AB Calculus Summer Review Packet

Due the first day of school

The problems in this packet are designed to help you review topics from Algebra2 and

PreCalculus that are important for your success in AB Calculus.

Please attempt the problems on your own without any notes and SHOW ALL WORK!

In addition, do not use your calculator for these problems. When you come across

topics that require a little review, feel free to look at your old notes or ask a friend for

help. If you want to check your work with a calculator, that is fine, too.

Bring finished packet with you to your AB Calculus class on the first day of school. You

will be assessed on these skills during the first week of school as part of your first

quarter grade.

Enjoy your summer. I am looking forward to seeing you in September. If you have any

questions, please contact me at gthornton@pmschools.org

Simplify:

1. 2 x

x 2.

− + −

−

52 2

2

x x

x 4

3. x x

4. 2

1 53

4

12

xy

x y

5. 2

273 6. ( ) ( ) ( )

− + −

−

23 1 2 1

1

x x x

x

7. − +3 1x −

2

3 1x 8. −

+

1 1

x h x

1

9. Graph the equation = 3 − and answer y x x

the following questions.

a. Is (3, 2) on the graph?

b. Is (2, 6) on the graph?

c. Is the function even, odd, or neither?

d. What is the y-intercept?

e. Find the x-intercepts

10. Write an equation of a line, in point-slope form, with the given conditions:

a. Through the point (2, 4) and is parallel to 2x + 3y – 8 = 0.

b. Perpendicular to the line 2x + 3y – 8 = 0, going through the point (1, 2).

c. Containing the points (1, -3) and (-5,2).

11. Solve:

a. x4 − 9x2 + 8 = 0 for x b. xy y 1 y` for`+ = + y`

2

12. Given ( ) =f x x − 3 −5 , find f (1) − f (5)

13. Given f x = 2 3x + , find ( + 2) − f (2) ( ) x − 4 f x .

f x+ Δx − f x ( )14. Find if f x( ) 8 2 + .( ) = x 1

Δx

x15. Given f x = , g(x) =( ) x +3 , and h x( ) = x2 + 5 , find: x + 3

a. h g x ( ( )) b. 𝑓 ∘ 𝑔 −2 c. f f( (3) )

16. Sketch the graph of each function:

⎧2x, (−∞,−1)⎧1, x ≤ 0 ⎪⎪ 2f xa. ( ) = ⎨− b. f (x) = ⎨2x , ⎡⎣−1,2 )1, x > 0⎩ ⎪

⎪−x +3, (2,∞)⎩

3

17. Without a calculator, determine the exact value of each expression:

π π 3π a. sin b. cos c. sin

2 3 4

7π 2πd. cos e. tan f. sin 0 6 3

π π g. tan h. cos i. sinπ

2 4

18. Aseven-foot ladder, leaning against a wall, touches the wall x feet above the ground.

Write an expression, in terms of x, for the distance from the foot of the ladder to the

base of the wall.

19. Find the surface area of a box of height h, whose base dimensions are p and q, and that

satisfies the following condition:

a. The boxisclosed.

b. The box has an open top.

c. The box has an open top and a square base with side length p.

4

20. If ( ) = x2 , describe in words what the following transformations would do to f x .f x ( )

a. f x − b. ( −( ) 4 f x 4)

c. f x + d. − ( +( ) 3 f x 2)

21. Which of the following expressions are identical?

a. cos2 x b. (cos x )2 c. cos x2

22. Which of the following expressions are identical?

a. (sin x)−1 b. arcsin x c. sin x−1 d. 1

sin x

23. Solveforx:

a. lne3 = x b. lnex = 4 c. lnx + lnx = 0

d. e ln 5 = x ln1 − = f. ln6 + lnx −e. lne x ln2 = 3

g. ln(x + 5) = ln(x −1)−ln(x +1)

5

24. For each question, determine the following:

lim ( ) lim f x f xf x ( ) lim ( )x→1− x→1+ x→1

a. b.

⎧x2 −1, x < 1 c. d. ( ) = ⎨f x

4 − x x, ≥1⎩

3x2 + 525. Identify the vertical and horizontal asymptotes in the graph of y = .

x2 − 4

26. Determine all points of intersection (using algebra)

a. parabola = 2 + 3x − 4 and the line y = 5y x x +11

b. y = cos x and y = sin x in the first quadrant.

6

27.Foreachfunction,makeaneatsketch.(nocalculator!)

a. =y x b. = 3y x

c. = xy e

d. = lny x e. = 2xy f. = 1/y x

g. = −2 4y x

h. = + +2 4 3y x x

i. = siny x

j. = −2y x k. = − 24y x l. = + −3 2y x

28.Simplify,usingpositiveexponentsonly.Donotrationalizethedenominator.

a.( )

−

− 34

4 16

4

x

x b.

− − − −

⎛ ⎞+ +⎜ ⎟⎝ ⎠

12

2 1 1 2

1 2 1x x y y

8