AP Calculus AB
Summer Packet Assignment 2015
On the following pages, you will find a selection of mathematical topics that you will need to be familiar with before you
start Calculus AB. Your calculus teacher will be expecting you to walk into class, on the first day of school, already
familiar enough with this material to pass an assessment involving any of it.
All work should be done on a separate piece of paper and should be easy to follow. In Calculus, your process is as
important as your answer, so you must show your process clearly. Bald answers rarely, if ever, receive credit.
Trigonometric Expressions
Evaluate the following trig expressions without using a calculator. Feel free to draw the two special right triangles
and/or the basic graphs of sine, cosine and tangent to help you with this task.
1. sin 0°
π
2. cos 6
3. tan 45°
π
π
5. sec 2
6. cot 120°
3π
4
8. cos 150°
9. tan π
10. csc 210°
11. sec 225°
12. cot
13. sin 270°
14. cos
4. csc 3
7. sin
16. csc
11π
6
4π
3
5π
3
15. tan 315°
17. sec 2π
18. cot 390°
Equation of a Line
Use the given information to write the equation of each line.
3
slope-intercept form or
𝑦 = 𝑚𝑥 + 𝑏
or
point-slope form
𝑦  𝑦1 = 𝑚(𝑥  𝑥1 )
2
3
1. Slope = 4
y-intercept = 7
2. Slope = 4
y-intercept = −1
3. Slope = −
4. Point P = (2, 4)
Point Q = (5, 4)
5. Point P = (1, −4)
Point Q = (8, 5)
6. Point P = (3, 9)
Point Q = (3, 7)
7. Point P = (−3, −2)
y-intercept = 4
8. Point P = (4, 0)
y-intercept = 3
9. Point P = (2, −3)
y-intercept = −3
y-intercept =
4
5
Solve Quadratic Equations – Factoring Methods
Solve for the x-intercepts of the following quadratic equations using factoring methods. Do not use a calculator.
1. y = x 2 + 3x + 2
2. y = 2x 2 − 4x − 96
3. y = 6x 3 + x 2 − 2x
4. y = x 2 − 9
5. y = x 3 − x
6. y = 25x4 − 144
7. y = −2 + 6x 2 − 4x
8. y = x − 20 + 12x 2
9. y = 34x − 8x 2 − 21
Solve Quadratic Equations – Use Quadratic Formula
Find the solutions to the following quadratic equations such that x   (x is a Real number), using the quadratic
formula. Leave answers as simplified radicals or round answers to thousandths place.
1. y = 2x 2 + 3x − 3
2. y = x 2 − 2x − 6
3. y = 3x 2 − 12x + 15
Simplify Radicals and Exponents
Simplify the radical expressions without using a calculator. No rounded decimal answers. Only exact value answers.
1. √45
2. √20
3. √32 + √50
4. √3 + √3
5. 3√15 + 4√15
6. √11 − √11
7. √2 ∙ 5√6
8. 3√3 ∙ 2√5
9. √10x ∙ 2√2x
10. √
1
4
16
9
11. √
13. √16 + 9
16.
2
25 ⁄3
(yes, without a calculator)
+
√8
√2
3
15. √84
14. √16 ∙ 9
17.
3
16 ⁄2
12. √11 ∙ √11
(yes, without a calculator)
18.
−1
36 ⁄2
3.
6
√3
(yes, without a calculator)
(yes, without a calculator)
Rationalize Denominators
Rewrite the fraction removing the radical from the denominator.
1.
1
√2
2.
2
√2
Simplify Expressions -- Properties of Radicals and Exponents
Use the properties of exponents to simplify each given expression into a single term. If the radical cannot be simplified
then state, “the expression is already simplified.” (Leave answers with fractional exponents when appropriate.)
x2
1. √x ∙ x 2
2.
4. √x 4 + x 2
5. √x 4 ∙ x 2
7. x
3⁄
5
2∙ x ⁄2
8.
√x
5x2
6
15x ⁄5
3.
x3
4x
6.
√x 8
9.
√5x10 + 4x10
Simplify Expressions – Factoring Techniques
Use algebraic principles to simplify the given expressions.
1. sin2 x − sin x cos x
2.
x3 −x
x2 +x
3.
x2 +4x+3
(x+3)2
Evaluate Function Expressions
Evaluate the given expressions. Given:
𝑓(𝑥) = √x 2 − 2x and 𝑔(𝑥) = 2𝑥 − 1.
1. f(−1/2) = ?
2. g(7) = ?
3. f(3) + g(3) = ?
4. f(3t) = ?
5. f( g(x) ) = ?
6. f( g(t + 5) ) = ?
Graphs of Basic Functions
Write the basic equation and draw a sketch of the following ten functions. Make sketches without using a calculator.
Your scale and neatness matter.
