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AP Calculus
Section P.1 – The Real Numbers OUTLINE
Definition:
Ordered –
What is the Trichotomy Property?
Order, inequalities and intervals can be thought of algebraically and geometrically using interval
notation, inequality notation and number line graphs.
What are bounded intervals?
What is the difference between open and closed?
Express the set of numbers between negative 6 and positive 4, including negative 6 but not
including positive 4 using all three types of notations (interval notation, inequality notation and
number line).
What are unbounded intervals?
What interval represents the entire set of real numbers?
• Represent the set of real numbers less than or equal to 2 using all three notations.
• Represent the set of real numbers greater than negative 1 using all three notations.
• Represent the set of real numbers greater than or equal to negative 3, but less than 7
using all three notations.
• Represent the set of real numbers less than 5, but that does not include the numbers
between 0 and negative 2 inclusive using all three notations.
Assignment: P.1 exercises, #6 – 36 multiples of 3, 47 – 52 all
AP Calculus
Section P.4 – Lines
All about lines:
Increments
If a particle moves from the point ( x1, y1 ) to the point ( x2 , y2 ) , the increments in its coordinates are
Dx = x2 - x1 and Dy = y2 - y1 .
Definition of slope
Let P1 ( x1, y1 ) and be P2 ( x2 , y2 ) points on a nonvertical line, L. The slope of L is
m=
rise Dy y2 - y1
=
=
.
run Dx x2 - x1
Point-slope Equation (this one is even more important than the familiar slope-intercept equation)
The equation
y = m ( x - x1 ) + y1
is the point-slope equation of the line through the point with slope m.
Exercises:
Find the coordinate increments from A to B.
1. A (1, 2) B ( -1,-1)
2. A ( -3,1)
B ( -8,1)
Let L be the line determined by points A and B.
a) Plot A and B
b) Find the slope of L
c) Draw the graph of L.
3. A (1,- 2) B ( 2,1)
4. A ( 2,3) B ( -1,3)
Write an equation for (a) the vertical line and (b) the horizontal line through the point P.
5. P ( 2,3)
(
6. P 0,- 2
)
Write the point-slope equation for the line through the point P with slope m.
7. P ( 2,3) m = 4
8. P ( 0,3) m = 2
9. P ( -1, 1) m = -1
Write an equation for the line through P that is (a) parallel to L, (b) perpendicular to L.
10. P ( 0, 0) L : y = -x + 2
11. P ( -2, 2) L : 2x + y = 4
12. P ( -2, 4)
æ
è
L:x=5
1ö
2ø
13. P ç -1, ÷ L : y = 3
No textbook assignment for this section.
AP Calculus
Appendix A.1 – Radicals and Rational Exponents OUTLINE
Pg. 839 – 843
Define each term and give an example for each term (p. 839):
square root:
cube root:
Define each term:
nth root:
principal nth root:
Example:
35  243
What is the principal 5th root of 243?
radical expression:
index:
radicand:
Real and Principal Roots
Write this fact using a radical expression:
Summarize the differences when finding or identifying the nth roots when n is even compared to
when n is odd (p. 839).
If there is no index in a radical expression, what assumption can we make?
EXAMPLES (work out each sample problem):
a)
121
b)
4
81
256
c)
5
 32
d)
6

1
64
Properties of Radicals (p. 840)
In the table below, write out the 6 properties of radicals and complete each example so that it
illustrates the property given.
Property
Example
1.
80  16  5
=
2.
3
125

27
3
3
=
3.
5
4.

5.
4
813 
6.
4
 124

3
 123

9
11 =
 16

9

In each property, u and v represent:
m and n represent:
BE VERY CAREFUL! What is the difference between property 4 and property 6?
Simplifying Radical Expressions (pages 840, 841)
EXAMPLES (work out each sample problem):
a)
5
96 
b)
3
 1080 
c)
75z 9 =
d)
6
p6
=
q6
e)
3
 54x 9
Rationalizing Denominators (p. 841)
EXAMPLES (work out each sample problem):
a)
7
=
5
b)
2
x
=
1
c)
d)
=
5
y2
4
w
=
8v 3
Rational Exponents (p. 841)
Explain why x 2  x :
1
Write the formal definition for Rational Exponents (p. 841):
What does the numerator in a rational exponent represent?
What does the denominator in a rational exponent represent?
EXAMPLES (write the radical expressions using rational exponents and vice versa):
115 
a)
7
b)
5  x 
3
2
=
c) 6a 2  4 a 3 =
d) p 
9
4
Summarize the procedures used to simplify radicals (p. 843):
Book assignment: Pages 843 and 844 – #3-81 (multiples of 3 and 4)
SHOW ALL WORK FOR THESE PROBLEMS!
A LIST OF ANSWERS WITHOUT WORK WILL NOT BE ACCEPTED!
AP Calculus
Appendix A.2 Polynomials and Factoring OUTLINE
Pg. 845 – 850
Adding, Subtracting, and Multiplying Polynomials
Write the definitions of:
Polynomial in x
Degree (of a polynomial)
Lead coefficient
Monomial
Binomial
Trinomial
Standard form
Like terms

 
Add the following polynomials: 3x 3  5x 2  7 x  2   2 x 3  8x 2  3x  9

 


Subtract the following polynomials: 2 x 3  6 x 2  11x  16  x 3  2 x 2  4 x  18
Expand each product and simplify the polynomial:
a) 2x  8x  7
b)
2x
3

 2 x 2  3x  2 x 3  2 x

Special Products (p. 846)
Special Binomial Products: Let u and v be real numbers, variables, or algebraic expressions.
1.
2.
3.
4.
5.
Using Special Products
Use the special products from above to expand each product below without performing any
multiplication or simplification.
2
a) 4 x  5
b)
5a  22
c)
8z  38z  3
d)
6 x  23
Factoring Polynomials Using Special Products (p. 847)
What does it mean for a polynomial to be completely factored?



