Fox Lane High School Department of Mathematics

Transcription

1 Fo Lane High School Department of Mathematics June 08 Hello Future AP Calculus AB Student! This is the summer assignment for all students taking AP Calculus AB net school year. It contains a set of problems along with their answers. These problems focus on skills that you have previously learned and are essential to your success in AP Calculus. The first few days of class will be devoted to answering questions on this material. You will then have a test that assesses your mastery of this content. Your assignment is to review these questions to ensure you have the necessary prerequisite skills to be successful in AP Calculus. If after reviewing the problems you are confident in your ability to do well on the eam, there is no need to work the problems as nothing will be collected. However, it is strongly suggested that you practice these questions and seek additional help if necessary. The first test of the year is an ecellent opportunity to start the year off well, so make sure you study until you are confident on these questions. You may me at the address shown below if you have any questions. Please keep in mind that it may take me some time to get back to you depending on my schedule over the summer, so please don t wait until the last minute to ask for help. PLEASE NOTE: This assignment is to be done WITHOUT the use of a calculator. Over half of the AP Calculus eam needs to be completed without a calculator, so start getting used to that now. In addition, a calculator will not be allowed on the test you will take on this material. You may also find it helpful to use the following websites as a resource while working through the problems: follow links for Algebra and Trigonometry Online Math Videos can be found at the following sites: If for some reason you misplace this assignment, you can access it on my website located at: Just follow the links on the left for AB Calc, then click on summer assignment. Enjoy your summer and see you in September! Mr. Swart: jswart77@bcsdny.org

2 Required Skills (note: sample questions on these skills are found on the following pages). Simplifying comple fractions. Working with rational (fraction) and negative eponents without a calculator. Simplifying rational epressions (adding, subtracting, multiplying, and dividing algebraic fractions) 4. Factoring quadratic epressions. Using function notation and composite function notation 6. Finding and y intercepts of a function 7. Solving systems of equations (i.e., finding intersection points) algebraically 8. Finding the domain and range of functions and epressing answers in interval notation 9. Finding the inverse of a function 0. Writing the equation of a line using point slope form. Solving all types of equations including: a. Rational (fraction) equations b. Equations with square roots and cube roots c. Quadratic equations (when factorable and not factorable) d. Eponential equations (including those with a base of e) e. Logarithmic equations (including those with ln). Evaluating trig and inverse trig functions without a calculator. Proving trig identities 4. Sketching basic functions without a calculator. These include: y =, y =, y =, y =, y = e, y = ln, y =, y =. Graphing the functions listed in item 4 after a shift has been applied. 6. Graphing and evaluating piecewise functions 7. Rationalizing a binomial denominator (i.e., get the radical out of a denominator of a fraction)

3

4 Simplify each epression using only positive eponents where applicable r r 0. ( ) b. y y 4 4. y 8y Simplify and epress answer without eponents:. ( ) Simplify each of the following:. + h Factor Completely: Simplify and Epress Answer Without Eponents: 0.. 4a

5 Let f ( ) = + and g ( ) =. Find each of the following: 4. f ( ) =. g ( ) = 6. ( ) f t + = ( ) ( ) 7. f g ( ) = 8. ( ) g f m + = 9. ( + ) ( ) f h f h Let f ( ) = sin. Find the eact value of each of the following: 0. π f =. π f = f g h =, = +, =. Find each of the following: Let ( ) ( ) ( ) ( ). h f ( ) =. ( ) ( ) f ( g ) = 4. ( ) g h =

6 . If f ( ) = 9 +, find the value of: ( + ) ( ) f h f h 6. If f ( ) =, find the value of: ( + ) ( ) f h f h To find points of intersection, solve the system of equations using either the substitution or elimination methods. Remember to find the coordinates of each intersection point by using one of the original equations. See eample below: y = + Find the intersection point(s) of the two curves: y 9 = 0 Using the substitution method and substituting the top equation into the bottom one: + 9 = = = = 4 4 ( ) ( )( ) ( )( ) = = 0 + = 0 = = If = y = ( ) + = 4 Point =,4 If = ( ) y = ( ) + = 0 Point =,0 ( )

7 Find the and y intercepts for each: 7. y = 8. y = + 9. y = y = 4 Find the intersection point(s) of the graphs of the given equations: 4. + y = 8 4 y = y = 6 + y = 4 INTERVAL NOTATION: recall that 4 < epressed in interval notation is (,4] epressed in interval notation is (,] and Find the domain and range for each function. Write your answer in INTERVAL notation. 4. f ( ) = 44. f ( ) = + 4. f ( ) = sin 46. f ( ) = Find the Inverse of each function: f = 47. f ( ) = ( )

8 49. Use point-slope form to write the equation of a line passing through the point (, ) with a slope of Find the equation of the line passing through the points (,6) and ( 9, ).. Find the equation of a line passing through the point (,8 ) and parallel to the line y =. 6. Write the equation of a line that passes through the point (, ) and is perpendicular to the line 7 y = 8.. Determine the equation of a line passing through the point (, ) with an undefined slope. 4. Determine the equation of a line passing through the point ( 4,) with a slope of 0.. Find the equation of a line with an -intercept of (,0 ) and a y-intercept of ( 0, ). You need to be able to solve equations that are in many different forms. Practice by solving the following equations. 6. = = = + 9. ( ) ( ) = = 0 6. Solve for : 8.76 = 9