1. Linear Function
2. Quadratic Function
3. Cubic Function
4. Square Root Function
5. Logarithmic Function
6. Reciprocal Function
7. Exponential Function
8. Sine Function
9. Cosine Function
10. Absolute-Value of a Linear Term Function
<Answers – Odd numbered problems>
Some adjustments have been made to problems. There is a possibility in the transition some answers were not updated.
90
Evaluate trig ratios
Notes: cos = adj./hyp. and cot = adj./opp.;
Ex(a):  = /6 is quadrant 1 ,
convert to degrees
cos(/6) = cos(30) =
π
6
∙
180
π
= 30° ;
and
2
√3
-1
60
2
Ex(b):  = 4/3 is quadrant 3, cot(4/3) = cot(240) = √3
1. 0 ,
3. 1
5. undefined value ,
7. 1/√2 ,
9. 0 ,
− √𝟑
11.  √2 ,
Lines
3. 𝑦 = (2/3)𝑥 + (4/5) ,
1. 𝑦 = 4𝑥 + 7 ,
7 9
6. 𝑚 =
(3−3)
is undefined , 𝑥 = 3 ,
7. 𝑚 =
5. 𝑚 =
4+2
0+3
5+4
8 1
13. 1 ,
9
9
7
7
= , 𝑦 =
𝑥
= 2 , 𝑦 = 2𝑥 + 4 ,
15. 1 ,
37
7
30
1
0
√𝟑
2
17. 1
,
9. 𝑚 =
3+3
0−2
= 0, 𝑦 = 3
Solve using quadratic factoring
1. 0 = (𝑥 + 1)(𝑥 + 2) , 𝑥 = 1 and 𝑥 = 2 ,
1
2
3. 0 = 𝑥(6𝑥 2 + 𝑥  2)  0 = 𝑥(2𝑥  1)(3𝑥 + 2) , 𝑥 = 0, 𝑥 = , 𝑥 = ,
2
3
5. 0 = 𝑥(𝑥 2  1)  0 = 𝑥(𝑥 + 1)(𝑥  1)  𝑥 = 0, 𝑥 = 1, 𝑥 = 1 ,
1
7. 0 = 2(3𝑥 2  2𝑥  1)  0 −= 2(3𝑥 + 1)(𝑥  1) , =
and 𝑥 = 1 ,
3
9. 0 = 8𝑥 2  34𝑥 + 21  0 = (4𝑥3)(2𝑥7)  𝑥 =
3
4
and 𝑥 =
7
2
Solve using quadratic formula
1. x =
−3 ± √33
4
,
3. x =
12 ± √−36
6
 no real solution
Simplify radicals and exponents
1. 3√5 ,
11.
√16
3. 4√2 + 5√2 = 9√2 ,
=
√9
4
√8
𝑎𝑛𝑑
3
√2
5. 7√15 ,
= √4 = 2  4/3 + 2 = 10/3 ,
7. 5√12 = 10√3 ,
9. 2√20x 2 4x√5 ,
3
13. 5 , 15. √8 = 2  24 = 16 , 17. √16 = 4  43 = 64
Rationalize denominators
1.
1
√2
∙
√2
√2
=
√2
2
,
3.
6
√3
∙
√3
√3
= 2√3
Simplify expressions- Properties of Exponents and Radicals
1. x
1/2
x = x
2
1/2+2
=x
5/2
,
3.
1 x3
4 x
=
1 2
x
4
, 5. √x 4 ∙ x 2 = √x 6 = x 3 , 7. x
8⁄
2
= x4 ,
9. √9x10 = 3x 5
Simplify expressions-Factoring
1. sinx (sinx  cosx) ,
3.
(𝑥+3)(𝑥+1)
(𝑥+3)(𝑥+3)
=
𝑥+1
𝑥+3
Evaluate function expressions
1
5
√5
1. f (− ) = √ =
2
4
2
3. √3 + 5,
4. √(3t)2 − 2(3t) = √3t (3t − 2) ,
5. √(2x − 1)2 − 2(2x − 1) = √(2x − 1) (2x − 3) ,  If you don’t know how we got √(2x − 1) (2x − 3) you need to do some
studying. This process of factoring is VERY, VERY, VERY IMPORTANT.
6. Start with 𝑔(𝑡 + 5) = 2(𝑡 + 5)  1  𝑔(𝑡 + 5) = 2𝑡 + 9. Then work 𝑓( 2𝑡 + 9).
Graph equations
The pictures of the basic functions can be created using a graphing calculator, if needed. However, try to draw them on your own first
though. Here are the basic equations for each function name.
2
3
1. linear y = x ,
2. quadratic y = x ,
3. cubic y = x ,
4. square-root y = √𝑥 ,
5. logarithmic y = ln(x) ,
1
x
6. reciprocal y = ,
7. exponential y = e ,
8. sine y = sin(x) ,
9. cosine y = cos(x) ,
10. absolute y = | x |
x