P-CEP. Mathematics. From: P-CEP Mathematics Department To: Future AP Calculus BC students Re: AP Calculus BC Summer Packet

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1 P-CEP Mathematics From: P-CEP Mathematics Department To: Future AP Calculus BC students Re: AP Calculus BC Summer Packet Dear future AP Calculus BC student: There are certain math skills that have been taught to you over the previous years that are necessary to be successful in calculus. If you do not have these skills, you will consistently struggle to correctly solve problems net year, even though you understand the calculus concepts. It is frustrating for students when they are doing calculus but are tripped up by the algebra. This summer packet is intended to help you brush up on and possibly relearn these topics. On the following pages are problems from many different important topics. All problems should be done on a separate sheet of paper, in order, and all necessary work must be shown. Your work will be due on the first day of class and will be graded as a 0-point quiz. To make the most of this packet and to start the semester off right, we recommend you spend some quality time with the packet this summer. Do not try to finish it before school is out for the summer we want the topics to be fresh in your minds in the fall! Do not attempt to do it all the night before the first day of classes you will find it a daunting task! We also recommend that you do not rely on your calculator. Almost all problems should be possible to solve using paper, pencil and your brain. Another don t don t fake your way through these problems. If you find yourself needing some assistance, please go to one of the websites listed below. In some cases, these sites have full instructions on certain techniques. We cannot emphasize enough that prospective AP Calculus AB students are notoriously weak on these topics, even students who have been very successful in their other math classes. Use the websites. Really. For Algebra Topics: Beginning Algebra Topics: Fractional and Negative Eponents Equations of Lines Solving for Indicated Variable Intermediate Algebra Topics: Domain Solving Inequalities o Absolute value o Quadratic

2 Even and Odd Functions Function Transformation Factor theorem (p over q method/synthetic division) Special Factorization Solving Quadratic by Factoring or Quadratic Formula Equations of Circles Advanced Algebra Topics: Asymptotes Comple Fractions Composition of Functions Solving Rational Equations Logarithms Inverse Functions For Trigonometry Topics: Right Triangle Trigonometry Solving Trigonometric Equations Trigonometric Identities For Other Topics: The Difference Quotient Area/Geometric Probability Parametric and Polar Equations Vectors Sequences and Series For Calculus Topics:

3 AP Calculus BC Summer Packet Name INSTRUCTIONS: Where applicable, put your solutions in interval notation. Do not use any calculator (ecept on Topics :# & :#). Please do all work on separate paper and do the problems in order. SHOW ALL WORK! We ve given you the answers we care about the process that gets you there. Simplify, using positive only eponents.. Topic : Fractional and Negative Eponents... y y Find the domain of the following functions.. y 8. y 9 9 Topic : Domain. y. y log Topic : Solving Inequalities Write the following absolute value function as a piece-wise function.. y Solve the following absolute value inequalities... Solve the following quadratic inequalities sin sin, 0 Solve the following rational inequality. 7. Topic : Even and Odd Functions Show algebra to determine if the relation is even, odd, or neither.. f 7. f. f Topic : Function Transformation If f, describe in words what the following would do to the graph of :. f. f. f. f. f. f

4 Topic : Factor theorem (p over q method/synthetic division) Use the p over q method and synthetic division to factor the polynomial P. Then solve P 0.. P() 0 Topic 7: Special Factorization Factor completely.. 7 y y 0y. 9y.. ( ) ( ) ( ) ( ) 7. ( 0 ) ( ) Solve each equation. Topic 8: Solving by Factoring or Quadratic Formula Topic 9: Asymptotes For each function, find the equations of both the vertical and horizontal asymptote(s), if they eist.. y. y 9 8. y Simplify. Topic 0: Comple Fractions. y y. 9. y y y y.. 0 Topic : Composition of Functions If f, g(), and h, find the following:. f g. h f. g f h. g f. g g. f h

5 Topic : Rationalizing Denominators and Numerators For problem # rationalize the denominator. For problem # rationalize the numerator.. (). Topic : Solving Rational Equations Solve equation for Topic : Right Triangle Trigonometry. If cos, in quadrant II,. If cot, in quadrant III, find sin and tan. find sin and cos.. A kite is 00m above the ground. If there are 00m of string out, what is the angle (in radians) between the string and the horizontal? (Assume that the string is perfectly straight.) Topic : Solving Trigonometric Equations Solve each equation on the interval [0, ). Please use a calculator to complete #.. cos cos. sin. sin sin. sin cos sin 0. 8 cos cos. sin cos 0 Simplify. Topic : Logarithms. log log log. log 9 log. log Solve each equation for the indicated variable.. a y b z c Topic 7: Solving for Indicated Variable, for a. A r rh, for r 0. 0, for

