Name:_____________________________
AP Calculus AB
Pre-Calculus Skills
Review Packet
1
Name:_____________________________
Preparing for Calculus
Welcome to an exciting year of AP Calculus! Yeah, we’re getting started with you a bit early,
but we think that it is important to start the year off learning Calculus rather than reviewing
things you should already know. If we are able to do that we’ll have about 4 or 5 weeks before
the AP test in May to review, work practice tests, and discuss problems that have appeared on
former AP tests. Having that much time is a HUGE advantage.
But to get that, you’ll need to sacrifice a bit of your time this summer. I know that may not sound
like something you want to do, but we’ve asked our students to do this for years. For the most
part, they have. Without a doubt, they have all appreciated the opportunity to thoroughly review
at a normal pace in April, rather than cramming at the last minute.
Certainly the reward of making sure you know these basic pre-calculus skills before stepping foot
into a Calculus classroom should be motivation enough. However, we, as a human race, tend to
gravitate toward more direct rewards. In light of that, we are going to offer some extra credit to
those of you who complete these assignments. You’ll get either 1 or 2 points of effort extra credit
for each assignment you complete (depending on the length of the assignment). Your work must
be neat, organized, complete, and detailed. A final answer is generally not sufficient if you want
to earn this extra credit. You must finish all of the problems on an assignment using “SWOOP”
(Show Work On Own Paper) to earn the extra credit. Please copy the problem down as well.
We’ve attached some graph paper for problems you need to make graphs for.
There is one other thing we’d like you to do in preparation of next year’s class. Please purchase
and/or otherwise secure the following items before the first day of class next year. Bring them
with you on the first day of class.
1. A 3-inch, 3-ring binder. Yup, that’s the big one. Do NOT buy a smaller one. You won’t
have enough room to store everything we give you.
2. A graphing calculator. We strongly recommend either a TI-83+, TI-84, or TI-89.
3. One dozen pencils (yes, we’re serious about this).
If you have any questions on the review material, or about the class in general, feel free to email
us at anytime. We can’t guarantee an immediate response, but we’ll get back to you with an
answer at some point.
Now, on to your review problems. Remember be thorough and detailed with your work. Consult
the lesson notes that accompany this hand-out, as much as necessary.
2
Name:_____________________________
Topic A
Simplifying Expressions
Simplify.
(2x – 3)(x + 5)
(x3 + 1)(2x2 – 3x – 2)
(2x + 4)2
(x2 + 3x + 2)2
1.
2.
3.
4.
Simplify. Leave no negative exponents in your final answer.
5.
6.
7.
8.
9.
10.
(3xy5)4
(7xy2)0
x-3
3x-2
(3x-2)3
(3x2)-4
 x3 
11.  4 
y 
5
Simplify by rationalizing the denominator.
8
3
12.
10 5
5
4x  2
14
x 1
13.
Simplify by rewriting with no radical signs.
15.
3
x2
16.
5
x8
Simplify by rewriting with no fractional exponents.
17. x
18. x
6
4
5
9
3
Name:_____________________________
Topic B
Fraction Basics
Simplify.
1.
10 1

