Advanced Calculus Summer Packet

Transcription

1 Advanced Calculus Summer Packet PLEASE DO ALL YOUR WORK ON LOOSELEAF PAPER! KEEP ALL WORK ORGANIZED This packet includes a sampling of problems that students entering Advanced Calculus should be able to answer. The questions are organized by topic: A Super-Basic Algebra Skills T Trigonometry F Higher-Level Factoring L Logarithms and Eponential Functions R Rational Epressions and Equations G Graphing In Calculus, it s rarely the calculus that ll get you; it s the algebr Students entering Calculus absolutely must have a strong foundation in algebr Most questions in this packet were included because they concern skills and concepts that will be used etensively in Calculus. Others have been included not so much because they are skills that are used frequently, but because being able to answer them indicates a strong grasp of important mathematical concepts and more importantly the ability to problem-solve. This packet will be collected on the first FULL DAY of school. It is etremely important for all students to review the concepts contained in this packet and to be prepared for an assessment of these prerequisite skills to take place within the first week of school. Date to be given on first FULL DAY of school. Students whose scores show they were not prepared for the assessment probably either a) don t have the mathematical prerequisite skills necessary for success in Calculus, or b) don t have the work ethic necessary for success in Calculus. With this being said, we will work through any prerequisite skills that need to be built up while learning Calculus methods and skills along the way. Please feel free to contact me by this summer if you have any questions. Coach Swagart jswagart@baysideacademy.org

2 A: Super-Basic Algebra Skills A1. True or false. If false, change what is underlined to make the statement true. 1 T F 1 b. T F c. ( + ) = + 9 T F 1 1 T F e. ( + 1) = 16( + ) T F f. 5 T F g. If ( + )( 10) =, then + = or 10 =. T F A. More basic algebr If 6 is a zero of f, then is a solution of f () = 0. b. Lucy has the equation ( + 6) 8 = 16. She multiplies both sides by ½. If she does this correctly, what is the resulting equation? c. Simplify 10 Rationalize the denominator of 1 1 e. If f () = + + 5, then f ( + h) f () = (Give your answer in simplest form.) 1 f. A cone s volume is given by V r h. If r = h, write V in terms of h. g. Write an epression for the area of an equilateral triangle with side length s. h. Suppose an isosceles right triangle has hypotenuse h. Write an epression for its perimeter in terms of h.

3 T: Trigonometry Do your best to answer all without the use of your unit circle first! If you must, do use your unit circle. It is advantageous to evaluate all trigonometry without a Unit Circle. T1. Use the following directions for the given angles: a) Convert each angle in degrees to radians, or radians to degrees. b) Determine a positive and negative coterminal angle. c) Determine the reference angle. 150 b. 150 c. 0 e. 5 6 f. 7 T. Find the value of each epression, in eact form. sin b. cos 11 c. 6 tan sec 5 e. csc 7 f. cot 5 6 T. Find the value(s) of in [0, ) which solve each equation. sin b. cos 1 c. tan sec e. csc is undefined f. cot 1 T. Find the value(s) of in [0, ) which solve each equation. Solve by factoring. cos 5 9cos b. cos cos = 0 c. sin cos sin 0 tan sin tan 0 T5. Graph each function, identifying - and y-intercepts, if any, and asymptotes, if any. y = -sin () b. y = + cos c. y = tan 1 y = sec + 1 e. y = csc () f. y = cot

4 F1. Solve by factoring = 0 F: Higher-Level Factoring b. + 6 = 0 c. ( 6) + ( 6) 10 = = + 8 F. Solve by factoring. You should be able to solve each of these without multiplying the whole thing out. (In fact, for goodness sake, please don t multiply it all out!) ( + ) ( + 6) + ( + )( + 6) = 0 b. ( ) ( 9) + ( ) 5 ( 9) = 0 c. ( + 11) 5 ( + 5) ( 1) + ( + 11) ( + 5) ( 1) = ( 1) ( ) F. Solve. Each question can be solved by factoring, but there are other methods, too. 1 a ( a ) (a ) 0 b. 6 0 c. 6 1 ( 1) 1 1

5 L: Logarithms and Eponential Functions L1. Epand as much as possible. ln y b. ln y c. ln ln y L. Condense into the logarithm of a single epression. ln + 5ln y b. ln a 5ln c. ln ln ln ln (contrast with part c) L. Solve. Give your answer in eact form and rounded to three decimal places. ln ( + ) = b. ln + ln = 1 c. ln + ln ( + ) = ln ln ( + 1) ln ( ) = ln L. Solve. Give your answer in eact form and rounded to three decimal places. e + 5 = 1 b. = 8 1 c. 100e ln = 50 = 1 (need rounded answer only on d)

6 R: Rational Epressions and Equations R1. b. c. f ( ) Function Domain Horiz. Asym., if any ( ) 8 f ( ) 6 f ( ) skip 10 8 Vert. Asym.(s), if any R. Write the equation of a function that has asymptotes y = and = 1 b. an asymptote = 0, and no horizontal asymptote R. Find the -coordinates where the function s output is zero and where it is undefine For what real value(s) of, if any, is the output of the function f ( ) e equal to zero? undefined? 6 1 b. For what real value(s) of, if any, is the output of g ( ) equal to zero? undefined? cos sin R. Simplify completely. (Don t worry about rationalizing) b. 1 (Your final answer should have just one numerator and one denominator) c. 5 1 ( ) 1 ( ) 5 (Don t worry about rationalizing)

7 G: Graphing G1. PART of the graph of f is given. Each gridline represents 1 unit. Complete the graph to make f an EVEN function. (Use the graph to the right to complete the graph. Do it in RED ink!) b. What are the domain and range of feven? c. What is feven(-)? Complete the graph to make f an ODD function. (Use the graph to the right to complete the graph. Do it in BLACK ink!) e. What are the domain and range of fodd? f. What is fodd(-)? G. The graphs of f and g are given. Answer each question, if possible. If impossible, eplain why. Each gridline represents 1 unit. f -1 (5) = b. f (g(5)) = c. (g f )() = Solve for : f (g ()) = 5 e. Solve for : f () = g () For parts f i, respond in interval notation. f. For what values of is f () increasing? g. For what values of is g () positive? h. Solve for : f () < i. Solve for : f () g() G. Given the graph of y = f () (dashed graph), sketch each transformed graph. b. c. y = f ( + ) y = f () y = f () y = f () + 1