AP Calculus – Summer Packet
Congratulations on making the choice to take Calculus! We’re going to work very hard, but have so
much fun too! There are certain skills that have been taught to over your years as a math student. If
you do not have these skills down you will find that you will consistently get problems incorrect next
year, even though you understand the calculus concepts! You will hear me say a lot: “The calculus
isn’t the hard part; it’s the ALGEBRA that’s the hard part!”
The topics I see students consistently struggle with are in this packet. These are skills that are used
continually in Calculus. If you don’t know how to solve some of the problems, get online and look up
websites that will help you or find a friend to help. Don’t fake your way through these problems! As I
said, students are notoriously weak in them, even students who have achieved high grades in math
prior to AP calculus.
Please do not do this packet the week after school gets out. Wait until mid-summer. I want these
techniques to be relatively fresh in your mind in the fall. At the same time, do not wait until the night
before school starts in August! These will take time!
To receive full credit, all work and thinking will be shown and recorded (if you need more room feel
free to use a separate piece of paper). Do as much as you can without a calculator. Half of the AP
test is taken without a calculator! This packet is due the first day back to school. It will be graded. You
need to get off to a good start, so spend some quality time on this packet during the summer.
If you have questions about any of these problems or techniques please feel free to contact me at:
schlagel_megan@svvsd.org
Have a wonderful summer! I cannot wait to meet you!
TOPICS:
1. Negative and Fractional Exponents
2. Special Factorization
3. Domain and Functions
4. Solving Inequalities
5. Solving Quadratic Equations
6. Asymptotes
7. Complex Fractions
8. Composition of Functions
9. Solving Rational equations
10. Basic Right angle trig
11. Solving Trig Equations
12. Log and exponent rules and equations
Sincerely,
Mrs. Megan Schlagel
Topic 1: Fractional and Negative Exponents
Simplify using only positive exponents
3
3
4
4.
(16𝑥 2
7.
2𝑥−1
2(2𝑥+1)−2(2𝑥−1)
−4(2𝑥+1)−3 (
)
(2𝑥+1)2
𝑦)
1
2
2. −5 (2) (−9)(4 − 9𝑥)−2
1. −3𝑥 −3
−2
3. 2 (2−𝑥) ((2−𝑥)2 )
1
5. −
8.
−
𝑥 2
2
sin(√𝑥)
3
1
−
(2𝑥+5) 2
2
3
2
6.
1
√4𝑥−16
4
√(𝑥−4)3
4
1
−
9. (𝑥 −2 + 𝑥 −1 𝑦 −1 + 𝑦 −2 )
1
2
Topic 2: Special Factorization
Factor completely
1. 𝑥 3 + 8
2. 𝑥 3 − 8
3. 27𝑥 3 − 125𝑦 3
4. 𝑥 4 + 11𝑥 2 − 80
5. 𝑎𝑐 + 𝑐𝑑 − 𝑎𝑏 − 𝑏𝑑
6. 2𝑥 2 + 50𝑦 2 − 20𝑥𝑦
7. 𝑥 2 + 12𝑥 + 36 − 9𝑦 2
8. 𝑥 3 − 𝑥𝑦 2 + 𝑥 2 𝑦 − 𝑦 3
9. (𝑥 − 3)2 (2𝑥 + 1)3 + (𝑥 − 3)3 (2𝑥 + 1)2
Topic 3: Domain and Functions
If 𝑓(𝑥) = 𝑥 2 − 1, describe in words what the following would do to the graph of 𝑓(𝑥):
1. 𝑓(𝑥) − 4
2. 𝑓(𝑥 − 4)
3. −𝑓(𝑥 + 2)
4. 5𝑓(𝑥) + 3
5. 𝑓(2𝑥)
6. |𝑓(𝑥)|
Find the domain of the following functions:
𝑥 2 −4
√2𝑥−9
2𝑥+9
7. 𝑦 = 2𝑥+4
8. 𝑦 =
10. 𝑦 = log(2𝑥 − 12)
11. 𝑦 = √tan 𝑥
3
√𝑥−6
9. 𝑦 = √𝑥 2
−𝑥−30
𝑥
12. 𝑦 = cos 𝑥
13. Identify each graph by its family of function and write the general equation.
Topic 4: Solving Inequalities
Solve the following by factoring and making appropriate sign charts.
