Prerequisites for Calculus

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Prerequisites for Calculus

• Dr. Ching I Chen

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1.1 Lines (1) Increments
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1.1 Lines (2, Example 1) Increments
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1.1 Lines (3) Slope of a Lines
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1.1 Lines (4) Parallel and Perpendicular Lines

• Parallel lines Lines have the same slope.
• Perpendicular lines L1 (slope m1), L2 (slope
  m2) satisfy m1m2 -1

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1.1 Lines (5, Example 2) Parallel and
Perpendicular Lines
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1.1 Lines (6, Example 3) Parallel and
Perpendicular Lines
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1.1 Lines (7) Equations of Lines
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1.1 Lines (8, Example 4) Equations of Lines
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1.1 Lines (9, Example 5) Equations of Lines
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1.1 Lines (10) Equations of Lines
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1.1 Lines (11, Example 6) Equations of Lines
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1.1 Lines (12) Equations of Lines
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1.1 Lines (13, Example 7) Equations of Lines
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1.1 Lines (14, Example 8) Applications
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1.1 Lines (15, Example 9) Applications
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1.2 Function and Graphs (1) Functions

• The values of one variable often depend on the
  values for another

• Temperature of water boils depends on elevation
• Area of a circle depends on the circles radius

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1.2 Function and Graphs (2) Functions
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1.2 Function and Graphs (3, Example 1) Functions
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1.2 Function and Graphs (4) Domains and Ranges

• Domain and range depend on the type of the
  function

• natural domain there is no restriction for D
  and R, such as y(x) x, y(x) 3x

• One has to the ability to judge domain and range

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1.2 Function and Graphs (5) Domains and Ranges
finite interval
Half open finite interval
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1.2 Function and Graphs (6) Domains and Ranges
Infinite interval
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1.2 Function and Graphs (7, Example 2) Functions
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1.2 Function and Graphs (8) Viewing and
Interpreting Graphs

• When you look at a function, how do you know the
  tendency of the dependent variables and
  independent variables ?
• On the other hand, what is the value of x with
  respect to y ?
• General speaking, the simply way is to illustrate
  the value of x and y
• However, you still do not know the variation of x
  with respect to y.
• The easily way is to plot the relation of x and y
  in a manner of graph such that realizes the
  relation between x and y.

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1.2 Function and Graphs (9) Viewing and
Interpreting Graphs
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1.2 Function and Graphs (10, Example 3) Viewing
and Interpreting Graphs
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1.2 Function and Graphs (11) Viewing and
Interpreting Graphs
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1.2 Function and Graphs (12) Viewing and
Interpreting Graphs
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1.2 Function and Graphs (13) Viewing and
Interpreting Graphs
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1.2 Function and Graphs (14) Even Functions and
Odd Functions - Symmetry
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1.2 Function and Graphs (15, example 4) Even
Functions and Odd Functions - Symmetry
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1.2 Function and Graphs (16, Example 5)
Functions Defined in Pieces
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1.2 Function and Graphs (17, Example 6)
Functions Defined in Pieces
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1.2 Function and Graphs (18, Example 7) Absolute
Value Functions
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1.2 Function and Graphs (19) Composite Functions

• Suppose that some of the outputs of a function g
  can be used as inputs of a function f.
• We can then link g and f to form a new function
  whose inputs x are inputs of g and whose outputs
  are the numbers f(g(x)).

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1.2 Function and Graphs (20, Example 8)
Composite Functions
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1.2 Function and Graphs (21, Exploration 1-1)
Composite Functions
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1.2 Function and Graphs (22, Exploration 1-2)
Composite Functions
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1.2 Function and Graphs (23, Exploration 1-3)
Composite Functions
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1.3 Exponential Functions (1)Exponential Growth
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1.3 Exponential Functions (2)Exponential Growth
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1.3 Exponential Functions (3)Exponential Growth
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1.3 Exponential Functions (4)Exponential Growth
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1.3 Exponential Functions (5)Exponential Growth
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1.3 Exponential Functions (6, Example
1)Exponential Growth
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1.3 Exponential Functions (7, Example
2)Exponential Decay
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1.3 Exponential Functions (8)Exponential Decay
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1.3 Exponential Functions (9, Example
3)Exponential Decay
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1.3 Exponential Functions (10, Example
4)Exponential Decay
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1.4 Parametric Equations (1)Relations

• A relation is a set of ordered pairs (x, y) of a
  real numbers. The graph of relation is the set
  of points in the plane that correspond to the
  ordered pairs of the relation.

• However, if x and y are functions of a third
  variable t, called parameter, then we can use the
  parametric mode of a grapher to obtain a graph of
  the relation.

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1.4 Parametric Equations (2, Example 1)Relations
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1.4 Parametric Equations (3)Relations
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1.4 Parametric Equations (4, Exploration1-1)Circl
es
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1.4 Parametric Equations (5, Exploration1-2)Circl
es
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1.4 Parametric Equations (6, Exploration1-3)Circl
es
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1.4 Parametric Equations (7, Exploration1-4)Circl
es
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1.4 Parametric Equations (8, Example 2)Ellipses
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1.4 Parametric Equations (9, Exploration
2-1)Ellipses
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1.4 Parametric Equations (10, Exploration
2-2)Ellipses
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1.4 Parametric Equations (11, Exploration
2-3)Ellipses
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1.4 Parametric Equations (12, Exploration
2-4)Ellipses
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1.4 Parametric Equations (13, Exploration
2-5)Ellipses
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1.4 Parametric Equations (14, Example 3)Lines
and Other Curves
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1.4 Parametric Equations (15, Exploration
3-1)Lines and Other Curves
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1.4 Parametric Equations (16, Exploration
3-2)Lines and Other Curves
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1.4 Parametric Equations (17, Exploration
3-3)Lines and Other Curves
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1.4 Parametric Equations (18, Example 4)Lines
and Other Curves
Is it the only solution?
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1.5 Functions and Logarithms (1) One-to-One
Functions
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1.5 Functions and Logarithms (2) One-to-One
Functions
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1.5 Functions and Logarithms (3, Example 1)
One-to-One Functions
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1.5 Functions and Logarithms (4) Inverses

• A one-to-one function is just one input
  corresponding to one output.
• On the contrary, one-to-one function can be
  reversed to send outputs back to the inputs from
  which they came.

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1.5 Functions and Logarithms (5) Inverses

• This example suggests, composing a function with
  its inverse in either order sends each output
  back to the input from which if came.
• In other word, the result of composing a function
  and its inverse in either order is the identity
  function, the function that assigns each number
  to itself.

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1.5 Functions and Logarithms (6) Inverses

• Computing f ? g and g ? f . If (f ? g )(x)
  (g ? f )(x) x, then f and g are inverse of one
  another.

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1.5 Functions and Logarithms (6) Inverses
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1.5 Functions and Logarithms (7, Exploration 1-1)
Inverses
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1.5 Functions and Logarithms (8, Exploration 1-2)
Inverses
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1.5 Functions and Logarithms (9, Exploration 1-3)
Inverses
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1.5 Functions and Logarithms (10, Exploration
1-4) Inverses
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1.5 Functions and Logarithms (11) Finding
Inverses

• How do we find the graph of the inverse of a
  function ?

To find the value of f at x, we start at x, go up
to the curve, and then over to the y-axis.
The graph of f is also graph of f--1. To find
the x that gave y, we start at y and go over to
the curve and down to the x-axis.
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1.5 Functions and Logarithms (12) Finding
Inverses

• For f -1, the input-output pairs are reversed.
  To display the graph of in the usual way, we have
  to reverse the the pairs by reflecting the graph
  across the line yx and interchanging the letters
  x and y

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1.5 Functions and Logarithms (13) Finding
Inverses
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1.5 Functions and Logarithms (14, Example 2)
Finding Inverses
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1.5 Functions and Logarithms (15, Example 3)
Finding Inverses
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1.5 Functions and Logarithms (16) Logarithmic
Functions
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1.5 Functions and Logarithms (17) Properties of
Logarithms
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1.5 Functions and Logarithms (18, Example 4)
Properties of Logarithms
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1.5 Functions and Logarithms (19) Properties of
Logarithms
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1.5 Functions and Logarithms (20, Exploration 2)
Properties of Logarithms
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1.5 Functions and Logarithms (21, Exploration 2)
Properties of Logarithms
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1.5 Functions and Logarithms (22) Properties of
Logarithms
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1.5 Functions and Logarithms (23, Example 5)
Properties of Logarithms
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1.5 Functions and Logarithms (24, Example 6)
Applications
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1.5 Functions and Logarithms (25, Example 7)
Applications
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1.6 Trigonometric Functions (1) Radian Measures

• A circle with radius r, there are six basic
  trigonometric functions

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1.6 Trigonometric Functions (2) Graphs of
Trigonometric Functions

• The basic trigonometric functions are based on
  the radius 1, since q 0 to 2p

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1.6 Trigonometric Functions (3) Radian Measures

• The basic trigonometric functions are based on
  the radius 1, since q 0 to 2p

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1.6 Trigonometric Functions (4) Graphs of
Trigonometric Functions

• The basic trigonometric functions are based on
  the radius 1, since q 0 to 2p

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1.6 Trigonometric Functions (5, Exploration
1-1,2) Graphs of Trigonometric Functions
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1.6 Trigonometric Functions (6, Exploration 1-3)
Graphs of Trigonometric Functions
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1.6 Trigonometric Functions (7, Exploration 1-4)
Graphs of Trigonometric Functions
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1.6 Trigonometric Functions (8, Exploration 1-5)
Graphs of Trigonometric Functions
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1.6 Trigonometric Functions (9, Exploration 1-5)
Graphs of Trigonometric Functions
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1.6 Trigonometric Functions (10) Periodicity
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1.6 Trigonometric Functions (11, Example 1) Even
and Odd Trigonometric Functions
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1.6 Trigonometric Functions (12) Even and Odd
Trigonometric Functions
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1.6 Trigonometric Functions (13) Transformations
of Trigonometric Graphs

• The rules for shifting, stretching, shrinking,
  and reflecting the graph of a function apply to
  the trigonometric functions as follows

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1.6 Trigonometric Functions (14, Example 2)
Applications
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1.6 Trigonometric Functions (15, Example 3-a)
Applications
Fig1.45
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1.6 Trigonometric Functions (16, Example 3-b)
Applications
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1.6 Trigonometric Functions (16, Example 4)
Inverse Trigonometric Functions
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1.6 Trigonometric Functions (17) Inverse
Trigonometric Functions

• The inverse of the restricted sine function is
  called inverse sine function. The inverse sine
  of x is the angle whose sine is x. It is denoted
  by sin-1x or arcsin x.

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1.6 Trigonometric Functions (18) Inverse
Trigonometric Functions
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1.6 Trigonometric Functions (19) Inverse
Trigonometric Functions
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1.6 Trigonometric Functions (20) Inverse
Trigonometric Functions
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1.6 Trigonometric Functions (21) Inverse
Trigonometric Functions
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1.6 Trigonometric Functions (22, Example 5)
Inverse Trigonometric Functions