11 6
2.
2 10

3 13
3.
3 6

7 5
4.
5 1

12 2
5.
x 5

4 6
6.
5 x

6 6
7.
2 4

11 7
8.
3x  6
3
12.
4
1
6
6x 1
9.
3x
2 1
10.
1 3
2
11.
5
1 1
2
3
1 1
6
Topic C
Solving Algebraic Equations
Solve.
1. 2x2 + 5 = 13
2. 5x + 2(x + 4) = -6
3. 5x + 5 = 2x – 1
4. 80 = 10(3t + 2)
5. 5[2 + 3(x+1)] = 4
6. 5x2 – 10 = 2x2
7.
x2
5  7
3
8. 4(x2 + 5) = 2(x2 + 1)
Solve the following for r.
9. SA = 4r2
10. V = h( - r)2
4
Name:_____________________________
Topic D
Factoring
Factor.
1. 5x4 – 20x3
2. –2x3 + 6x2 – 10x
3. x2 + 8x +15
4. 2x2 + 34x – 220
5. 12x2 + 23x + 10
6. 3x2 + 10x – 8
7. x2 + 6x + 9
8. x2 – 25
9. 4x2 – 16
10. 2x2 – 12xy + 18y2
Solve by using the zero product rule.
11. x2 + 8x + 12 = 0
12. x2 + 10 = -7x
13. 2x4 + 6x3 + -4x2
Topic E
Inequalities
Solve, then graph, each inequality.
1. 16 – 7y > 10y – 4
2. –5x > -80
3. –3(x + 8) – 5x > 4x – 9
4. | 3x – 2 | < 4
5. | p – 2 | > 3
6. | x – 5 | > 10
Topic F
More Fractions
Simplify and reduce.
1. 4 x 
4.
3x  2
3
x  3 x 1

3x
2x
2.
4x  2 6x

3
x 1
3.
x 1 2x  2

x
x
5.
3
4

x 1 x  2
6.
2 x  1 3x

x
x2
5
Name:_____________________________
Topic G
Linear Equations and Systems
1. Determine the equation of a line with a slope of
2
and a y-intercept of (0, 4).
3
2. Determine the equation of a line with a slope of 2 and a y-intercept of –1.
3. Determine the equation of a line that goes through (1, 6) and (-3, 5).
4. Determine the equation of a line that goes through (-1, -5) and that is parallel to 3x + y = -4.
5. Determine the equation of a line that goes through (1, 1) and that is perpendicular to –2x + y = 6.
Find the slope and y-intercept of each line.
6.
5
x 1  y
2
7. 5x + 4y = 4
Solve the following systems, the first by substitution and the second by elimination. How could
you check your answer with a graphing calculator?
8. x + y = 3
5x – y = -27
9. 3x + 3y = 15
2x + 6y = 22
Topic H
Trigonometry
Use the diagram at the right to express each ratio.
1. sin (A)
2. cos (B)
3. tan (A)
4. sec (B)
A
m
n
p
5. cot (A)
6. csc (B)
Prove the following trig identities. SHOW WORK!
7. tan   sin 2 = 2sin2
8. sin   tan  + cos  = sec 
6
B
Name:_____________________________
Topic H (cont’d.)
Solve for x without the help of a calculator. Give only the principal value.
9. cos

3
x
10. sin x 
1
2
11. cos
7
x
6
12. cos x  1
Graph the following equations. Use the graph paper provided.
13. y  2  3sin[ 1 ( x   )]
14. y  3  cos(4 x)
2
Topic I
Solving Trig Equations
Solve the following without the help of a calculator. Answers should be exact (no decimal
approximations, please). Restrict x to: -2 < x < 2.
1. sin

