Curve Sketching: Role of first and second derivatives

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Curve Sketching: Role of first and second derivatives - PowerPoint PPT Presentation

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 ) &lt; f(x 2 ) whenever x 1 &lt;x 2

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Presentation Transcript
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