Chapter 1.1 Notes
1
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Chapter1: Preview and Review
1.1 Preliminaries
*NOTE this is not a complete list of material you are expected to know
to be prepared for this course*
Real number line:
Interval notation
( a, b)
and [ a, b ]
ì
ï a if a ³ 0
The absolute value of a real number a is a = ïí
ï
ï
î-a if a < 0
Example: Graph the function y = 4 x - 8
Chapter 1.1 Notes
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A vertical line has no slope (undefined). All other lines have a slope given
by the equation
y y 2  y1 rise

m

x x2  x1 run
POINT-SLOPE form:
y - y1 = m( x - x1 )
SLOPE-INTERCEPT form: y = mx + b , b is the y-intercept
GENERAL form:
Ax + By + C = 0
Example: A line has a slope of 2 and goes through the point (3, 4). What
is the equation of the line? Graph the line and find the intercepts?
The quantities x and y are proportional ( y µ x ) if they are linearly related
( y = mx or y  b  mx ).
Chapter 1.1 Notes
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Radians and Degrees: 360 = 2p
A few common angles
180 = p
p
2
p
60 =
3
90 =
Example: Convert
17p
to degrees and convert 50 to radians
12
Trigonometric Functions
r
y
sin q =
r
x
cos q =
r
y
tan q =
x
r
csc q =
y
r
sec q =
x
x
cot q =
y
y
q
x
Pythagorean Theorem:
x2 + y2 = r 2
Chapter 1.1 Notes
4
p
2
p
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p
3
2
1
p
6
1
4
1
3
p
p
Example: (a) What is cos ? (b) What is sin ?
3
4
Some Formulas and Identities
tan q =
sin q
cos q
cot q =
cos q
1
=
sin q tan q
csc q =
1
sin q
sec q =
1
cos q
sin 2 q + cos 2 q = 1
tan 2 q + 1 = sec 2 q
sin (-q ) = - sin q
cos (-q ) = cos q
sin ( x + y ) = sin x ⋅ cos y + cos x ⋅ sin y
cos ( x + y ) = cos x ⋅ cos y + sin x ⋅ sin y
4
Chapter 1.1 Notes
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f ( x) = a x , a > 0, a ¹ 1 is called an exponential function
If a > 1, then this is exponential growth function where
8
y
7
 The domain is (-¥, ¥)
3
6
2x
x
5
 The range is (0, ¥)
4
3
2
1
x
-2
-1
1
2
3
-1
If 0 < a < 1, then this is exponential decay function where
8
0.25x
y
7
 The domain is (-¥, ¥)
6
5
 The range is (0, ¥)
0.5
x
4
3
2
1
x
-3
-2
-1
1
-1
Properties to remember: Given that a and b are positive,
a x+ y = a x a y
(a x ) = a xy
y
(ab) = a xb x
x
2
Chapter 1.1 Notes
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log a x = y  a y = x
Example: Evaluate the following
(a) log 2 64
æ1ö
(b) log 6 çç ÷÷÷
çè 36 ø
(c) log16 4
If x, y > 0 then
log a ( xy ) = log a x + log a y
Two special logarithms:
and
log a ( x y ) = y log a x
log10x = log x
logex = ln x = ln x
Two useful formulas:
log a (a x ) = x
a loga x = x , for x > 0
log a x =
ln x
ln a
Chapter 1.1 Notes
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Example: Express the given quantity as a single logarithm
(a) log 2 x + 5log 2 ( x + 1) + 12 log 2 ( x -1)
(b) ln x + a ln y - b ln z
Example: Solve each equation for x
(a) log ( x + 1) = 3
(b) e3 x-4 = 2
(c) ln x + ln ( x -1) = 1
(d) log x + log ( x + 1) = log 6
Chapter 1.1 Notes
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A complex number is a number of the form
z = a + bi
where a and b are real numbers and i 2 = -1. The real part is a and the
imaginary part is b.
Example: Find
(a) (7 - 2i ) - (9 + 6i )
(b) (7 - 2i )(9 + 6i )
If z = a + bi is a complex number, then its conjugate is z = a - bi
Example: Let z = 4 - 3i . Compute
(a) z
(b) z z
A quadratic equation is an equation in the form ax 2 + bx + c = 0 . Where
a, b, and c are real numbers and a ¹ 0 . The solution to this equation is
-b  b 2 - 4ac
x=
2a
Example: Solve 3 x 2  2 x  5