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Curve Sketching: Role of first and second derivatives. Increasing/decreasing functions and the sign of the derivative Concavity and the sign of the second derivative. Increasing/Decreasing functions. If f is defined on an interval then f is increasing if f(x 1 ) < f(x 2 ) whenever x 1 <x 2

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## Curve Sketching: Role of first and second derivatives

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**Curve Sketching: Role of first and second derivatives**• Increasing/decreasing functions and the sign of the derivative • Concavity and the sign of the second derivative**Increasing/Decreasing functions**• If f is defined on an interval then • f is increasing if f(x1) < f(x2) whenever x1<x2 • f is decreasing if f(x1) > f(x2) whenever x1<x2 • f is constant if f(x1) = f(x2) for all points x1 and x2**Sign of first derivatives**• Fact: If f is continuous on [a,b] and differentiable in (a,b), then • f is increasing on [a,b] if f (x)>0 for all x in (a,b) • f is decreasing on [a,b] if f (x)<0 for all x in (a,b) • f is constant on [a,b] if f (x)=0 for all x in (a,b)**Examples**• Find the intervals on which the following functions are increasing or decreasing**Concavity**• Concavity measures the curvature of a function. Concave up=“holds water”. Concave down=“spills water”. • If f is differentiable on an open interval I then f is concave up if f is increasing there, and f is concave down if f is decreasing there • Conclusion: f concave up on I if f >0 on I and f concave down on I if f <0 on I**Examples**• Find the intervals on which the following functions are concave up and concave down**Inflection Points**• If f changes concavity at x0 then f has an inflection point at x0 (i.e. f changes sign at x0) • Examples: Find the inflection points of the following functions**More examples**• Find the inflection points of the following functions and sketch their graphs

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