AP Calculus I and Calculus I Summer 2013 Summer Assignment Review Packet ARE YOU READY FOR CALCULUS?

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1 Name: AP Calculus I and Calculus I Summer 0 Summer Assignment Review Packet ARE YOU READY FOR CALCULUS? Calculus is a VERY RIGOROUS course and completing this packet with your best effort will help you to succeed. This packet is to be completed and is due on the first day you return to school in September. All work for Part I is to be done neatly and orderly on your own paper while Parts II and III are to be completed in this packet.you will be tested on this material in your AP Calculus I or Calculus I class in September. This review packet is meant to help you review the algebra, geometry, and pre-calculus (math analysis) concepts you will need in order to have success in your Calculus eperience. This packet contains questions assessing topics THAT YOU HAVE ALREADY BEEN TAUGHT. These topics will NOT be re-taught during your Calculus course. If there are any topics in this packet that you do not remember how to do you should work on them over the summer using any resources available (working groups, asking for help from family and/or friends, tutoring, internet sites, review workbooks, independent study, ) Do not wait until the end of the summer to look at this packet. You should work on this packet a little at a time. This will allow you to take the necessary time to review topics that you are having trouble remembering how to do. You are responsible to review these concepts if you have not mastered them yet. Here are some websites that may be of some help: Calculus Tetbook Appendices:(click on Appendices and then choose D., D., or D.) Precalculus Lessons at: The Greatest Integer Function at: Quiz Questions and Review at: And of course there is always Ask Dr. Math at:

2 Note: Ecept for # 4 and 8 calculator use is NOT permitted to solve any of the problems. Part I. Simplify: (a) f ( ) 9 7 (b) 8 (c) 5 5 (d) 9. Rationalize the denominator: (a) (b) 4 5 (c) 5. Write each of the following epressions in the form c a p b q where c, p and q are numbers: (a) b a (b) 9ab (c) a b b (d) ab a b b a (e) b a (f) a b b a 4. Solve for (do not use a calculator): (a) 5 ( + ) = 5 (b) (c) log = (d) log = log 4-4 log 5 5. Simplify: (a) log 5 + log ( - ) - log ( - ) (b) log log (c) log 5 6. Simplify: (a) log 0 0 (b) log 0 (c) log log Solve the following equations for the indicated variables. (a) y z (b) V = (ab + bc + ca), for a, for a a b c (c) A r rh, for r (d) A = P + nrp, for P (e) - yd = y + d, for d (f) 0 4, for

3 8. For the equations (a) y = (b) + + y = 0 (c) 9y - 6y = 0 complete the square and reduce to one of the standard forms y - b = A( - a) or - a = A(y - b). 9. Factor completely: (a) (b) (c) (d) 4-0. Find all real solutions to: (a) = 0 (b) 7-64 = 0 (c) (d) 0. Solve for : (a) sin cos ; 0 (b) cos sin sin ; - (c) tan sec cos ;. Without using a calculator, evaluate the following: (a) cos 0 (e) cos 9 4 (b) sin 5 4 (f) sin (c) tan - (-) (g) tan 7 6 (d) sin- (-) (h) cos - (-).) Given the graph of sin, sketch the graphs of: (a) sin (b) sin 4 (d) cos (e) sin (c) sin

4 4. Solve the equations: (a) = 0 (b) (c) 0 5. Find the remainders on division of (a) by + (b) by + 6. (a) The equation = 0 has a solution =. Find all other solutions. (b) Solve for, the equation = 0. (All solutions are rational and between ±.) 7. Solve the inequalities: (a) 0 (b) (c) 0 8. Solve for : (a) 4 (b) 5 - = 8 (c) + = + 9. Determine the equations of the following lines: (a) the line through (-,) and (,-4); (b) the line through (-,) and perpendicular to the line - y + 5 = 0; (c) the line through (,) and the midpoint of the line segment from (-,4) to (,). 0. (a) Find the point of intersection of the lines: - y - 7 = 0 and + 5y + = 0. (b) Shade the region in the y-plane that is described by the system of inequalities. y 7 0 5y 0. Find the equations of the following circles: (a) the circle with centre at (,) that passes through the point (-,-); (b) the circle that passes through the origin and has intercepts equal to and on the - and y- aes, respectively.

