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Calculus I Chapter 4(2) Increrasing Decreasing First Derivative Test

Calculus I Chapter 4(2) Increrasing Decreasing First Derivative Test. Increasing & Decreasing Functions. A function is Increasing between two points if the y-value of the second point is greater than the y-value of the first point.

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Calculus I Chapter 4(2) Increrasing Decreasing First Derivative Test

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  1. Calculus IChapter 4(2)IncrerasingDecreasingFirst Derivative Test

  2. Increasing & Decreasing Functions A function is Increasing between two points if the y-value of the second point is greater than the y-value of the first point. A function is Decreasing between two points if the y-value of the second point is lower than the y-value of the first point.

  3. Give the intervals where the function is increasing, decreasing, and constant a b c d e f g h i j Increasing intervals: Decreasing Intervals: Constant Intervals:

  4. Derivatives and increasing & Decreasing Functions A function is Increasing at a point if the derivative is positive at the point (tangent line is going up) A function is Decreasing at a point if the derivative is Negative at the point (tangent line is going down)

  5. Steps for finding where a function is increasing & decreasing Find the Critical Values (when f’(x) = 0) Graph the Critical values on a number line Check the intervals in f’(x) to see if Positive – Increasing Negative - Decreasing

  6. 2 Find the increasing intervals and the decreasing intervals for: 2 Find Critical Values (derivative = 0) Put Critical Value(s) on a number Line: F(x) is Increasing F(x) is Decreasing Put a # from the left side (1) into the derivative Put a # from the right side (3) into the derivative Positive means increasing Negative means Decreasing (-,2) is Increasing, (2, ) is Decreasing

  7. First Derivative Test If f’(x) changes from negative to positive at some point c, then f(x) is a relative minimum. Positive Derivative Negative Derivative Relative Minimum If f’(x) changes from positive to negative at some point c, then f(x) is a relative maximum. Relative Maximum Negative Derivative Positive Derivative

  8. Steps for finding relative max & min Find the Critical Values (when f’(x) = 0) Graph the Critical values on a number line Check the intervals in f’(x) to see if Positive – Increasing Negative - Decreasing Increasing to Decreasing - Max Decreasing to Increasing - Min

  9. 0 4 Find the local extrema, the increasing intervals and the decreasing intervals for: Find Critical Values (derivative = 0) Put Critical Value(s) on a number Line: F(x) is Increasing F(x) is Decreasing F(x) is Increasing Pick -1 Pick 1 Pick 5 (-,0) U(4,) isIncreasing, (0, 4) is Decreasing (0, 15) Max, (4, -17) Min

  10. -6 0 Find the local extrema, the increasing intervals and the decreasing intervals for: Find Critical Values (derivative = 0) Put Critical Values on a number Line: F(x) is Decreasing F(x) is Increasing F(x) is Decreasing Pick -7 Pick -1 Pick 1 (-,-6) U(0,) isDecreasing, (-6, 0) is Increasing x = 0 Vertical Asymptote, (6, -1/12) Min

  11. Concavity A graph is Concave Up on an interval if f’ is increasing on the interval. A graph is Concave Down on an interval if f’ is decreasing on the interval. f(x)

  12. Draw the derivative graph and indicate where the graph is concave up and concave down and where it is increasing and decreasing. f(x) a b c d e f g Concave Up Concave Down (a, c) (c, e) U (e,f) Increasing Decreasing (b, d) U (e ,f) (a, b) U (d ,e) U (f ,g)

  13. Second Derivative and Concavity A graph is Concave Up on an interval if is positive on the interval. A graph is Concave Down on an interval if is negative on the interval. f(x)

  14. Inflection Point An Inflection Point is a point where the concavity changes This is where the second derivative = 0 or is undefined

  15. Steps for finding where a function is Concave Up or Down Find the second derivative and set it = 0 Graph the resulting values on a number line Check the intervals in to see if Positive – Concave up Negative – Concave Down

  16. -1 1 Find the inflection points and where concave up & down: Find Second Derivative Values Inflection Put Inflection points on a number Line: Concave Up Concave Down Concave Up Pick -2 Pick 0 Pick 2 (-,-1) U (1, ) is Concave Up, (-1,1) is Down

  17. Second Derivative Test If for some value of c, and then c is a minimum point on the graph of f(x) Concave Up Relative Minimum If for some value of c, and then c is a Maximum point on the graph of f(x) Relative Maximum Concave Down

  18. Find the Extrema and inflection points and give the concavity intervals for: Find both Derivatives and set =0 Put all Points on a number Line and finish:

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