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Additional Applications of the Derivative Chaper Three

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§3.1 Increasing and Decreasing Function Increasing and Decreasing Function Let f(x) be a function defined on the interval a

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§3.1 Increasing and Decreasing Function Tangent line with negative slopef(x) will be decreasing Tangent line with positive slopef(x) will be increasing

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§3.1 Increasing and Decreasing Function Intermediate 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 interval If for every x on some interval I, then f(x) is decreasing on the interval If for every x on some interval I, then f(x) is constant on the interval How to determine all intervals of increase and decrease for a function ? How to find all intervals on which the sign of the derivative does not change.

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§3.1 Increasing and Decreasing Function Procedure for using the derivative to determine intervals of increase and decrease for a function of f. Step 2. Choose a test number c from each interval a

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Example. Find the intervals of increase and decrease for the function Solution: The number -2 and 1 divide x axis into three open intervals. x 1 Risingf is increasing2x>1 Fallingf is deceasing0-2

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§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

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§3.1 Critical Points Relative extrema can only occur at critical points! Critical Numbers and Critical Points A number c in the domain of f(x) is called a critical number if either or does not exist. The corresponding point (c,f(c)) on the graph of f(x) is called a critical point for f(x).

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§3.1 Critical Points Not 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.

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§3.1 Critical Points Not 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.

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§3.1 The First Derivative Test The First Derivative Test for Relative Extrema Let c be a critical number for f(x) [that is, f(c) is defined and either or does not exist]. Then the critical point (c,f(c)) is A relative maximum if to the left of c and to the right of c A relative minimum if to the left of c and to the right of c Not a relative extremum if has the same sign on both sides of c c c c c

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Example Solution Find all critical numbers of the function and classify each critical point as a relative maximum, a relative minimum, or neither The derivative exists for all x, the only critical numbers are Where that is, x=0,x=-1,x=1. These numbers divide that x axis into four intervals, x 1 Choose a test number in each of these intervals -1 min --------++++++--------+++++ + 0 max1 min Thus the graph of f falls for x<-1 and for 0 1 x=0 relative maximum x=1 and x=-1 relative minimum

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§3.1 Sketch the graph A Procedure for Sketching the Graph of a Continuous Function f(x) Using the Derivative Step 1. Determine the domain of f(x). Step 2. Find and each critical number, analyze the sign of derivative to determine intervals of increase and decrease for f(x). Step 3. Plot the critical point P(c,f(c)) on a coordinate plane, with a “cap” at P if it is a relative maximum or a “cup” if P is a relative minimum. Plot intercepts and other key points that can be easily found. Step 4 Sketch the graph of f as a smooth curve joining the critical points in such way that it rise where, falls where and has a horizontal tangent where

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Example Solution Sketch the graph of the function The derivative exists for all x, the only critical numbers are Where that is, x=0, x=-3. These numbers divide that x axis into three intervals, x 0. Choose test number in each interval (say, -5, -1 and 1 respectively) -3 neither -------- ++++++-------- 0 min Thus the graph of f has a horizontal tangents where x is -3 and 0, and it is falling in the interval x<-3 and -3

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f(-3)=19 f(0)=-8 Plot a “cup” at the critical point (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

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Example Solution 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 is the maximum revenue Critical number t=7 divides the domain into two intervals x<=t<7 and 7

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

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§3.2 Concavity Concavity If the function f(x) is differentiable on the interval a

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§3.2 Concavity A 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.

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Note Don’t confuse the concavity of a graph with its “direction” (rising or falling). A function may be increasing or decreasing on an interval regardless of whether its graph is concave upward or concave downward on the interval.

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§3.2 Concavity and the second Derivative How to characterize the concavity of the graph of function f(x) in terms of the second derivative? A function f(x) is increasing where its derivative is positive. Thus, the derivative function must be increasing where its derivative is positive. Similarly, on interval a

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§3.2 Concavity and the second Derivative Second Derivative Procedure for Determining Intervals of Concavity for a Function f. Step 1. Find all values of x for which or is not continuous, and mark these numbers on a number line. This divides the line into a number of open intervals. Step 2. Choose a test number c from each interval a

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to be continued

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-------- ++++++ -------- ++++++ 01 Type of concavity Sign of

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§3.2 Inflection points

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-------- ++++++ 0 No inflection 1 inflection Type of concavity Sign of to be continued

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------- - ++++++ 0 inflection Type of concavity Sign of

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

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

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-1.5 min -------- ++++++ 1 Neither ++++++ to be continued

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-------- ++++++ -2/3 inflection 1 inflection Type of concavity Sign of ++++++ to be continued

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b.Find all critical numbers of the function c.Classify each critical point as a relative maximum, a relative minimum, or neither e.Find all inflection points of function a. d.d. Review

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to be continued

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--------++++++--------++++++ to be continued

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++++++-------- ++++++ to be continued

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§3.2 The Second Derivative Test

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to be continued

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3 Max ++++++-------- t 0 4

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