a) L 9 If, then find f (5) SW 0 Find the slope of the line tangent to y ln at 4 SW The position of a particle moving along a straight path at any time t is given by s ( t) t 4t 4 Find the acceleration of the particle when t 4 SW

5 The graph of the second derivative of a function f is shown below Determine whether each of the following statements are true or false I The graph of f has an inflection point at II The graph of f is concave down on the interval (,3) III The graph of the derivative of f is increasing at 3 Evaluate 4n lim n 0000n Determine where has a relative maimum SW 5 If y 0, find d 6 Evaluate lim csc 0 7 If y cos sin, find d 8 If y arctan(cos ), find d 9 If y 3, find at SW d 30 If y y8, find at (,) SW d

6 3 Let 3 3 Determine the values of c that satisfy the Mean Value Theorem for () f on [ 0,3] SW 3 If ln( ), find f () 33 The graph of f () is shown below Determine how many points of inflection the graph has from this picture 34 The area of a circular region is increasing at a rate of 96 square meters per second When the area of the region is 64 square meters, how fast, in meters per second, is the radius of the region increasing? SW 35 A particle moves along the -ais so that at any time t its position is given by the particle at rest? SW ( t) te t For what values of t is 36 Let f () be a function that is continuous on [,3] such that f (0) does not eist, f ( ) 0, and f ( ) 0 for all ecept 0 Sketch a possible graph of f () 37 If g () is a second degree polynomial such that g ( 0) 3, g ( ) 0, and g ( 0) 4, then determine which of the following could be g () : SW I g ( ) 3 3 II g ( ) 4 3 III g ( ) 4 3 IV g ( ) 3 V g ( ) Given the functions f () and g () are both continuous and differentiable, and they are defined at the values given in the table below If h( ) g( ), find h (4) SW f () f () g () g () Determine where the graph of y has a point of inflection SW

7 40 The graph of the derivative of f () is shown below Which of the following is true about the function f ()? I f () is decreasing at 0 II f () has a local maimum at III f () is concave up at 4 The graph of the function f () is shown above At which point on the graph is f ( ) 0 and f ( ) 0? 4 The graph of the derivative of f () is shown below At what value does f () have a local maimum?

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9 FREE RESPONSE SECTION Graphing calculators are allowed On the final eam, you will be given the following si free response questions You must choose TRHEE to answer completely Problem (Unit Limits and Continuity) There is a function f () with all of the following characteristics: 3 f () has eactly two roots at and f () has two vertical asymptotes with equations 7 and lim lim 3 lim eists, f (3) eists, but lim f ( 3) A Write the function f () B Evaluate f () 3 Problem (Units 3 & 4 Derivative Rules) Let the functions f (), g (), h (), and j () and their respective derivatives be continuous and differentiable for all real numbers The table at the right contains certain values of the functions at a If h( ) and j( ) g( ), g( ) answer the following questions A Complete the table of values Show work that supports your answers B If p( ) g( ) j( ), evaluate p (a) Function When a f () f () g () g () h () h () j () 3 j () 0 Problem 3 (Units 3 & 4 Derivative Rules) Let the functions f (), g (), h (), j() and k () be defined as follows: f ln ( ) e, ( ) sin(tan( )) 3 4 g, h( ), e cos j( ), k( ) cos sin Evaluate f (), g ( ), h (), j (0), and k ( ), and list them in order from least to greatest Show all work

10 Problem 4 (Units 3 & 4 Derivative Rules) Two functions f () and g () and their derivatives f () and g () are continuous and differentiable for all real numbers Select values of the functions are given in the table below 0 3 f () 4 3 f () 0 g () 3 g () A Evaluate f g () B Evaluate fg () C Evaluate g ( ) D Approimate g (4) Show work that leads to your solution 6 Problem 5 (Unit 4 Derivative Rules) Consider the curve given by y y e cos e sin A Show that Show work to support your answer d y B Find the equation of the tangent line at ( 0,) Show work to support your answer d y C Find the value of d at ( 0,) Show work to support your answer Problem 6 (Unit 5 Graphical Analysis) Use the function 3 3a 8 7c d to answer the following questions A Determine the value of a such that f () has a point of inflection when Show work to support your answer Rewrite your function after you have determined the value of a B Determine the value of c such that f () has a etrema when 5, 3 Show work to support your answer Rewrite your function after you have determined the value of c C Determine the value of d such that ( 0,5) is a solution of f () function after you have determined the value of d Show work to support your answer Rewrite your