3
x
5. cos x 
1
2
2. cos
5
x
6
6. sin x 
3
2
3. sin
5
x
4
7. sin x  1
4. cos

x
6
8. cos x  0
Solve the following algebraically. Use a calculator for the final step. SHOW WORK!
9. 5 sin 3(x + 4) = 2
10. 2 cos2(2x + 5) = 1
Solve the following graphically. Give only the two solutions that are closest to the origin.
11. cos (3x) = sin (x3)
12. sin2(x + 1) = log (x)  tan (x)
7
Name:_____________________________
Topic J
Quadratic Equations
For exercises 1-3, determine the following:
A. Orientation (opens up/down)
B. Coordinates of the vertex
C. Number of roots
1. y = x2 + 2x + 3
2. y = -2x2 + 5
3. y = x2 + 6x + 9
Find the roots by factoring.
4. y = x2 – 3x – 10
5. y = x2 – 2x + 1
Find the roots by using the quadratic formula.
6. y = x2 + 5x – 1
7. y = -3x2 + 2x + 10
Find the roots by completing the square.
8. y = x2 + 2x – 4
9. y = x2 – 10x – 1
Graph the following by plotting the x-intercepts (roots) and the vertex. Use the graph paper
provided. Indicate the coordinates of the points on your graph.
10. y = x2 + x – 6
11. y = x2 + 5x – 2
Topic K
Absolute Value Equations
Solve the following algebraically.
1. | 4x + 1 | = 20
2. | x2 – 3 | = 8
3. | -x – 4 | = 10
Rewrite the following as piece-wise functions.
4. y = | 4x – 5 |
5. y = | 8 – 3x |
Graph the following. Use the graph paper provided. Indicate the coordinates of at least two points.
6. y = | x2 + 2x – 3 |
7. y = | cos (2x) |
8
8. y = | x – 1 |
Name:_____________________________
Topic L
Rational Equations
For exercises 1-5, solve the equation algebraically. For exercise 6, solve the equation graphically.
For all of the exercises, state all domain restrictions. SHOW WORK!
x2  9x  8
2.
0
x 1
3x  6
0
1.
x4
4.
6x  4
3
x2
5.
sin x
0
ln x 2
3.
2 x3  3x 2
0
x3  4 x
6.
cos( x  1)
 10
log( x  2)
Topic M
Logarithmic and Exponential Equations
Solve the following without the help of a calculator.
1. log 3 81 = x
2. log x 25 = 2
3. log x = 3
Consider the function, y = log 3 x for the following. Do NOT use a calculator.
4. Give the inverse of this function.
5. Give 3 solutions for this function.
6. Graph (on the graph paper provided) and label the function and its inverse.
Rewrite the following strictly in terms of log 3 and/or log 2.
7. log 12
8. log 18
9. log (0.666666)
Use the change of base formula and your calculator to evaluate the following.
10. log 4 5
11. log 7 100
Solve the following algebraically. Use your calculator in the final step. SHOW WORK!
12. e3x = 19
13. log (3x2 – 2) = 5
15. ln (2x2 – 2x + 5) = 6
9
14. 5 (x - 2) = 250
Name:_____________________________
Topic N
Functions and Relations
State the domain and range of each of the following functions.
1. y 
x2
4. y  log x
7.
x2  5
x2  6 x
10. y = x2 + 6x – 1
13.
2. y 
x2  5x  6
5. y = log (3x – 6)
8.
x2  3
log x
11.
3.
y   x2  5x
6. y = ln (x2 + 7x + 12)
9. y = (x + 2)(x + 5)(x2 – 3)
x
x 1
12. y = (x + 1)(x – 2)(x – 3)2
3x
3 x
Decide whether the following functions are even, odd, or neither by graphing or using the definitions
provided in your lesson notes. State which method you use to decide.
14. y = x4
15. y = 3x + 5
16. y = x2 + 4
17. y = sin x
18. y = sin (x2)
19. y = sin (3x)
20. y = sin (3x + 5)
21. y = | x |
22. y = | x3 |
23. y = x2 + 2x + 1
24. y = x2 + 2x + 2
25. y = cos x
26. y = cos (ax); where a is any integer
27. y = x3 + 4
28. y = (x + 4)3
29. log x
30. y = log | x |
31. y = 3x
32. y = 3 |
33. y = sin | x |
34. y = x2 + cos x
35. y = x + x3
36. y = x2 + x3
10
x|
Name:_____________________________
Topic O
Limits and Continuity
Use the graphs below for exercises 1-21.
1. Name the x values for which f (x) is discontinuous on (-3, 3).