1. 𝑥 2 − 16 > 0
4. 2𝑠𝑖𝑛2 𝑥 ≥ sin 𝑥
0 ≤ 𝑥 ≤ 2𝜋
2. 𝑥 2 + 6𝑥 − 16 > 0
3. 𝑥 2 − 3𝑥 ≥ 10
5. |𝑥 − 3| > 12
6. |3𝑥 − 4| > −2
Write the following absolute value expressions as piecewise expressions
7. 𝑦 = |2𝑥 − 4|
8. 𝑦 = |6 + 2𝑥| + 1
Graph the following piecewise functions
𝑥,
𝑥 < −2
10. 𝑓(𝑥) = {𝑥 + 2, − 2 ≤ 𝑥 ≤ 2
−𝑥 + 5,
𝑥>2
2𝑥 + 5, 𝑥 ≤ 0
9. 𝑓(𝑥) = { 2
𝑥 − 4, 𝑥 > 0
9
2
y
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
-14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
y
x
1 2 3 4 5
6 7 8 9 10 11 12 13 14
1
-14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1
-2
-2
-3
-3
-4
-4
-5
-5
-6
-6
-7
-7
-8
-8
-9
-9
x
1 2 3 4 5
6 7 8 9 10 11 12 13 14
Topic 5: Solving Quadratic Equations
Solve each equation
1. 7𝑥 2 − 3𝑥 = 0
2. 4𝑥(𝑥 − 2) − 5𝑥(𝑥 − 1) = 2
3. 𝑥 2 + 6𝑥 + 4 = 0
4. 2𝑥 2 − 3𝑥 + 3 = 0
5. 2𝑥 2 − (𝑥 + 2)(𝑥 − 3) = 12
6. 𝑥 + 𝑥 =
7. 𝑥 4 − 9𝑥 2 + 8 = 0
8. 𝑥 − 10√𝑥 + 9 = 0
9. 𝑥 2 − 𝑥 = 6
1
1
1
13
6
Topic 6: Asymptotes
For each function, find the equation of both vertical asymptotes(s) and horizontal asymptotes (if they
exist).
𝑥
1. 𝑦 = 𝑥−3
𝑥 2 −2𝑥+1
4. 𝑦 = 𝑥 2 −3𝑥−4
𝑥 2 −𝑥−6
7. 𝑦 = 𝑥 3 −𝑥 2 +𝑥−6
𝑥+4
2. 𝑦 = 𝑥 2 −1
𝑥 2 −9
5. 𝑦 = 𝑥 3 +3𝑥 2 −18𝑥
2𝑥 3
8. 𝑦 = 𝑥 3 −1
𝑥+4
3. 𝑦 = 𝑥 2 +1
2𝑥 2 +6𝑥
6. 𝑦 = 𝑥 3 −3𝑥 2 −4𝑥
√𝑥
9. 𝑦 = 2𝑥 2 −10
Topic 7: Complex Fractions
Simplify the following
1.
4.
7.
𝑥
1
2
𝑥−
3 4
−
𝑥 𝑦
4 3
−
𝑥 𝑦
𝑥 −3 −𝑥
𝑥 −2 −1
2.
1
+4
𝑥
1
−2
𝑥
5.
2
3𝑥
4
𝑥−
9𝑥
8.
𝑥
1+𝑥
+
1−𝑥
𝑥
1−𝑥
𝑥
+
𝑥
1+𝑥
1−
3.
1
𝑥
1
𝑥+
𝑥
𝑥−
6.
𝑥2 −𝑦2
𝑥𝑦
𝑥+𝑦
𝑦
9.
