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Math 1304 Calculus I

Math 1304 Calculus I. Chapter 1. Functions and Models. Sections Covered in Chapter 1. 1.1: Four Ways to Represent a Function 1.2: Mathematical Models and Essential Functions 1.3: New Functions from Old - composition 1.4: Graphing functions 1.5: Exponential Functions 1.6: Inverse Functions.

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Math 1304 Calculus I

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  1. Math 1304 Calculus I Chapter 1. Functions and Models

  2. Sections Covered in Chapter 1 • 1.1: Four Ways to Represent a Function • 1.2: Mathematical Models and Essential Functions • 1.3: New Functions from Old - composition • 1.4: Graphing functions • 1.5: Exponential Functions • 1.6: Inverse Functions

  3. Section 1.1 • Four Ways to Represent a Function • Covers functions: • Definition • Terminology • Conceptualization • Ways to represent

  4. Definition of Function • Definition: A function is a rule that assigns to each element in one set exactly one element in another set.

  5. Terminology • Domain – set of values for which the rule is defined • Range – set of values that the rule produces as output • Argument: input to the rule • Value of: output from the rule • Variables • independent variable: input to the rule • dependent variable: output from the rule

  6. Conceptualization: arrow diagram f(x) x f(a) a A B

  7. Conceptualization: Machine Input Output

  8. Ways to represent functions • Verbally – use a language • Numerically – use a table • Visually – use a diagram • Algebraically – use a formula • Implicit: as formula that gives a relation between argument and value • Explicit: value is given directly by a formula in terms of the argument

  9. Examples • See book for plenty of examples

  10. Real Functions • Note: In this case we study real-valued functions of a real variable. • In other courses we study functions between other types of sets. • Calculus III, functions can go from subsets of n-dimensional space to subsets of m-dimensional space. • In Modern Algebra, functions often go between arbitrary finite sets. • Sometimes they go between sets of whole numbers.

  11. Graph of a Function • The graph of a real-valued function of a real variable is a curve in the real plane.

  12. Vertical Line Test • Vertical line test – a curve in the xy-plane is the graph of a function if and only if no vertical line intersects the curve more than once. Is a function Not a function

  13. Concepts • Symmetry - odd or even functions – • even functions satisfy: f(-x) = f(x) • and odd functions satisfy: f(-x) = -f(x) • Order - increasing/decreasing functions preserve or reverse order. • Increasing: x < y f(x)<f(y) • Decreasing: x < y f(x)>f(y)

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