basic calculus

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2. CHAPTER
0
Preliminaries
Slide 2
0.1 POLYNOMIALS AND RATIONAL FUNCTIONS
0.2 GRAPHING CALCULATORS AND COMPUTER
ALGEBRA SYSTEMS
0.3 INVERSE FUNCTIONS
0.4 TRIGONOMETRIC AND INVERSE TRIGONOMETRIC
FUNCTIONS
0.5 EXPONENTIAL AND LOGARITHMIC FUNCTIONS
0.6 TRANSFORMATIONS OF FUNCTIONS

3. 0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
The Real Number System and Inequalities
Slide 3
The set of integers consists of the whole numbers and
their additive inverses: 0, ±1,±2,±3, . . . .
A rational number is any number of the form p/q , where
p and q are integers and q ≠ 0. For example, 2/3 and −7/3
are rational numbers.
Notice that every integer n is also a rational number, since
we can write it as the quotient of two integers: n = n/1.

4. 0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
The Real Number System and Inequalities
Slide 4
Irrational numbers are all those real numbers that cannot
be written in the form p/q , where p and q are integers.
Recall that rational numbers have decimal expansions that
either terminate or repeat. By contrast, irrational numbers
have decimal expansions that do not repeat or terminate.

5. 0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
The Real Number System and Inequalities
Slide 5
We picture the real numbers arranged along the number
line (the real line). The set of real numbers is denoted by
the symbol .

6. 0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
The Real Number System and Inequalities
Slide 6
For real numbers a and b, where a < b, we define the
closed interval [a, b] to be the set of numbers between a
and b, including a and b (the endpoints).
That is, .
Similarly, the open interval (a, b) is the set of numbers
between a and b, but not including the endpoints a and b,
that is, .

7. THEOREM
0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
1.1
Slide 7

8. EXAMPLE
0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
1.2 Solving a Two-Sided Inequality
Slide 8

9. EXAMPLE
Solution
0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
1.2 Solving a Two-Sided Inequality
Slide 9

10. EXAMPLE
0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
1.3 Solving an Inequality Involving a Fraction
Slide 10

11. EXAMPLE
Solution
0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
1.3 Solving an Inequality Involving a Fraction
Slide 11

12. DEFINITION
0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
1.1
Slide 12

13. 0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
The Real Number System and Inequalities
Slide 13
Notice that for any real numbers a and b,
|a · b| = |a| · |b|, although |a + b| ≠ |a| + |b|, in
general.
However, it is always true that |a + b| ≤ |a| + |b|. This is
referred to as the triangle inequality.
The interpretation of |a − b| as
the distance between a and b is
particularly useful for solving
inequalities involving absolute values.

14. EXAMPLE
0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
1.7 Solving Inequalities
Slide 14
Solve the inequality |x − 2|< 5.

15. EXAMPLE
Solution
0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
1.7 Solving Inequalities
Slide 15

16. THEOREM
0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
1.1
Slide 16
The distance between the
points (x1, y1) and (x2, y2) in
the Cartesian plane is given by

17. EXAMPLE
0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
1.8 Using the Distance Formula
Slide 17
Find the distance between the points (1, 2) and (3, 4).

18. EXAMPLE
Solution
0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
1.8 Using the Distance Formula
Slide 18

19. DEFINITION
0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
1.2
Slide 19
For x1 ≠ x2, the slope of
the straight line through
the points (x1, y1) and (x2, y2)
is the number
When x1 = x2 and y1 ≠ y2,
the line through (x1, y1)
and (x2, y2) is vertical and the slope is undefined.
Notice that a line is horizontal if and only if its slope is zero.

20. EXAMPLE
0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
1.10 Using Slope to Determine if Points Are
Colinear
Slide 20
Use slope to determine whether the points (1, 2), (3, 10)
and (4, 14) are colinear.

21. EXAMPLE
Solution
0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
1.10 Using Slope to Determine if Points Are
Colinear
Slide 21
Since the slopes are the same, the points must be
colinear.

22. 0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
POINT-SLOPE FORM OF A LINE
Slide 22

23. EXAMPLE
0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
1.12 Finding the Equation of a Line Given Two
Points
Slide 23
Find an equation of the line through the points (3, 1) and
(4, −1), and graph the line.

24. EXAMPLE
Solution
0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
1.12 Finding the Equation of a Line Given Two
Points
Slide 24

25. THEOREM
0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
1.2
Slide 25
Two (nonvertical) lines are parallel if they have the same
slope.
Further, any two vertical lines are parallel.
Two (nonvertical) lines of slope m1 and m2 are
perpendicular whenever the product of their slopes is −1
(i.e., m1 · m2 = −1).
Also, any vertical line and any horizontal line are
perpendicular.