Explain why x 4  16  x 2  4 x 2  4 is NOT a complete factorization of the polynomial
x 4  16 .
Factor each polynomial completely:
a) 3x 3  3x 2  9 x
a) a 4 b  ab 4
b) 49 y 2  4
c) 36 g 2  g  5
2
d) 25d 2  30 x  9 Should be 25d 2 + 30d + 9
e) 81x 2  36 xy  2 y 2
f)
x 3  27
g) 125 x 3  8
h) x 2  9 x  14
i)
z 2  5 z  24
j) 14t 2  33t  15
Factor each polynomial by grouping (p. 860):
x 3  4 x 2  5 x  20
x 6  3x 4  x 2  3
Steps for Factoring Polynomials (p. 860):
Book assignment: P. 851 (2 – 78 even, 83, 87, 88, 90, 91)
SHOW ALL WORK FOR THESE PROBLEMS!
A LIST OF ANSWERS WITHOUT WORK WILL NOT BE ACCEPTED!
AP Calculus
Appendix A.3 – Fractional Expressions OUTLINE
Pg. 852 – 855
Reducing Rational Expressions (p. 852)
Under what circumstances can a rational expression be reduced? What does it mean for a
rational expression to be in reduced form?
Examples (work out each sample problem):
a) Write in reduced form:
16 x 5 y 2
28 xy 9
b) Write in reduced form:
x 2  5x
x 2  3x  10
Operations with Rational Expressions (p. 853)
Explain the differences between the first two operations listed in the green box on page 853.
Explain why the fourth operation (fraction division) works as shown.
Examples (work out each sample problem):
a) Perform the operations indicated and simplify completely
x 2  8 x  15 x 3  3x  10
 2
x 2  25
x  4x  3
b) Perform the operations indicated and simplify completely
x3  8
x 2  2x  4

2 x 2  11x  14
4 x 2  49
c) Perform the operations indicated and simplify completely
x
x3

x  2 x 1
What is the LCD (least common denominator) in a rational expression (p. 854)?
Example (work out the sample problem):
Perform the indicated operations and completely simplify the result.
1
3
4
 2
 2
x  3 x  2x  3 x  x
Compound (complex) Rational Expressions (p. 854)
What is a compound or complex fraction?
To simplify a complex fraction multiply both the numerator and denominator of the fraction by
the LCD of all fractions within the expression. Here is how Example 6 would work if we use
this method:
7
3
x2
When we look at the fractions in the numerator and
1
1
x3
denominator, we see that the LCD is x  2x  3 .
So to simplify:
7 

7
3 
   x  2  x  3
x  2
x2  
1
1 

1
1 
   x  2  x  3
x3 
x 3
3 x  2  x  3  7 x  3

1 x  2  x  3  1 x  2 
3

3x  3x  18  7 x  21
x  x  6  x  2

3 x 2  10 x  3
x 2  2x  8
3
2
Since the polynomials in the numerator and denominator don’t share any common factors (try
factoring the two polynomials yourself to confirm this fact), we cannot simplify the expression
any further.
Examples (work out the sample problems):
Perform the indicated operations and completely simplify the results.
1
1
2
 2 
2
ab
a
b
1
1
 2
2
a
b
2
1
1
 x
x2 x2
x 1
Book assignment: P. 856 - #3, 6, 8, 12, 13, 18, 20, 21, 26, 36-42, 45 – 69 multiples of 3 and 4,
72, 74
SHOW ALL WORK FOR THESE PROBLEMS!
A LIST OF ANSWERS WITHOUT WORK WILL NOT BE ACCEPTED!
Name:
Date:
AP Calculus
Section P.5 – Solving Equations Graphically, Numerically and Algebraically OUTLINE
Pg. 44 - 52
Definition:
Quadratic Equation –
What is the zero product (factor) property?
Solve the equation graphically, and confirm by using factoring: 6x 2  7 x  5  0
Solve algebraically by extracting square roots: (3x  1) 2  16
How can any quadratic equation be written in the form of (ax  b) 2  c ?
Solve by completing the square: 4 x 2  20x  17  0
State the Quadratic Formula:
Solve using the quadratic formula, and state the solutions in simplest form and as decimal
approximations: 2 x 2  8x  5
State the four ways to solve a quadratic equation algebraically:
1.
2.
3.
4.
Explain how to solve the equation graphically: x 3  x  1  0
Explain how to solve the equation using tables: x 3  x  1  0
Solve the equation algebraically, and verify by finding intersections: 2 x  5  7
Assignment: Pg. 50 #3-44 multiples of 3 and 4, 59-61
Name:
Date:
AP Calculus
Section P.6 – Complex Numbers OUTLINE
Pg. 53 – 58
Define the imaginary unit (i):
Definition:
Complex Number –
What is the difference between a complex number and an imaginary number?
Definitions:
Equal (complex numbers) –
Sum of two complex numbers –
Difference of two complex numbers –
What is the additive identity element of the set of complex numbers?
What is the additive inverse for any complex number?
Definition:
Complex Conjugate –
Why is the conjugate of a complex number significant?
How is the conjugate of a complex number used to perform division of complex numbers?
Illustrate with an example.
Explain how the discriminant of a quadratic equation can be used to determine the nature of the
solutions of a quadratic equation.
Describe the Complex Plane. Illustrate below by graphing the following complex numbers:
2+6i
7–2i
–8+0i
0+3i
Definition:
Absolute Value of a Complex Number –
Find the absolute value of the 4 complex numbers you plotted above:
Assignment: P.6 Exercises, #3-44 multiples of 3 and 4; 55-57