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CHAPTER 3 SECTION 3.3 INCREASING AND DECREASING FUNCTIONS AND THE FIRST DERIVATIVE TEST

CHAPTER 3 SECTION 3.3 INCREASING AND DECREASING FUNCTIONS AND THE FIRST DERIVATIVE TEST. Definitions of Increasing and Decreasing Functions. A function is increasing when its graph rises as it goes from left to right. A function is decreasing when its graph falls as it goes from left to right.

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CHAPTER 3 SECTION 3.3 INCREASING AND DECREASING FUNCTIONS AND THE FIRST DERIVATIVE TEST

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  1. CHAPTER 3SECTION 3.3INCREASING AND DECREASINGFUNCTIONS ANDTHE FIRST DERIVATIVE TEST

  2. Definitions of Increasing and Decreasing Functions

  3. A function is increasing when its graph rises as it goes from left to right. A function is decreasing when its graph falls as it goes from left to right. dec inc inc

  4. The inc\dec concept can be associated with the slope of the tangent line. The slope of the tan line is positive when the function is increasing and negative when decreasing

  5. Theorem 3.5 Test for Increasing and Decreasing Functions

  6. Find the Open Intervals on which f is Increasing or Decreasing

  7. Find the Open Intervals on which f is Increasing or Decreasing

  8. Find the Open Intervals on which f is Increasing or Decreasing

  9. Find the Open Intervals on which f is Increasing or Decreasing

  10. Find the Open Intervals on which f is Increasing or Decreasing tells us where the function is increasing and decreasing.

  11. Guidelines for Finding Intervals on Which a Function Is Increasing or Decreasing

  12. Theorem 3.6 The First Derivative Test

  13. Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs • Example 1: Graph the function f given by • and find the relative extrema. • Suppose that we are trying to graph this function but • do not know any calculus. What can we do? We can • plot a few points to determine in which direction the • graph seems to be turning. Let’s pick some x-values • and see what happens.

  14. Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs • Example 1 (continued):

  15. Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs • Example 1 (continued): • We can see some features of the graph from the sketch. • Now we will calculate the coordinates of these features • precisely. • 1st find a general expression for the derivative. • 2nd determine where f (x) does not exist or where • f (x) = 0. (Since f (x) is a polynomial, there is no • value where f (x) does not exist. So, the only • possibilities for critical values are where f (x) = 0.)

  16. Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs • Example 1 (continued): • These two critical values partition the number line into • 3 intervals: A (– ∞, –1), B (–1, 2), and C (2, ∞).

  17. Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs • Example 1 (continued): • 3rd analyze the sign of f (x) in each interval.

  18. Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs • Example 1 (concluded): • Therefore, by the First-Derivative Test, • f has a relative maximum at x = –1 given by • Thus, (–1, 19) is a relative maximum. • And f has a relative minimum at x = 2 given by • Thus, (2, –8) is a relative minimum.

  19. Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs • Example 3: Find the relative extrema for the • Function f (x) given by • Then sketch the graph. • 1st find f (x).

  20. Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs • Example 3 (continued): • 2nd find where f (x) does not exist or where f (x) = 0. • Note that f (x) does not exist where the denominator • equals 0. Since the denominator equals 0 when x = 2, • x = 2 is a critical value. • f (x) = 0 where the numerator equals 0. Since 2 ≠ 0, • f (x) = 0 has no solution. • Thus, x = 2 is the only critical value.

  21. Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs • Example 3 (continued): • 3rdx = 2 partitions the number line into 2 intervals: • A (– ∞, 2) and B (2, ∞). So, analyze the signs of f (x) in both intervals.

  22. Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs • Example 3 (continued): • Therefore, by the First-Derivative Test, • f has a relative minimum at x = 2 given by • Thus, (2, 1) is a relative minimum.

  23. Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs • Example 3 (concluded): • We use the information obtained to sketch the graph below, plotting other function values as needed.

  24. STATEMENT II IS CORRECT BECAUSE SPEED IS THE ABSOLUTE VALUE OF THE VELOCITY FUNCTION!!!!! USING YOUR GRAPHING UTILITY THE ABSOLUTE VALUE OF v(t) increases over that interval!!!!

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