UAlbany AMAT 106 - Increasing and Decreasing Functions

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Slide 1Increasing and Decreasing FunctionsExampleTest for Intervals Where f(x) is Increasing and DecreasingCritical NumbersApplying the TestSlide 7Relative Maximum or MinimumExampleAnother Example w/disagreementRelative ExtremumFirst Derivative TestRelative Extrema(cont.)Slide 15Higher DerivativesNotation for Higher Order DerivativesConcavity of a GraphConcavity (Second Derivative)(cont.)A specific point2 Examples of Concave Up Functions2 Examples of Concave Down FunctionsTest for ConcavityFinding the Concavity of a function at Open intervalsInflection pointSecond Derivative TestSlide 28ExampleSlide 30Curve Sketching Procedure(cont.)(cont.)I N C R E A S I N G A N D D E C R E A S I N G F U N C T I O N SChapter 5Section 1Increasing and Decreasing FunctionsA function is increasing if the graph goes up (the y-value) from left to right and decreasing if its graph goes down from left to right.Let f be a function defined on some interval. Then, for any two numbers x1 and x2 in the interval, f is increasing on the interval iff(x1) < f(x2) whenever x1 < x2 and f is decreasing on the interval iff(x1) > f(x2) whenever x1 < x2ExampleTest for Intervals Where f(x) is Increasing and DecreasingSuppose a function f has a derivative at each point in an open interval, thenif for each x in the interval, f is increasing on the intervalif for each x in the interval, f is decreasing on the intervalif for each x in the interval, f is constant on the interval0)( xf0)( xf0)( xfCritical NumbersThe critical numbers for a function f are those numbers c in the domain of f for which or doesn’t exist. A critical point is a point whose x-coordinate is the critical number c and whose y-coordinate is f(c).Note: Not all critical numbers will be an extreme point (mentioned in the next section).0)( cf)(cfApplying the Test1. Locate the critical numbers for f on a number line, as well as any points where f is undefined. These points determine several open intervals.2. Choose a value of x in each of the intervals determined in Step 1. Use these values to decide whether or in that interval.3. Use the test on the previous 2 slides to decide whether f is increasing or decreasing on the interval.0)( xf0)( xfR E L AT I V E E X T R E M AChapter 5Section 2Relative Maximum or MinimumLet c be a number in the domain of a function f. Then, f(c) is a relative maximum for f if there exists an open interval (a, b) containing c such that f(x) ≤ f(c) for all x in (a, b).Also, f(c) is a relative minimum for f if there exists an open interval (a, b) containing c such that f(x) ≥ f(c) for all x in (a, b).If c is an endpoint of the domain of f, we only consider x in the half-open interval that is in the domain.Examplex1 would be a relative max (as well as x3) since it would have the highest y-value for all x in the open interval. x2 would be a relative min as well as x4.Another Example w/disagreementThere is disagreement on calling an endpoint (x4) a maximum or a minimum. However, we may define it has a relative min (or relative max) because this is an applied calculus book which should be considered a max or a min of the function. This will be mentioned in Chapter 6.Relative ExtremumIf a function f has a relative extremum (max or min) at c, then c is a critical number or c is an endpoint of the domain.NOTE: We can not say the reverse way.First Derivative TestLet c be a critical number for a function f. Suppose that f is continuous on (a, b) and differentiable on (a, b) except possibly at c, and that c is the only critical number for f in (a, b).1. f(c) is a relative maximum of f if the derivative is positive in the interval (a, c) and negative in the interval (c, b)2. f(c) is a relative minimum of f if the derivative is negative in the interval (a, c) and positive in the interval (c, b))(xf)(xfRelative Extrema(cont.)The last 2 that we saw shows that there is a critical point but the critical point is not a relative extreme point.H I G H E R D E R I VAT I V E S, C O N C AV I T Y, A N D T H E S E C O N D D E R I VAT I V E T E S TChapter 5Section 3Higher DerivativesIf a function f has a derivative , then the derivative of , if it exists, is the second derivative of f, written . The derivative of , if it exists, is called the third derivative of f, and so on. By continuing this process, we can find fourth derivatives and other higher order derivatives.ffffNotation for Higher Order DerivativesThe second derivative of y = f(x) can be written using any of the following notations:The third derivative can be written in a similar way. For n ≥ 4, the nth derivative is written  Note: is not the same as  )(Dor , ),(2x22xfdxydxf)()(xfn)()4(xf)(4xfConcavity of a Graph•The following examples are graphs of increasing functions•The only difference between these 2 graphs is its concavityConcavity (Second Derivative)The second derivative gives the rate of change of the first derivative (i.e. the rate at which the rate of change is changing). It indicates how fast the function is increasing or decreasing. The rate of change of the derivative affects the shape of the graph.(cont.)A function is said to be concave up on an interval (a, b) if the graph of the function lies above its tangent lines at each point of (a, b).A function is said to be concave down on an interval (a, b) if the graph of the function lies below its tangent lines at each point of (a, b).A specific pointA point where a graph changes concavity is called an inflection point.2 Examples of Concave Up Functions2 Examples of Concave Down FunctionsTest for ConcavityLet f be a function with derivatives of and existing at all points in an interval (a, b). Then f is concave up on (a, b) if for all x in (a, b) and concave down on (a, b) if for all x in (a, b).ff0)( xf0)( xfFinding the Concavity of a function at Open intervalsThe way of finding the concavity of a function at open intervals is done the same way as the first derivative:First, find the values of c that will make or doesn’t exist.If for all x in the open interval (a, b), then the function is concave up.If


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UAlbany AMAT 106 - Increasing and Decreasing Functions

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