26. EXAMPLE
0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
1.14 Finding the Equation of a Perpendicular
Line
Slide 26
Find an equation of the line perpendicular to y = −2x + 4
and intersecting the line at the point (1, 2).

27. EXAMPLE
Solution
0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
1.14 Finding the Equation of a Perpendicular
Line
Slide 27
The slope of y = −2x + 4 is −2.
The slope of the perpendicular
line is then −1/(−2) = ½.

28. DEFINITION
0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
1.3
Slide 28
A function f is a rule that
assigns exactly one element
y in a set B to each element
x in a set A. In this case, we write y = f (x).
We call the set A the domain of f. The set of all values f(x)
in B is called the range of f , written {y|y = f (x), for some
x in set A}.

29. DEFINITION
0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
1.3
Slide 29
Unless explicitly stated otherwise, whenever a function f
is given by a particular expression, the domain of f
is the largest set of real numbers for which the
expression is defined.
We refer to x as the independent variable and to y as the
dependent variable.

30. 0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
Functions
Slide 30
By the graph of a function f, we mean the graph of the
equation y = f (x).
That is, the graph consists of all points (x, y), where x is in
the domain of f and where y = f (x).
Notice that not every curve is the graph of a function,
since for a function, only one y-value can correspond to a
given value of x.

31. 0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
Functions
Slide 31
We can graphically determine whether a curve is the
graph of a function by using the vertical line test:
If any vertical line intersects the graph in more than
one point, the curve is not the graph of a function
since, in this case, there are two y-values for a given
value of x.

32. EXAMPLE
Solution
0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
1.16 Using the Vertical Line Test
Slide 32
Determine which of the curves correspond to functions.

33. EXAMPLE
Solution
0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
1.16 Using the Vertical Line Test
Slide 33

34. DEFINITION
0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
1.4
Slide 34
A polynomial is any function that can be written in the
form
where a0, a1, a2, . . . , an are real numbers (the
coefficients of the polynomial) with an ≠ 0 and n ≥ 0 is an
integer (the degree of the polynomial).

35. DEFINITION
0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
1.5
Slide 35
Any function that can be written in the form
where p and q are polynomials, is called a rational
function.

36. EXAMPLE
0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
1.18 A Sample Rational Function
Slide 36
Find the domain of the function

37. EXAMPLE
Solution
0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
1.18 A Sample Rational Function
Slide 37

38. 0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
Functions
Slide 38
The square root function is defined in the usual way.
When we write we mean that y is that number
for which y2 = x and y ≥ 0.
In particular, . Be careful not to write erroneous
statements such as . In particular, be careful to
write
Similarly, for any integer n ≥ 2, whenever yn = x,
where for n even, x ≥ 0 and y ≥ 0.

39. EXAMPLE
0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
1.19 Finding the Domain of a Function
Involving a Square Root or a Cube Root
Slide 39

40. EXAMPLE
Solution
0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
1.19 Finding the Domain of a Function
Involving a Square Root or a Cube Root
Slide 40

41. 0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
Functions
Slide 41
A solution of the equation f (x) = 0 is called a zero of the
function f or a root of the equation f (x) = 0.
Notice that a zero of the function f corresponds to an x-
intercept of the graph of y = f (x).

42. EXAMPLE
0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
1.20 Finding Zeros by Factoring
Slide 42
Find all x- and y-intercepts of f (x) = x2 − 4x + 3.

43. EXAMPLE
Solution
0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
1.20 Finding Zeros by Factoring
Slide 43
y-intercept x-intercepts
For the quadratic equation ax2 + bx + c = 0 (for a ≠ 0), the
solution(s) are also given by the familiar quadratic
formula:

44. THEOREM
0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
1.3
Slide 44

45. REMARK
0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
1.3
Slide 45
Polynomials may also have complex zeros. For instance,
f (x) = x2 + 1 has only the complex zeros x = ±i, where i
is the imaginary number defined by .
We confine our attention in this text to real zeros.

46. THEOREM
0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
1.4
Slide 46
(Factor Theorem)

47. EXAMPLE
0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
1.22 Finding the Zeros of a Cubic Polynomial
Slide 47

48. EXAMPLE
Solution
0.1
POLYNOMIALS AND RATIONAL
FUNCTIONS
1.22 Finding the Zeros of a Cubic Polynomial
Slide 48
By calculating f (1), we can see that one zero of this
function is x = 1.
Factor: