Determine the Composite Function

AP Calculus AB Summer Assignment 2017

Name: ______

Directions: Complete this packet of review material in its entirety. Any student enrolled in AP Calculus AB should be competent with the material presented in this packet. You may use notes or other resources to help you complete this packet of review material. You should not work together with other students nor receive extensive help from a tutor.

Bring the completed packet with you on the first day of school,

This packet will be checked for completeness and conceptual comprehension. All problems should be done or well attempted. Show work on every problem in the space provided. Write neatly.

You will be assessed separately on the topics presented in this packet within the first two weeks of school. You should use this packet as a study guide for the assessment. Prior to the assessment you will be given an opportunity in class to ask questions pertaining to the problems and concepts represented in this packet.

If you have significant trouble completing this packet you should contact your guidance counselor to reconsider your course placement.

I hope you are looking forward to a fun and challenging year in AP Calculus AB! See you in September!

Please contact me with any further questions. Mr. Cassidy [email protected] 1. Given: f (x)  x 2  3x  4 , find the following: Show all work. a) f (1) b) f (x  2)

f (x  2)  f (2) f (x  h)  f (x) c) d) ,h  0 x h

2. Sketch the graph of the equation. 3. Write an equation of the line that passes 3 y  2  x 1 through the point (2, 1) and is 2

a) parallel to 4x – 2y = 3

b) perpendicular to 4x – 2y = 3

4. Use point- slope form to write the equation of the line that passes through the points (2,-1) and (-4, 3) 1 5. Let f (x)  , g(x)  x 2 1, and h(x)  x  5 . 1 x a) State the domain and range of each function.

b) Find the following and state the domain of each.  g  i. h 1 (x) ii.  f  g(x) iii.  x  h 

iv.  f ° g(x) v. g(h(x))

6. Assume g(x) and h(x) are unknown functions. However it is known that g(0) 1, g(1)  3, g(2)  5 , g(7)  2 , h(1)  7 , h(2)  1, h(5)  0 . Evaluate:

a) (g ° h)(2) b) g(g(h(1))) c) h(g(h(5)))

d) g 1 (5) e) (g 1 ° h 1 )(1) f) (h 1 ° g 1 )(3)

7. If F(x)  f ° g ° h , identify a set of possible functions for f, g, and h. a) F(x)  2x  2 b) F(x)  4sin(1 x) 8. The domain of function f is [-6, 6]. Complete the graph of f given that f is a) even b) odd

9. Algebraically determine whether the functions are even, odd, or neither. Show all work. x2 x3  x a) g(x)  x4  2x  2 b) h(x)  c) g(x)  x2  4 2x2 1

10. Consider the function f (x)  x3  2x 2  x . a) Describe the end behavior of the function. (Consider the leading coefficient test.)

b) Find the zeros of the function and their multiplicity.

c) Sketch a graph of the function without using your graphing calculator. 11. Sketch a possible graph of the situation. a) The speed of an airplane as a function b) The value of a new car as a function of of time during a 5-hour flight. time over a period of 8 years.

12. For the following rational functions, state the equations of the vertical and horizontal asymptotes. 2x 2 1 3x 2x 2 11x 15 a) f (x)  b) f (x)  c) f (x)  x 2  4 x 4 16 2x  3

2  P  13. Let log P  x , log Q  y , and log R  z . Express log   in terms of x, y, and z. 10 10 10 10  3   QR 

14. Solve the equation: log6 x  1 log 6 (x 1) 15. The mass m kg of a radio-active substance at time t hours is given by m  4e0.2t . If the mass is reduced to 1.5 kg., how long does it take?

16. The function f is given by f (x)  ex11  8 . Find f 1 (x) and its domain.

17. The graph of y = f(x) and 6 transformations (a, b, c, d, e, g) are given. Match each of the transformation to one of the functions (i – vi) listed below.

i) y  f (x  5) _____ ii) y  f (x)  5 _____ iii) y   f (x)  2 _____

iv) y   f (x  4) _____ v) y  f (x  6)  2 _____ vi) y  f (x 1)  3 _____

18. Sketch  in standard position and find EXACT values for the 6 trig functions of  . 5 a)   495 b)     6 19. Find the values of the other 5 trig functions under the given conditions. 6 sec  and tan  0 5

20. Find two degree angles 0    360 sec  2

21. Solve the given equations on the interval 0,2 . Give the answers in radians. Show all work. a) sin2x  3  sin2x b) 3sec 2 x  4

22. Evaluate each expression. Give the answer in radians. Reminder: The range for inverse trig functions is restricted to the following intervals:         y = arcsinx  , y = arccosx 0,  y = arctanx  ,   2 2   2 2       y = arccscx  , , y≠0 y = arcsecx 0, , y≠ y = arccotx 0,   2 2  2  2 3    a) arctan 3 b) arccos1 c) arc csc  d) arc cot(1)  3 

 1  e) arc sec(1) f) arcsin(1) g) arc sec(2) h) arcsin   2 

 3   2      i) arctan(1) j) arc cot  k) arccos  l) arc csc 2  3   2  23. Sketch a graph and use it to evaluate the limit. If the limit does not exist, state the reason why. You may use your graphing calculator to help with sketching the graphs. x3 , x  2 x x a) lim b) lim c) lim x2 5, x  2 x4 x  4 x x  4

2 1 d) lim e) limcos f) lim x  2 x0 x 2 x0 x x2

24. Use the cancellation method to evaluate the following limits. Show all work. 1 1 t3  8  a) lim b) 5  x 5 t2 t  2 lim x0 x 25. Graph and solve the following quadratic inequalities: a.) x2  6x  8  0 b.) x2  4x  60

26. Simplify the following expressions 4 x 2 y 3  x 4 y 2 a.) b.) 2xy3 2x 4 y 5

3 2 1 1 3 5x y 2 c) 2x 2 (3x  4x 2 ) d.) 10xy

27. Rewrite the following as simplified expressions over a common denominator

3x 2  x x  2 2x 1 a.)  b.)  x  4 x x  4 x  3