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(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

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

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

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

Examples

  • Find the intervals on which the following functions are increasing or decreasing


Concavity

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


Examples1

Examples

  • Find the intervals on which the following functions are concave up and concave down


Inflection points

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

More examples

  • Find the inflection points of the following functions and sketch their graphs


Applied example crop yield

Applied Example: Crop Yield


Fish length

Fish Length


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