Download CHAPTER 1: PRELIMINARIES

MAT 132: CALCULUS WITH ANALYTIC GEOMETRY I
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CHAPTER 1: PRELIMINARIES
1.1
Real Numbers and the Real Line
Real Numbers
Much of calculus is based on properties of the real number system. Real numbers are
numbers that can be expressed as decimals.
Rules for Inequalities
If a, b and c are real numbers, then:
1.
2.
3.
4.
a  ba c  bc
a  b a c  bc
a  b and c  0  ac  bc
a  b and c  0  bc  ac
Special case: a  b  b  a
5.
a 0
6.
If a and b are both positive or both negative, then a  b 
1
0
a
1 1

b a
Intervals
A subset of the real line is called an interval if it contains at least two numbers and
contains all the real numbers lying between any two of its elements.
Types of intervals
Notation
Finite:
Set description
Type
(a , b )
x a  x  b
Open
a, b
x a  x  b
Closed
a, b)
x a  x  b
Half-open
(a, b
x a  x  b
Half-open
MAT 132: CALCULUS WITH ANALYTIC GEOMETRY I
Infinite:
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(a ,  )
x x  a
Open
a, )
x x  a
Closed
(, b)
x x  b
Open
(, b
x x  b
Closed
(, )
 (set of all real numbers)
Both open & closed
Example: Attend lecture.
Absolute Value
The absolute value of a number x, denoted by x , is defined by the formula
 x, x  0
x 
 x, x  0
Absolute Values and Intervals
If a is any positive number, then
1.
x  a if and only if x  a
2.
x  a if and only if  a  x  a
3.
x  a if and only if x  a or x  a
4.
x  a if and only if  a  x  a
5.
x  a if and only if x  a or x  a
Example: Attend lecture.
MAT 132: CALCULUS WITH ANALYTIC GEOMETRY I
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PROBLEM SET 1.1
Inequalities
Solve the following inequalities.
1.
 2x  4
5.
2.
8  3x  5
6.
3.
5x  3  7  3x
7.
4.
3(2  x )  2(3  x )
8.
1
7
 7x 
2
6
6  x 3x  4

4
2
4
1
( x  2)  ( x  6)
5
3
x  5 12  3x


2
4
2x 
Absolute Value
Solve the following equations.
1.
y 3
4.
1 t  1
2.
y3  7
5.
8  3s 
3.
2t  5  4
6.
s
1  1
2
9
2
Solve the following inequalities.
1 1

x 2
1.
x 2
9.
3
2.
x 2
10.
2
4 3
x
3.
t 1  3
11.
2s  4
4.
t  2 1
12.
5.
3y  7  4
13.
1
2
1 x  1
6.
2y  5  1
14.
2  3x  5
7.
8.
z
1  1
5
3
z 1  2
2
15.
16.
s3 
r 1
1
2
3r
2
1 
5
5
MAT 132: CALCULUS WITH ANALYTIC GEOMETRY I
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Quadratic Inequalities
Solve the following inequalities.
1.
2.
3.
4.
1.2
x2  2
4  x2
4  x2  9
1
1
 x2 
9
4
5.
6.
7.
(x  1) 2  4
8.
x2  x  2  0
( x  3) 2  2
x2  x  0
Lines, Circles and Parabolas
Slope
The constant
rise y y 2  y1


