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2.10 Applications of the Derivative

Learn how to analyze functions using the first and second derivatives, identify relative extrema, concavity, and sketch graphs effectively. Explore concepts such as increasing/decreasing intervals, sign diagrams, inflection points, and vertical/horizontal asymptotes. Master curve sketching techniques step by step.

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2.10 Applications of the Derivative

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  1. 2.10 Applications of the Derivative • Applications of the First Derivative • Applications of the Second Derivative • Curve Sketching

  2. Increasing/Decreasing A function f is increasing on (a, b) if f (x1) <f (x2) whenever x1 < x2. A function f is decreasing on (a, b)if f (x1) >f (x2) whenever x1 < x2. Increasing Decreasing Increasing

  3. Increasing/Decreasing/Constant

  4. Sign Diagram to Determine where f (x) is Inc./Dec. Steps: • Find all values of x in the domain of f for which is discontinuous and identify open intervals with these endpoints. 2. Test a point c in each interval to check the sign of f is increasing on that interval. a. If f is decreasing on that interval. b. If

  5. Example Determine the intervals where is increasing and where it is decreasing. + - + 0 4 f is decreasing on f is increasing on

  6. Relative Extrema A function f has a relative maximum at x = c if there exists an open interval (a, b) containing c such that for all x in (a, b). A function f has a relative minimum at x = c if there exists an open interval (a, b) containing c such that for all x in (a, b). Relative Maximums Relative Minimums

  7. Critical Points of f A critical point of a function f is a point in the domain of f where (horizontal tangent lines, vertical tangent lines and sharp corners)

  8. The First Derivative Test 1. Determine the critical points of f. • Determine the sign of the derivative of f to the left and right of the critical point. left right f(c) is a relative maximum f(c) is a relative minimum No change No relative extrema

  9. Example Find all the relative extrema of Relative max. f (0) = 1 Relative min. f (4) = -31 + - + 0 4

  10. Example Find all the relative extrema of or Relative max. Relative min. + + - - + + -1 0 1

  11. Concavity Let f be a differentiable function on (a, b). 1. f is concave upward on (a, b) if is increasing on (a, b). That is, for each value of x in (a, b). 2. f is concave downward on (a, b) if is decreasing on (a, b). That is, for each value of x in (a, b). concave upward concave downward

  12. Determining the Intervals of Concavity • Determine the values for which the second derivative of f is zero or undefined. Identify the open intervals with these endpoints. 2. Determine the sign of in each interval from step 1 by testing it at a point, c, on the interval. f is concave up on that interval. f is concave down on that interval.

  13. Example Determine where the function is concave upward and concave downward. – + 2 f concave down on f concave up on

  14. Inflection Point A point on the graph of f at which concavity changes is called an inflection point. To find inflection points, find any point, c, in the domain where is undefined. If changes sign from the left to the right of c, Then (c,f (c))is an inflection point of f.

  15. Example Determine where the function is concave upward and concave downward and find any inflection points. - + + f concave up on f concave up on 0 1 f concave down on (0,1) Inflection points f (1) = -4 and

  16. The Second Derivative Test 1. Compute 2. Find all critical points, c, at which If Then f has a relative maximum at c. f has a relative minimum at c. The test is inconclusive.

  17. Example Classify the relative extrema of using the second derivative test. Critical points: x = 0, 1, 2 Relative max. Relative mins.

  18. Vertical Asymptote The line x = a is a vertical asymptote of the graph of a function f if either is infinite. Horizontal Asymptote The line y = b is a horizontal asymptote of the graph of a function f if

  19. Finding Vertical Asymptotes of Rational Functions If is a rational function, then x = a is a vertical asymptote if Q(a) = 0 but P(a) ≠ 0. Ex. f has a vertical asymptote at x = 5.

  20. Finding Horizontal Asymptotes of Rational Functions Ex. 0 0 Divide by the highest power of x 0 f has a horizontal asymptote at

  21. Curve Sketching Guide 1. Determine the domain of f. 2. Find the intercepts of f if possible. 3. Look at end behavior of f. 4. Find all horizontal and vertical asymptotes. • Determine intervals where f is inc./dec and show results on a the f’ number line. 6. Find the relative extrema of f. • Determine the concavity of f and show the results on the f’’ number line. 8. Find the inflection points of f. 9. Sketch f, use additional points as needed.

  22. Example Sketch: 1. Domain: (−∞, ∞). 2. yIntercept: (0, 1) 3. • No Asymptotes 5. f inc. on (−∞, 1) U (3, ∞), dec. on (1, 3). 6. Relative max.: (1, 5); relative min.: (3, 1) 7. f concave down (−∞, 2); up on(2, ∞). 8. Inflection point: (2, 3)

  23. Sketch:

  24. Example Sketch: 1. Domain: x≠ −3 2. Intercepts: (0, −1) and (3/2, 0) 3. • Horizontal: y = 2; Vertical: x = −3 5. f is increasing on (−∞,−3) U (−3, ∞). 6. No relative extrema. f is concave up on (−∞,−3) and f is concave down on (−3, ∞). 7. 8. No inflection points

  25. Sketch: y = 2 x = −3

  26. How to Sketch the Graph of a Rational Function f(x) = P(x)/Q(x), Where P(x) and Q(x) Have No Common Factors (p. 290)

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