Curve Sketching: Role of first and second derivatives

1. 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

2. 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

3. 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)

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

5. 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

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

7. 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

8. More examples • Find the inflection points of the following functions and sketch their graphs

9. Applied Example: Crop Yield

10. Fish Length