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Exponential Functions

Exponential Functions. Precalculus – Section 3.1. Definition. An exponential function is a function of the form We call b the base of the exponential function. a is a constant multiplier (think stretch/shrink). Requirements: b greater than zero and not equal to one

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Exponential Functions

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  1. Exponential Functions Precalculus – Section 3.1

  2. Definition An exponential function is a function of the form We call b the base of the exponential function. a is a constant multiplier (think stretch/shrink). Requirements: • b greater than zero and not equal to one • x is any real number

  3. Graph of an Exponential Function The graph of the function crosses the y-axis at the point (0,a). The graph also contains the point (1,a⋅b).

  4. Graph of an Exponential Function To get more points on the graph: As you increase the value of x by 1, you multiply the previous y-value by the base b. As you decrease the value of x by 1, you divide the y-value by b.

  5. Growth and Decay If the base is greater than 1… If the base is less than 1… …then the function grows. …then the function decays.

  6. Graph the exponential function.

  7. Assignment p. 206: 1-4, 7-10, 19-24

  8. The Natural Base Precalculus – Section 3.1

  9. The Natural Base A common choice for the base of an exponential function is e . e is the called the natural base because it naturally occurs in things such as: • compound interest • radioactive decay • science applications

  10. The Value of e The value of e to 15 decimal places… e = 2.718281828459045… Think of President Andrew Jackson (the guy on the $20 bill)! Good enough for precalculus use: 2.718281828

  11. Evaluate each function. Round to 3 decimal places. 1. 2.

  12. Graphs Graph the function.

  13. Assignment p. 206: 5,6, 25-30 You may want to scale your graphs to fit them on the paper. Tomorrow: using exponential functions to problem solve!

  14. Problem Solving with Exponential Functions Precalculus – Section 3.1

  15. Compound Interest To find interest that is compounded continuously: P = principle (amount invested) r = interest rate (as a decimal) t = time (in years)

  16. Example Find the current balance of a $7000 savings fund after 1 year if the interest is compounded continuously at 8%.

  17. Radioactive Decay To find the amount of a radioactive substance that remains after t years: N = initial quantity t = time (in years) H = half-life of substance (in years)

  18. Example A certain radioactive substance has a half-life of 825 years. Find the amount of substance that remains after 1000 years if the initial amount is 50 pounds.

  19. Assignment p. 207: 51-53, 55-58

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