run x x 2  x 1
is the slope of the nonvertical line P1 P2 .
m=
The equation
y  y1  m(x  x1 )
is the point-slope equation of the line that passes through the point (x1 , y1 ) and has
slope m.
Distance and Circles in the Plane
Distance Formula for Points in the Plane
The distance between P(x1 , y1 ) and Q(x 2 , y 2 ) is
d  (x ) 2  (y) 2  ( x 2  x 1 ) 2  ( y 2  y1 ) 2
Standard equation of a circle with center (h, k) and radius a is
(x  h) 2  ( y  k) 2  a 2
Parabolas
The Graph of y  ax 2  bx  c, a  0
MAT 132: CALCULUS WITH ANALYTIC GEOMETRY I
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The graph of the equation y  ax 2  bx  c, a  0 , is a parabola. The parabola opens
upward if a > 0 and downward if a < 0. The axis is the line
b
2a
The vertex of the parabola is the point where the axis and parabola intersect. Its xb
b
coordinate is x   ; its y-coordinate is found by substituting x  
in the
2a
2a
parabola’s equation.
x
Example: Attend lecture.
PROBLEM SET 1.2
Slopes, Lines, and Intercepts
In the following questions, write an equation for each line described.
1. Passes through (-1, 1) with slope -1
2. Passes through (2, -3) with slope ½
3. Passes through (3, 4) and (-2, 5)
4. Passes through (-8, 0) and (-1, 3)
5. Has slope -5/4 and y-intercept 6
6. Has slope ½ and y-intercept -3
7. Passes through (-12, -9) and has slope 0
8. Passes through (1/3, 4), and has no slope
9. Has y-intercept 4 and x-intercept -1
10. Has y-intercept -6 and x-intercepts 2
11. Passes through (5, -1) and is parallel to the line 2x  5y  15
12. Passes through (- 2 ,2) parallel to the line 2 x  5y  3
13. Passes through (4, 10) and is perpendicular to the line 6x – 3y = 5
14. Passes through (0,1) and is perpendicular to the line 8x – 13y = 13.
Circles
Graph the circles whose equations are given in the following equations. Label each
circle’s center and intercepts (if any) with their coordinate pairs.
1.
2.
3.
x 2  y 2  4x  4 y  4  0
x 2  y 2  8x  4y  16  0
x 2  y 2  3y  4  0
4.
5.
6.
x 2  y 2  4x  (9 / 4)  0
x 2  y 2  4x  4 y  0
x 2  y 2  2x  3
MAT 132: CALCULUS WITH ANALYTIC GEOMETRY I
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Parabolas
Graph the following parabolas. Label the vertex, axis, and intercepts in each case.
1.
2.
y  x 2  2x  3
y  x 2  4x  3
5.
6.
3.
y   x 2  4x
7.
4.
y   x 2  4x  5
8.
1.3
Functions and Their Graphs
y   x 2  6x  5
y  2x 2  x  3
1
y  x2  x  4
2
1
y   x 2  2x  4
4
Function
A function from a set D to a set Y is a rule that assigns a unique (single) element
f ( x )  Y to each element x  D .
Example: Attend lecture.
1.4
Identifying Functions
Linear Functions
A function of the form f ( x )  mx  c , for constants m and b, is called a linear function.
Power Functions
A function f ( x )  x a , where a is constant, is called a power function.
Polynomials
A function p is a polynomial if
p( x )  a n x n  a n 1 x n 1  ...  a 1 x  a 0
where n is a nonnegative integer and the numbers a 0 , a 1 , a 2 ,..., a n are real constants
(called the coefficients of the polynomial).
Rational Function
A rational function is a quotient or ratio of two polynomial:
p( x )
f (x) 
q( x )
where p and q are polynomials.
Algebraic Functions
An algebraic function is a function constructed from polynomials using algebraic
operations (addition, subtraction, multiplication, division, and taking roots.)
MAT 132: CALCULUS WITH ANALYTIC GEOMETRY I
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Even Function, Odd Function
A function y  f ( x ) is an
even function of x
if f ( x )  f ( x ) ,
odd function of x
if f ( x )  f ( x ) ,
for every x in the function’s domain.
Example: Attend lecture.
PROBLEM SET 1.3 & 1.4
Recognizing Functions
Identify each function as a constant function, linear function, power function, polynomial
(state its degree), rational function, algebraic function, trigonometric function,
exponential function, or logarithmic function. Remember that some functions can fall
into more than one category.
1.
2.
3.
a.
f ( x )  7  3x
b.
g( x )  5 x
c.
h(x) 
x2 1
x2 1
d.
r(x)  8 x
a.
F(t )  t 4  t
b.
G(t )  5 t
c.
H( z )  z 3  1
d.
R ( z)  3 z 7
a.
c.
4.
a.
c.
3  2x
y
x 1
y  tan x
1
y  log 5  
t
g( x )  2
1
x
5
2
b.
y  x  2x  1
d.
y  log 7 x
b.
f (z) 
d.
z5
z 1
 t 
w  5 cos  
2 6
MAT 132: CALCULUS WITH ANALYTIC GEOMETRY I
1.5
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Combining Functions
Sum, Differences, Products, and Quotients
Composite Functions
If f and g are functions, the composite function f  g (“f composed with g”) is defined by
(f  g)( x )  f (g( x )).
The domain of f  g consists of the numbers x in the domain of g for which g(x) lies in
the domain of f.
Example: Attend lecture.
1.6
Trigonometric Functions
Radian Measure:  radians = 180 
The Six Basic Trigonometric Functions
Identities
1.
2.
3.
cos 2   sin 2   1
1  tan 2   sec 2 
1  cot 2   csc 2 
Addition Formulas
cos( A  B)  cos A cos B  sin A sin B
sin( A  B)  sin A cos B  cos A sin B
Double-Angle Formulas
sin 2  2 sin  cos 
cos 2  cos 2   sin 2 
Half-Angle Formulas
cos 2  
1  cos 2
2
1  cos 2
2
Example: Attend lecture.
sin 2  