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Applied Calculus (MAT 121) Dr. Day Fri day Feb 17, 2012

In this lesson, we will explore how the first derivative of a function can help us determine if a function is increasing or decreasing and locate the maximum and minimum values of the function. We will also discuss the significance of critical numbers and the first derivative test.

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Applied Calculus (MAT 121) Dr. Day Fri day Feb 17, 2012

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  1. Applied Calculus (MAT 121)Dr. Day Friday Feb 17, 2012 • Using the First Derivative • Is a function increasing or decreasing? • Where are the max and min values of a function? MAT 121

  2. What does the first derivative tell us? Big Ideas • The direction a function is moving at any point—how it is changing at that point—is revealed in its derivative, because a function’s derivative gives us instantaneous rate of change (IROC) at a point. This connects with the increasing and decreasing nature of a function. • The peaks and valleys of a function—its maximum and minimum values and where they occur—can often be detected using the derivative. When a smoothly flowing function reaches a max or a min, its tangent line slope can be nothing except 0, because the tangent line is horizontal. MAT 121

  3. The First Derivative Reveals Whethera Function is Increasing and Decreasing • Connection: Sign of Derivative Direction of Change MAT 121

  4. The First Derivative Reveals Whethera Function is Increasing and Decreasing Example: Determine the intervals on which the function f(x) = x3 – 3x2 – 24x + 32 is increasing and on which it is decreasing. MAT 121

  5. The First Derivative Reveals Whethera Function is Increasing and Decreasing Example: Determine the intervals on which the function f(x) = x2/3 is increasing and on which it is decreasing. MAT 121

  6. The First Derivative Reveals Whethera Function is Increasing and Decreasing Example: Determine the intervals on which the function f(x) = x+ 1/xis increasing and on which it is decreasing. MAT 121

  7. A Function’s Maximum and Minimum Values • What do we mean by maximum and minimum? • How does the derivative connect with the existence of these extreme values? MAT 121

  8. A Function’s Maximum and Minimum Values • What does RELATIVE mean? MAT 121

  9. A Function’s Maximum and Minimum Values • How does the derivativeconnect with the existence of these extreme values? MAT 121

  10. A Function’s Maximum and Minimum Values • What if the derivative doesn’t exist at some point? MAT 121

  11. A Function’s Maximum and Minimum Values • First: Identify a function’s critical numbers. • Then: Look at the sign of the derivative near each critical number MAT 121

  12. A Function’s Maximum and Minimum Values Example: Determine the relative extrema for the function f(x) = x3 – 3x2 – 24x + 32. MAT 121

  13. A Function’s Maximum and Minimum Values The FirstDerivative Test MAT 121

  14. A Function’s Maximum and Minimum Values Example: Determine the relative extrema for the function f(x) = x + 1/x. MAT 121

  15. Assignments WebAssign 4.1 due Monday 2/20 Reminder: Check results for Test #2 MAT 121

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