4
2
+
𝑥−5 𝑥+2
2𝑥
+3
𝑥2 −3𝑥−10
Topic 8: Composition of Functions
If 𝑓(𝑥) = 𝑥 2 , 𝑔(𝑥) = 2𝑥 − 1 𝑎𝑛𝑑 ℎ(𝑥) = 2𝑥 , find the following
1. 𝑓(𝑔(2))
2. 𝑓(𝑔(2))
3. 𝑓(ℎ(−1))
4. ℎ(𝑓(−1))
5. 𝑔 (𝑓 (ℎ (2)))
6. 𝑓(𝑔(𝑥))
7. 𝑔(𝑓(𝑥))
8. 𝑔(𝑔(𝑥))
9. 𝑓(ℎ(𝑥))
1
Topic 9: Solving Rational Equations
Solve each equation for x
2
5
1
6
1. 3 − 6 = 𝑥
𝑥−5
2. 𝑥 + 𝑥 = 5
3
4. 𝑥+1 = 5
𝑥
2𝑥
5.
5𝑥
7. 𝑥−2 + 4−𝑥 2 = 𝑥+2
60
𝑥
60
3.
2
3
3
−
2
− 𝑥−5 = 𝑥
𝑥
𝑥+1
𝑥−1
2
=1
1
16
6. 𝑥+5 + 𝑥−5 = 𝑥 2 −25
𝑥−2
8. 2𝑥−6 − 𝑥 2 −6𝑥+9 = 3𝑥−9
9.
2𝑥+3
𝑥−1
10
= 𝑥 2−1 +
2𝑥−3
𝑥+1
Topic 10: Right Triangle Trig and Unit Circle
Solve the following problems
If point P is on the terminal side of 𝜃, find all 6 trig functions of 𝜃. Draw a picture.
1. (−2, 4)
2. (√5, −2)
5
3. If cos 𝜃 = 13 and 𝜃𝜖𝑄𝐼𝐼, find sin 𝜃 𝑎𝑛𝑑 tan 𝜃
4. If cot 𝜃 = 3 and 𝜃𝜖𝑄𝐼𝐼𝐼, find sin 𝜃 𝑎𝑛𝑑 cos 𝜃
Find the exact value of the following without calculators:
5. sin 135°
9. csc −
7𝜋
6
6. tan 270°
10. cot
13. 𝑠𝑖𝑛2 225° − 𝑐𝑜𝑠 2 300°
15. (4 cos 30° − 6 sin 120°)−2
5𝜋
4
2𝜋
7. cos 270°
8. cos
11. cos 𝜋
12. sec −
14. (6 sec 180° − 4 cot 90°)2
3
13𝜋
6
Topic 11: Solving Trig Equations
Solve each equation on the interval [0, 2𝜋)
1
1. sin 𝑥 = 2
2. 𝑐𝑜𝑠 2 𝑥 = cos 𝑥
3. 2 cos 𝑥 + √3 = 0
4. 4𝑠𝑖𝑛2 𝑥 = 1
5. 2𝑠𝑖𝑛2 𝑥 + sin 𝑥 = 1
6. 𝑐𝑜𝑠 2 𝑥 + 2 cos 𝑥 = 3
7. 2 sin 𝑥 cos 𝑥 + sin 𝑥 = 0
8. 8𝑐𝑜𝑠 2 𝑥 − 2 cos 𝑥 = 1
9. 𝑠𝑖𝑛2 𝑥 − 𝑐𝑜𝑠 2 𝑥 = 0
Topic 12: Logarithm and Exponential Rules and Equations
Simplify or expand using log rules:
1. log2 8
2. ln e
3
1
3. log 6
216
Solve:
5. 4 x 
1
64
8. 3  5log x  18
6. 5 x1  125
8. log7 (2x  3)  log7 (3x  6)
7. log x  2
9. 2 x  5
1
𝑥4 2
4. ln ( 𝑦 )