5 . For the circle + y + 6-4y + = 0, find: (a) the centre and radius; (b) the equation of the tangent at (-,5).. A circle is tangent to the y-ais at y = and has one -intercept at =. (a) Determine the other -intercept. (b) Write the equation of the circle. 4. 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. 5. (a) Find the domain of the function f ( ) 5 (b) Find the domain and range of the functions: i) f ( ) 7 ii) g( ) 6. Let f ( ), show that, 0 f ( ), 0 Then, find the domain and range of f ( ). 7. Simplify f ( h) f ( ) h, where (a) f ( ) (b) f ( ) (c) f ( ) 8.) The graph of the function y f ( )is given as follows: Determine the graphs of the functions: (a) f ( ) (b) f ( ) (c) f ( ) (d) f ( )

6 9. Sketch the graphs of the functions: (a) g( ) (b) h ( ) 0. (a) The graph of a quadratic function (a parabola) has -intercepts - and and a range consisting of all numbers less than or equal to 4. Determine an epression for the function. (b) Sketch the graph of the quadratic function y = Find the inverse of the functions: (a) f ( ) (b) f ( ) 5 (c) f ( ), 0. A function f ( ) has the graph to the right. Sketch the graph of the inverse function

7 . Epress in terms of the other variables in the picture. 4.)(a) Find the ratio of the area inside the square but outside the circle to the area of the square in the picture (a) below. (b) (c) (d) (e) Find a formula for the perimeter of a window of the shape in the picture (b) above. A water tank has the shape of a cone (like an ice cream cone without ice cream). The tank is 0m high and has a radius of m at the top. If the water is 5m deep (in the middle) what is the surface area of the top of the water? Two cars start moving from the same point. One travels south at 00km/hour, the other west at 50 km/hour. How far apart are they two hours later? A kite is 00m above the ground. If there are 00m of string out, what is the angle between the string and the horizontal. (Assume that the string is perfectly straight.)

8 5. Using the trig identities: (a) Show that sin cos can be simplified to csc cos sin (b) Rewrite sin so that it is not in fractional form. (c) Show that sec can be simplified to sin sec 6. Use Binomial Theorem Epansion (do NOT use repeated multiplication) to epand the following: (a) ( ) 4 (b) ( - y) 6 (c) ( u + b) 0 7. Determine the right-hand and left-hand behavior of the graph of: (a) f ( ) f ( ) (b) 4 8. Use the Intermediate Value Theorem and a graphing calculator (or a sketch of the graph) to find intervals of length one in which the polynomial is guaranteed to have a zero. 4 (a) f ( ) (b) f ( ) 0 9. Two planes start from the same airport and fly in opposite directions. The second plane starts one-half hour after the first plane, but its speed is 80 kilometers per hour faster. Find the air speed of each plane if hours after the first plane departs the planes are 00 kilometers apart The path of a diver is given by y where y is the height in feet and is the horizontal 9 9 distance from the end of the diving board in feet. What is the maimum height of the dive?

9 Part II Graphical Analysis Much of Calculus deals with functions and their graphical characteristics. To facilitate the study of functions, it is important that you be able to quickly sketch the graph of a basic function or a transformation of a basic function without a calculator. Without using a calculator, graph each function below. As always, it is necessary to indicate the units and label the aes provided. Full credit will not be earned without units and aes completely labeled. f f ( ) ( ) y f ( ) f ( ) ( ) f ( ) f ( ) sin f ( ) cos f ( ) tan

10 f ( ) csc f ( ) sec f ( ) cot f ( ) f ( ) e f ( ) ln f ( ) f ( ) f ( ) (Greatest integer function)

11 Part III FILL IN THE TABLE BELOW. You may use the following abbreviations: R - the set of real numbers, Z - the set of integers, and N - the set of natural numbers. FUNCTION DOMAIN RANGE X-INTERCEPTS Y-INTERCEPTS f ( ) f ( ) y f ( ) f ( ) ( ) f ( ) f ( ) sin f ( ) cos f ( ) tan f ( ) csc f ( ) sec f ( ) cot f ( ) f ( ) e f ( ) ln f ( ) f ( ) f ( ) ODD OR EVEN SYMMETRY?