9 6. Solve: log = 4 6. Solve: log + log 6 log log = b b b b 64. Solve: log ( + ) log = log 6. Solve: ( ) 8ln = Solve: 60 = 0e 67. Solve: 4 = e 68. Solve: = 69. Solve for a and b: a + b = 6 a b = Find the zero of the function by hand, then check answer with graphing calculator: ( ) = 7 f e You should be able to find the eact values of trigonometric and inverse trig functions WITHOUT a calculator. You also need to be comfortable with performing algebraic operations with trig functions and using this to prove trig identities. 7. π sin 7. π cos 4 7. π tan 74. 4π csc 7. π sec 76. sin 77. arctan ( ) 78. arccos

10 79. cos 80. arcsin 8. Prove the identity: sin sin + = csc cos + cos 8. Prove the identity: cos sin = sin sin 8. Prove the identity: sec tan cos = You should be able to use composite function notation and be able to find inverse functions. 84. If ( f g )( ) = and f ( ) 6 = +, find g ( ). 8. If f ( ) = + and g ( ) a. f g b. g f =, find each of the following functions: You should be able to sketch functions without a graphing calculator. The functions you need to know include: y =, y =, y =, y =, y = e, y = ln, y =, y =. You also need to be able to shift these graphs as shown in the eamples below. 86A. Without using a calculator, make a rough sketch of each function below. Label all important features of each graph. a. y = b. y ( ) = + c. y ( ) = + d. y = 4 e. y = f. y = g. y = h. y = + i. y ln = j. y = ln ( )

11 86B. In addition to graphing without a calculator, you also need to be able to evaluate functions without a calculator. Specifically, you should know the following: Given f ( ) = ln, without a calculator, find: a. f ( ) b. ( ) f e c. f ( e ) 87. Given the graph of f ( ) below, sketch the graph of ( ) f on the same set of aes. 88. Sketch a graph of f ( ) = e. Then on the same set of aes, sketch a graph of f ( ) dashed line. as a 89. Sketch a graph of y = cos You should be comfortable with all aspects of piecewise functions, including graphing and evaluating them. 90. Graph the piecewise function:, f ( ) =, < +, >

12 9. Let f ( ) if 0 = + if > 0 a. Evaluate f ( ) and ( ) f. b. Sketch a graph of f. You should be able to rationalize a binomial denominator by using a conjugate pair. Rationalize the following (means get the radical out of the denominator): MIXED REVIEW: f = 9. Find the inverse of ( ) 96. The graph of a function f is given to the right. f. a. State the value of ( ) b. Estimate the value of f ( ). c. For what values of is f ( ) =? d. Estimate the values of such that f ( ) = 0. e. State the domain and range of f 97. Simplify: Simplify:

13 99. Simplify: a b b a + a b 00. Given f ( ) = 4 +, evaluate and simplify f ( ) Given f ( ) =, evaluate and simplify f ( + ) 0. Given f ( ) = 6 + a.) evaluate and simplify f ( + h) b.) evaluate and simplify f ( + h) f ( ) 0. Prove the following identity: tanθ + cotθ = secθ cscθ 04. a) sin b) sin c) π sin sin 0. Sketch a graph of y ln ( ) = + then use your graph to give the domain and range. 06. Given ( ) + <, f =, 6, > 6 a. find f ( 0) b. find f ( ) c. sketch a graph of f ( ) 07. Solve: 09. Solve: e = 08. Solve: ln + ln = + 4 y = e + 9 = 0. Solve the system: ln y = Evaluate the epressions: a. log8 b. log e. log0 000 c. ln d. e ln e. Solve: a. = 8 e b. e + 7 = e 8

14 ANSWERS: = 0. r r b. y = y. y h h ( + )( + 4) 9. 7( )( 4) a ( 4 )( + 4 )( + ) t m 40m ( + ) ,0 and ( 0, ) 8. (,0 ) and (,0 ) and ( 0, ) 9. ( 0,0 ) and ( 4,0) and ( 4,0 ) 40. ( 0,0 ) and (,0) and (,0 ) 4. (, ) 4. (,) and (, ) 4. Domain: (, ) Range: [, ) 44. Domain: [, ) Range: (,0] 4. Domain: (, ) Range: [,] 46. Domain: (,) (, ) Range: (,0) ( 0, ) 47. f ( ) 48. f ( ) = 49. y + = 7( ) =

15 equivalent to y = + ) 0. y = ( 9) or y 6 ( ) = + (Note, if multiplied out, these two equations are =. 6. y 8 ( ) 40 y =. = 4. y = 7 7. y = 6. = ± 7. = 8. = 4 ± 9. =, = 6. 4 ln 9 log 9 = = /8 ln 8.76 log e 66. ln ln 4 ln = a = 4, b = 70. = ln π π π π π sin sin sin ( + cos ) + sin ( cos ) sin sin + = = = = cos + cos cos + cos cos sin ( )( ) csc cos sin = sin sin = sin 8. ( ) sin sin sin 8. = = = sec tan cos cos cos cos cos 84. g ( ) 4 = 8.

16 86A. (i) y (j) y B. a. 0 b. c. 87. ( ) 88. f 89. ( 0, ) (,0 ) f ( )

17 f ( ) + = a b a. 6h h 6 6h b h h h 04. a. π b. π c. π 0. Domain: (, ), Range: (, ) 06. a. b. 07. = 8 + ln e

18 0. = y = e. a. / b. log log c. - d. e.. a. ± 8 e b. ( e ) ln 7 8