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Additional Applications of the Derivative

Chaper Three

3.1 Increasing and Decreasing FunctionIncreasing and Decreasing Function Let f(x) be a functiondefined on the interval ax1 f(x) is decreasing on the interval if f(x2)x1 Monotonic increasing Monotonic decreasing

3.1 Increasing and Decreasing FunctionTangent line with negative slope f(x) will be decreasing Tangent line with positive slope f(x) will be increasing

3.1 Increasing and Decreasing FunctionIntermediate value property A continuous function cannot change sign without first becoming 0. If for every x on some interval I,then f(x) is increasing on the intervalIf for every x on some interval I,then f(x) is decreasing on the intervalIf for every x on some interval I, then f(x) is constant on the interval

3.1 Increasing and Decreasing Function

- Example. Find the intervals of increase and decrease for the functionSolution: The number -2 and 1 divide x axis into three open intervals. x
- 3.1 Relative Extrema Relative (Local) Extrema The Graph of the function f(x) is said to be have a relative maximum at x=c if f(c) f(x) for all x in interval a
3.1 Critical PointsRelative extrema can only occur at critical points!

3.1 Critical PointsNot all critical points correspond to relative extrema!

Figure. Three critical points where f(x) = 0: (a) relative maximum, (b) relative minimum (c) not a relative extremum.

3.1 Critical PointsNot all critical points correspond to relative extrema!

Figure Three critical points where f(x) is undefined: (a) relative maximum, (b) relative minimum (c) not a relative extremum.

3.1 The First Derivative Test

- ExampleSolutionFind all critical numbers of the function and classify each critical point as a relative maximum, a relative minimum, or neither Thus the graph of f falls for x
3.1 Sketch the graphA Procedure for Sketching the Graph of a Continuous Function f(x) Using the Derivative Step 1. Determine the domain of f(x).

ExampleSolutionSketch the graph of the function

f(-3)=19 f(0)=-8 Plot a twist at (-3,19) to indicate a galling graph with a horizontal tangent at this point . Complete the sketch by passing a smooth curve through the Critical point in the directions indicated by arrow

ExampleSolution The revenue derived from the sale of a new kind of motorized skateboard t weeks after its introduction is given by million dollars. When does maximum revenue occur? What isthe maximum revenue

3.2 ConcavityIncrease and decrease of the slopes are our concern!Figure The output Q(t) of a factory worker t hours after coming to work.

3.2 Concavity

3.2 ConcavityA graph is concave upward on the interval if it lies above all its tangent lines on the interval and concave downward on an Interval where it lies below all its tangent lines.

Note Dont confuse the concavity of a graph with its direction(rising or falling). A function may be increasing or decreasing onan interval regardless of whether its graph is concave upward or concave downward on the interval.

3.2 Concavity and the second Derivative

3.2 Concavity and the second Derivative Second Derivative Procedure for Determining Intervals of Concavity for a Function f.

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3.2 Inflection points

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Note: A function can have an inflection point only where it is continuous.!!

3.2 Behavior of Graph f(x) at an inflection point P(c,f(c))

-1.5min --------++++++1 Neitherto be continued

--------++++++-2/3inflection1inflection Type of concavity++++++to be continued

Find all critical numbers of the function Classify each critical point as a relative maximum, a relative minimum, or neitherFind all inflection points of functiona.d.Review

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3.2 The Second Derivative Test

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As mentioned in the chapter two, if a graph of function has tangent with positive slope on the interval a