AP CALCULUS AB SUMMER PACKET

Transcription

1 AP CALCULUS AB SUMMER PACKET Congratulations on your decision to enroll in advance placement calculus! Choosing to take AP Calculus may have been a difficult decision; however the intellectual stimulus and challenges that drive the curriculum will be a reward in and of itself, regardless of the outcome in May. As students in an AP class you have chosen to hold yourself to higher standards. This is admirable, but carries with it the inevitable burdens of obligation. You are obligated to give more of your time to studying. You are also obligated, by virtue of being a future innovator and/or leader, to hold yourself to high ethical standards. To succeed in this course, you must commit to try every assignment, not copy answers from other students or the Internet, and never, ever give up, especially when it gets hard. If you can t make that commitment, I urge you to rethink your choice of this class. In this packet, there is a supply list, a list of important dates, a summer assignment, and formulas that you must memorize. I will be at the AP boot camp on August th to answer questions over the summer assignment and any additional questions that you may have. If you would like to reach me over the summer, you may tet or call me at I look forward to an incredible year! Mrs. De Leon Student Supplies: TI-89 Titanium Calculator o There are school calculators that you may rent on a first come first serve basis o The fee to rent the school calculator is a pack of 6 AAA batteries AP Practice Eam Spiral Book (to be provided the first day of school) o There is an $0 fee for this book to be paid during first quarter o You can choose to not write in the book to avoid the $0 fee Ream of paper Paper and Pencils Binder with sections o Notes o FRQ s o Multiple Choice Important Dates: AP Boot Camp: August th, 05 Summer Assignment Due: August 8 th, 05 Semester Eams: January th 4 th, 06 TBD in April 06: AP Calculus Practice Eam and Final Eam (-hours after school) 06 AP Calculus Eam: Thursday, May 5 th, 06 at 8am

2 AP Calculus Summer Assignment Success in calculus depends to a large etent on knowledge of the mathematics that precedes calculus: algebra, analytic geometry, functions, and trigonometry. The following problems are intended to diagnose weaknesses that you might have in these areas. Complete on a separate piece of paper. Use graph paper for the family of functions. Monomial Factors Factor as indicated: ) = ( ) ) e e + e = e ( ) 5) + 6 = ( ) 6) + = ( ) ) 4) 6 = ( ) + 4 = ( ) 7) sin + tan = sin( ) Binomial Factors Factor as indicated: 8) ( ) ( ) ( ) = ( )( ) 9) ( + 4)( + ) + 6( + 4) = ( + 4)( ) 0) + = + ( ) + ) ( ) ( + ) ( ) ( + ) = ( ) ( + )( ) 5 ) ( + ) ( ) + ( + ) ( ) = ( + ) ( ) Factoring Quadratic Epressions Factor: ) + 4) 9 5) Factor by Grouping Factor Completely: ) ) ) ) + 5 8) e + + e 9) ) sin ) 5cos 5sin + sin + cos 4) cos + 4 cos + 4 tan

3 Cancellation Reduce each epression to lowest terms: 5) ) 8) 9) 6 + ( ) 7) ( + ) ( ) + ( + ) ( + ) 4 (sin + cos ) 0) sin Quadratic Formula Solve for the indicated term: ) Solve 4 = 0 for ) Solve + = 0 for ) Solve cos + cos + = 0 for cos 4) Solve y (+ y ) = 0 for 5) Solve = 0 for Synthetic Division Use synthetic division to factor as indicated: 6) = ( )( ) 7) = ( + )( ) 8) = ( )( ) 9) = ( )( ) Special Products Factor Completely: 40) 7 4) + 4) ) Simplifying 4) Rewrite each if the following in simplest form: 45) ( )( + ) ( + ) + 48) + 46) ) ( ) )

4 Rationalizing Remove the sum or difference from the denominator by multiplying by the numerator and denominator by the conjugate of the denominator: 50) 5) 5) cos Analytic Geometry Complete the following problems: 5) Find an equation for the line that passes through the point (, 5) and a) has slope b) is parallel to the -ais c) is parallel to the y-ais d) is parallel to the line 4y = 54) Find an equation for the circle that has center (, 4) and passes through the point (, ). 55) Find the center and radius of the circle with equation + y 6 + 0y + 9 = 0. 56) Let A ( 7, 4) and B (5, ) be points in the plane. a) find the slope of the line that contains A and B. b) find an equation of the line that passes through A and B. What are the intercepts? c) find the midpoint of the segment AB. d) find the length of the segment AB. e) find the equation of the perpendicular bisector of AB. f) find an equation of the circle for which AB is the diameter. Trigonometry 57) Without using a calculator, find the eact values. a) tan b) sin 7 6 c) sec ) If sin = and secy = 5 4, where and y lie between 0 and, evaluate sin( + y ). 59) Prove the identities. a) tanθ sinθ + cosθ = secθ b) tan + tan = sin 60) Find all values of such that sin = sin and 0, 6) Sketch the graph of the function y = + sin without using a calculator.

5 Family of Functions Without your calculator, graph each function. Clearly label the units. Be able to identify key points and domain and range. 6) f ( ) = 67) f ( ) = sin 7) f ( ) = ln 6) f ( ) = 68) f ( ) = cos 7) f ( ) = 64) f ( ) = 69) f ( ) = tan 74) f ( ) = 65) f ( ) = 70) f ( ) = sec 75) f ( ) = 66) f ( ) =! "# $ %& 7) f ( ) = e 76) f ( ) = a

6 Functions 77) The graph of a function f is given. a) State the value of f ( ). b) Estimate the value of f (). c) For what values of is f() = 4? d) Estimate the values of such that f() = 0. e) State the domain and range of f. 78) If f ( ) =, evaluate the average rate of change (difference quotient) your answer. 79) Find the domain of the function. f ( + h) f () h and simplify a) f ( ) = + + b) f ( ) = + c) h( ) = ) How are graphs of the functions obtained from the graph of f? a) y = f ( ) b) y = f ( ) c) y = f ( ) + 8) Without using a calculator, make a rough sketch of the graph (use parent functions). a) y = + 4 b) y = + ( ) c) y = ( ) + d) y = 4 e) y = + f) y = g) y = h) y = + 8) Let f ( ) =, 0 +, > 0 a) Evaluate f(-) and f(). b) sketch the graph of f 8) If f ( ) = + and g( ) =, find each of the following functions. a) f(g()) b) g(f()) c) g(g(g()))

7 Formulas, Theorems, and Equations to Memorize Quadratic Formula: = b ± b 4ac a Volume of a Right Circular Cylinder = r h Surface Area of a Right Circular Cylinder = rh + r Area of a Trapezoid = h(b + b ) Area of a Parallelogram = b h Area of a Triangle = b h Area of a square = s Area of an Equilateral Triangle = s 4 Area of a Circle = r Volume of a Right Circular Cone = r h Surface Area of a Right Circular Cone = r + r h + r Equation of a Circle: ( h) + (y k ) = r Equation of an Ellipse: Pythagorean Identities: o sin + cos = ( h) (y k ) + = a b Volume of a Sphere = 4 r o o + tan = sec + cot = csc Surface Area of a Sphere = 4r Distance Formula: d = ( ) + (y y ) Trigonometric Angles to Memorize: Function Sin Cos 0 0 Tan 0 undefined 0 undefined 0