2. Name the x values for which g (x) is discontinuous on (-3, 3).
Determine whether the following limits exist. If they do, find the limit.
3. The limit of f (x), as x approaches –1.
4. The limit of f (x), as x approaches 1.
5. The limit of g (x), as x approaches –1.
6. The limit of g (x), as x approaches 1.
7. The limit of f (x) + g (x), as x approaches –1.
8. The limit of 2 f (x) + 3 g (x), as x approaches 0.
9. The limit of f (x)  g (x), as x approaches –1.
10. The limit of f (x)  g (x), as x approaches 2.
12. The limit of f (x)  g (x), as x approaches 0.
13. The limit of g (x)  f (x), as x approaches 0.
14. The limit of g (f (x)), as x approaches –2.
15. The limit of f (g (x)), as x approaches –1.
11
Name:_____________________________
16-17. The limit of f (x), as x approaches –1 from the left. From the right.
18-19. The limit of g (x), as x approaches –1 from the left. From the right.
19. The limit of g (x), as x approaches –1 from the right.
20. The limit of f (x + 2), as x approaches 0 from the left.
21. The limit of f (x2), as x approaches –1 from the left.
Find each limit.
22. The limit of x2 + 3, as x approaches 3.
23. The limit of log x, as x approaches 0 from the right.
24. The limit of x3, as x approaches -.
25-26. The limit of
27. The limit of
x  1 , as x approaches 1.
28-29. The limit of
30. The limit of
2
, as x approaches 3 from the right. From the left.
x 3
1
, as x approaches 0 from the right. From the left.
3x
1
, as x approaches 0.
3x
Graph the piecewise function (on the graph paper provided). Then find the limits below.
y=
3; if x < -2
1; if x = -2
x; if –2 < x < 0
2; if x = 0
x2; if x > 0
31. The limit of f (x), as x approaches –2.
32. The limit of f (x), as x approaches 0.
33-34. The limit of f (x), as x approaches –2 from the right. From the left.
35-36. The limit of f (x), as x approaches 0 from the left. From the right.
12
Name:_____________________________
Topic O (cont’d.)
Rewrite the following as a piecewise function to remove the discontinuity.
37. y 
x2  5x  6
x 2  10 x  16
38. y 
x2  5x
x2  4 x  5
39. y 
x2  7 x  6
x2  5x  4
Topic P
Asymptotes and End Behavior
Find the asymptotes (horizontal, vertical, and oblique) of the following functions. Check by graphing.
1. y 
x5
x 2  3x
2. y 
x2  2 x  3
x 1
3. y 
2 x2  4 x  9
5 x 2  10 x
4. y 
x3  x 2  3
x 1
5. y 
x2  2x  5
x3  4 x 2  3x
6. y 
x2  2 x  5
x2  4
7. y 
x2
x3
8. y 
x 2  7 x  12
x2  2x  1
9. y 
x2  1
x3  1
2 x 2  3x  4
10. y 
x 1
x3  2 x  1
12. y 
x2  5x
x6
11. y 
x2
Topic Q
Inverse Functions
Determine (algebraically) whether the two functions are inverses.
1. y = 2x + 5
y=½x+5
2. y = x2 + 4
y=
3. y = x3
x +4
y=
3
4. y = x2 + 2x
x
y = 2x + 2
Find the inverse of each function algebraically. Show your work.
5. y = 3x + 2
6. y = ½ x – 6
7. y = x2 + 5
13
8. y = log3 x
Name:_____________________________
Topic Q (cont’d.)
Graph the function (on the graph paper provided). Then, reflect the graph across the line y = x to graph
the inverse.
9. y = 3x – 5
11. y = x2 – 3x – 18
10. y = x2 + 4x + 3
12. y = (x + 3)3
13. Given the graph of a function, how can you determine whether the inverse is a function without
finding the inverse?
Topic R
Inverse Trig Functions
Evaluate.
1. sin 1
5. cot 1
1
2
3
3
9. tan 1 (cot
6. sec 1

4
2 3
3
7. sin 1 (cos
10. sec(tan 1 (1))
)
4. cos 1
3. csc 1 2
2. tan 1 ( 3)

3
)
8. cos 1 (sin
11. csc(csc 1 (2))
1
2
12. tan 1 (sin(cos 1 ))
Topic S
Basic Derivative Rules
Find the derivative of the following functions.
1. y = 3x + 5
2. y = 5x2
3. y = 3x2 + 5x – 1
4. y = sin (x)
5. y = tan (x)
6. y = cos (x2)
8. y = (3x2)(cos (x))
9. y =
7. y =
x3  x
14
x2  5x
3x  x3
3
2

6
)