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Math 1F03 – Lecture 2

Math 1F03 – Lecture 2. 1.5 Exponential Functions 1.6 Inverse Functions and Logarithms 2.2 Introduction to Limits. 1.5 Exponential Functions. An exponential function is a function of the form where is a positive real number called the base and is a variable called the exponent.

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Math 1F03 – Lecture 2

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  1. Math 1F03 – Lecture 2 1.5 Exponential Functions 1.6 Inverse Functions and Logarithms 2.2 Introduction to Limits

  2. 1.5 Exponential Functions An exponential function is a function of the form where is a positive real number called the base and is a variable called the exponent. Domain: Range:

  3. Graphs of Exponential Functions When a>1, the function is increasing. When a<1, the function is decreasing. y=0 is a horizontal asymptote

  4. Transformation of an Exponential Function Graph Recall: is a special irrational number between 2 and 3 that is commonly used in calculus Approximation:

  5. Laws of Exponents Examples:

  6. Exponential Models P. 57 #24. Suppose you are offered a job that lasts one month. Which of the following methods of payment do you prefer? • One million dollars at the end of the month. • One cent on the first day of the month, two cents on the second day, four cents on the third day, and, in general, cents on the nth day.

  7. 1.6 Inverse Functions The function is the inverse of if and . Each of and undoes the action of the other. Diagram: Some simple examples:

  8. What Functions Have Inverses? A function has an inverse if and only if it is a one-to-one function. A function f is one-to-one if for every y-value in the range of f, there is exactly one x-value in the domain of f such that y=f(x). Examples:

  9. Horizontal Line Test If every horizontal line intersects the graph of a function in at most one point, then the graph represents a one-to-one function.

  10. Finding the Inverse of a Function Algorithm: • Write the equation y=f(x). • Solve for x in terms of y. • Replace x by (x) and y by x. Note: The domain and range are interchanged Example: Find the inverse of the following functions. State domain and range.

  11. Graphs of and The graph of is the graph of reflected in the line Points (x,y) on become the points (y,x) on Example: Given sketch and .

  12. f

  13. 1.6 Logarithmic Functions The inverse of an exponential function is a logarithmic function, i.e. Cancellation equations: In general: For exponentials & logarithms:

  14. Understanding Logarithm Notation Examples:

  15. Graphs of Logarithmic Functions Recall: For inverse functions, the domain and range are interchanged and their graphs are reflections in the line Example: Graph

  16. Graphs of Logarithmic Functions

  17. The Natural Logarithm Domain: Range: Graph: The graph increases from negative infinity near x=0 (vertical asymptote) and rises more and more slowly as x becomes larger. Note: and

  18. Transformation of a Log Function Example: Graph State the domain.

  19. Laws of Logs For x,y>0 and p any real number: Example: Simplify, if possible. (a) (b) (c) (d)

  20. Some Exercises Solve the following equations for x. (a) (b) (c) (d)

  21. 2.2 The Limit of a Function Notations: means that the y-value of the function AT x=2 is 5 means that the y-value of the function NEAR x=2 is NEAR 4

  22. The Limit of a Function Definition: “the limit of as approaches , equals ” means that the values of (y-values) approach the number more and more closely as approaches more and more closely (from both sidesof ), but ignoring when .

  23. Limit of a Function Some examples: Note: f may or may not be defined ATx=a. Limits are only concerned with how f is defined NEAR a.

  24. Left-Hand and Right-Hand Limits means as from the left means as from the right ** The full limit exists if and only if the left and right limits both exist (equal a real number) and are the same value.

  25. Left-Hand and Right-Hand Limits For each function below, determine the value of the limit or state that it does not exist.

  26. Evaluating Limits We can evaluate the limit of a function in 3 ways: • Graphically • Numerically • Algebraically

  27. Evaluating Limits Example: Evaluate graphically.

  28. Evaluating Limits Example: Use a table of values to estimate the value of

  29. Evaluating Limits Example: Use a table of values to estimate the value of

  30. Infinite Limits Definition: “the limit of , as approaches , is infinity” means that the values of (y-values) increase without bound as becomes closer and closer to (from either side of ), but Definition: “the limit of , as approaches , is negative infinity” means that the values of (y-values) decrease without bound as becomes closer and closer to (from either side of ), but

  31. Infinite Limits Example: Determine the infinite limit. #30. #34. Note: Since the values of these functions do not approach a real number L, these limits do not exist.

  32. Vertical Asymptotes Definition: The line is called a vertical asymptote of the curve y=f(x) if either Example: Basic functions we know that have VAs:

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