AP Calculus AB Summer Packet (Due the 2nd day of class school year)

Transcription

1 AP Calculus AB Summer Packet (Due the 2nd day of class school year) Name: **Round answers to the nearest.001 ecept where eact answers are required.** **Selected answers are on the back. The graphs of these functions will be used routinely in the course. Practice these graphs until you can demonstrate them from memory. y =!!!!!!!!!!!!!!!!!!!y = 3!!!!!!!!!!!!!!!y = e!!!!!!!!!!!!!!!!!y = cos!!!!!!!!!!!!y = 2!!!!!!!!!!!!!!!!!y = 1 y = 1!!!!!!!!!!!!!!!!y = tan!!!!!!!!!!!!!!y = 2 3!!!!!!!!!!!!!!!!!y =!!!!!!!!!!!!!!!y = [ ]!!!!!!!!!!!!!!!!y = cot y =!!!!!!!!!!!!!!!y = ln!!!!!!!!!!!!!!!!y = sin!!!!!!!!!!!!!y = sec!!!!!!!!!!!!!y = csc 1. Match each above equation with its graph below: y = y = y = y = y = y = y = y = y = y = y = y = y = y = y = y = y = Important Trig Identities sin 2 + cos 2 = 1!!!!!!sin 2 = 2sin cos!!!!!!!!! tan = sec 2!!!!!!cos2 = cos 2! sin 2 1+ cot 2 = csc 2 2. Simplify: 3! 125 a.! 5 b. (Hint: graph) 2

2 3. a. Given g( ) =! 2 + 3, write the equation of the line through g (!2) and g( 3). b. Write the equation of the line that passes through the -intercept of 2 3! y normal! ( ) to that line. ( ) =!3 and is 4. Solve for. Eact answers mean eact, not decimal. a. e 2! 4e + 3 = 0 b. 2!+!cos2!=!3sin2 [eact answer(s)] 0! < 2 [eact answer(s)] Answer(s): Answer(s): c. ln!(!+!4)!+!ln!(!-!4)!=!9t (Solve for as a function of t.) Answer(s): (Before you move on, think about the domain of ln() ) 5. Given f ( ) = 3 + 1, find f ( + h)! f () h 6. In complete sentences, define the domain and range of a function.

3 7. Find the domain and range for the following functions. Use Interval Notation. a. f ( ) = % 9! 2 if 0 <3 $ &% e if <0 b. f ( ) = sin (Use your calculator) c. f ( ) = 2 +! 12! 3 d. f ( ) = 2 +! 12 (! 3) ( + 5) e. Graph g() = + 4 and compare it to the function in part c. How are their domain/ranges the same or different? What does g(3) have to do with the graph of f ()? 8. Given f $ ( ) = 4sin! 2 % & ' ( 2 a. Amplitude b. Period c. Graph one period. d. Write the equation of the line through the maimum point and minimum point of the period graphed.

4 9. Given the graph of y = f y ( ) below: a. graph y =! f ( ) b. graph y = f (!) y y 10. There eists a function y = f (!), such that f (!.4) =!!!.864!!and!!! f (!.6) =!!! a. If f() is an even function, find f(.4) = b If f() is an odd function, find f(.6) = 11. Given y = 2( + 3)(! 2) (! 3)( + 3) Identify: a. any holes b. any vertical asymptotes c. any horizontal asymptotes d. - intercepts e. y - intercepts f. Graph

5 12. A closed bo with a square base of side () and height (y) has a surface area of 100 sq. ft. a. Epress the volume of the bo as a function of. V() = b. Domain of V() = Range of V() = 13. At 8 am, the math dept. copier was making 100 copies per minute. Its counter read At 4 pm, the copier was only making 84 copies per minute (it gets tired) and the counter read Find the average number of copies made per minute from 8 am to 4 pm. 14. The area of the trapezoid at right is is 30 2 cm 2. Find the eact value of. 7 cm 6 cm 15. Water is poured into a conical container with a diameter of 20 cm and a height of 38 cm. If the volume of a cone is one-third the volume of a cylinder, write an equation for the volume of water as a function of the height of the water in the container. 20 cm 16. An investment of C dollars accumulates r% annual interest compounded quarterly. a.) Write a model to represent the value of the account after t years, in terms of C, r, and t. b.) If the value after 1 year is $2100 and the value after 3 years is $2400, find the values of r and C. Selected Answers on Back

6 2a b. 1,!! 0 3a. y =!! 3 3b. y = a. = ln 3,!0 4b.! 3, 2! 3, 4! 3, 5! 3 4c. e 9t h + h 2 6. Answers Vary 7a. D :(!,3)!!R :(0,3] 7b. D :!,! 0!!R :[.217,1) 7c. D :!,! 3!!R :!,! 7 7d. The hole in the graph of f() is not in the graph of g(). 8a. Amp=4 Period=4 8d. y =! a.! b a.!3, 5 % $ 3& ' 11b. = 3 11c. y = 2 11d. (2,0) 11e.! 0, 4 $ 3 % & 12a. V() = 1 ( 100! 23) 12b. 0, ( ) 12c. ( 0, ] copies/minute 14. = 10 2! V(h) = 25!h3 1083! 16a. A = C 1+ r $ 4 % & 4t 16b. r = 6.7%!!!!C = $