6 Determine the equation of each line: Topic 8: Equations of Lines. the line through (-, ) and (, -). the line through (-, ) and perpendicular to the line y 0. the line through (, ) and the midpoint of the line segment from (-, ) to (, ) Topic 9: Equations of Circles In # and #, for the circle y y 0, find:. the center and radius. the equation of the tangent at (-, ). A curve is traced by a point P, y which moves such that its distance from the point A(-, ) is three times its distance from the point B(, -). Determine the equation of the curve. Topic 0: The Difference Quotient Simplify f h f h, where:. f. f. f Find the inverse of each function:. f. f Topic : Inverse Functions Topic : Area. Find the ratio of the area inside the square but. Find a formula for the perimeter of a outside the circle to the area of the square in window of the shape in the picture the picture below. below. r r r. A water tank has the shape of a cone (like an ice cream cone without the ice cream). The tank is 0m high and has a radius of m at the top. If the water is m deep (in the middle) what is the surface are of the top of the water?

7 You should know the following identities:. sin sin. cos cos. sin cos. sin sin cos. cos cos sin. cos cos 7. cos sin Topic : Trigonometric Identities 8. cos cos 9. sin cos 0. cos y cos cos y sin sin y. sin y sin cos y cos sin y Please use a calculator to complete #. Topic : Vectors. Let u =, -, v =, -. Find u v.. Find a unit vector in the direction of v =, - and write your answer in component form.. Given that P =, - and Q = 7, -, find the component form and magnitude of the vector PQ.. Determine whether the vectors u and v are parallel, orthogonal, or neither. u =,, v = -0/, -/. Find a b, where a = i j, and b = -i + j.. Find the angle between the given vectors (in degrees) to the nearest tenth of a degree. u =, -, v =, - Topic : Parametric and Polar Eliminate the parameter, t.. t, y. 9 cos t, y 9sin t t. Find the rectangular coordinates of the point with the polar coordinates (-, ).. Find two polar coordinate pairs (in radians) to describe the point with rectangular coordinates (-, -).. Convert the rectangular equation to polar form: y. Convert the polar equation to rectangular form: r sin cos standard form.. Please leave your answer in Topic : Sequences and Series Find the eplicit form of the nth term of each sequence.. -, -, -,,, , -,, -, 8 Find the sum of the finite or infinite series.. n n. n n 0 ( ) n

8 Evaluate the following limits. 8. lim. lim 0 Topic 7: Limits. lim ( ) h 0 h h. sin( ) lim. lim f ( ), when f ( ),, Find the derivative of each function.. y. Topic 8: Basic Differentiation y ( )( ). y AP Calculus BC Summer Packet SOLUTIONS Topic : ) ( ) ) ( ) ) ( ) ) y y Topic : ) (-, -) (-, ) (, ) ) [9/, ) ) (-, -] [7, ) ) (, ) Topic : ), y ) [-, 7] ) (-, ) ) (-, -] [, ), ) [-, -] [, ) ) [0] [π/, π/] [π, π) 7), [, ) Topic : ) even ) odd ) neither Topic : ) translated units down ) translated units to the right ) reflected over the -ais and translated units left ) stretched vertically by a factor of and translated up units ) stretched horizontally by a factor of ½ ) no change (-, -) (, ), on [-, ] the graph would be reflected over the -ais Topic : ),,or Topic 7: ) ( y)( 9 y y ) ) ( )( ) ) ) ( + + y)( + y) ) ( ) ( ) ( ) ( y) ( ) ( ) ( ) ) 7) ( ) ( )

9 Topic 8: ) ) = - or ) = or 8 ) ),, or Topic 9: ) VA: none; HA: y = 0 ) VA: = 0, = -; HA: y = 0 Topic 0: ) y y ) VA: = ; HA: y = ) ), 0, ), -, ) y, -y 7 0 Topic : ) 9 ) ) ) ) ) Topic : ) ) 9 Topic : ) ) ) 7 ) i ) no solution Topic : ) sin ;tan 0 ) cos ;sin ) Topic : ) 0,, ) ) 0,,, ), 7,, ),, 8.,. 0 ),,,,, 7 Topic : ) log ( + ), > ) log ) Topic 7: ) a bc bc cy bz ) r h A h ) 7 Topic 8: ) y ) y ) y Topic 9: ) center is at (, ); radius = 0 ) y ) 8 8 8y 0y 0 Topic 0: ) ) ( h )( ) ) h Topic : ) f ( ) ) f ( )

10 Topic : ) ) r r ) 9 m Topic : ) 9, - ) 0, ), - ; 0 ) parallel ) -0 ).8 Topic : ) y ) y 8 ) (, -) ) (, );(, ) ) r ) ( ) ( y ) cos sin Topic : ) a n n 7 ) n a n ( )( ) ) ) Topic 7: ) - ) or does not eist ) ) ) Topic 8: ) y ' ) y ' ( ) ( ) ) y ' ( )