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Lesson 4.3 First and Second Derivative Theory

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Lesson 4.3 First and Second Derivative Theory

  1. 1. Section 4.3 First and Second Derivative Information
  2. 2. Test for Increasing or Decreasing Functions Let f be continuous on [a,b] and differentiable on (a,b).
  3. 3. Increasing/Decreasing To determine whether the function is increasing or decreasing on an interval, evaluate points to the left and right of the critical points on an f ’ numberline. f'(x) c + __ inc __ dec f'(x) c + __ inc dec
  4. 4. First Derivative Test Let c be a critical number of a function f that is continuous on an open interval containing c. If f is differentiable on the interval, except possibly at c, then f (c) can be classified as follows…
  5. 5. 1st Derivative Test f'(x) c + __ inc __ dec If the sign changes from + to - at c, then c is a relative maximum . Max f'(x) c + __ inc dec If the sign changes from - to + at c, then c is a relative minimum . Min
  6. 6. Concavity <ul><li>A curve is concave up if its slope is increasing , in which case the second derivative will be positive ( f &quot;(x) >0 ). </li></ul><ul><li>Also, the graph lies above its tangent lines. </li></ul><ul><li>A curve is concave down if its slope is decreasing , in which case the second derivative will be negative ( f &quot;(x) < 0 ). </li></ul><ul><li>Also, the graph lies below its tangent lines. </li></ul>
  7. 7. Test for Concavity Let f be a function whose 2 nd derivative exists on an open interval I.
  8. 8. Concavity Test To determine whether a function is concave up or concave down on an interval, determine where f &quot;(x) = 0 and f &quot;(x) is undefined . Then evaluate values to the left and right of these points on an f &quot; numberline. f &quot; ( x) c + __ ccu __ ccd f &quot;(x) c + __ ccu ccd
  9. 9. Inflection A point where the graph of f changes concavity, from concave up to concave down or vice versa, is called a point of inflection . At a point of inflection the second derivative will either be undefined or 0.
  10. 10. When the signs change on an f &quot; numberline, there is an inflection point. If the signs on the f &quot;(x) numberline do not change, then c is not an inflection point. f &quot;(x) c __ ccd ccd __ Not inflection
  11. 11. Second Derivative Test Let f be a function such that f ’(c) = 0 and the 2 nd derivative of f exists on an open interval containing c.

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