Applications of Derivatives

1. Applications of Derivatives Curve Sketching

2. What the First Derivative Tells Us: • Suppose that a function f has a derivative at every point x of an interval I. Then:

3. What This Means: • In geometric terms, the first derivative tells us that differentiable functions increase on intervals where their graphs have positive slopes and decrease on intervals where their graphs have negative slopes. • WHAT HAPPENS IF THE FIRST DERIVATIVE IS ZERO?

4. When The First Derivative is Zero • A derivative has the intermediate value property on any interval on which it is defined. • It will take on the value zero when it changes signs over that interval. • Thus, when the derivative changes signs on an interval, the graph of f(x) must have a horizontal tangent.

5. HOWEVER… • A derivative need not change sign every time it is zero. Consider • The derivative is • The derivative is zero at the origin but positive on both sides of the origin.

6. Relative Maxima and Minima • If the derivative changes sign, there may be a local max or min, as shown here. • More on this later.

7. Concavity • Concave down—”spills water” • Concave up—”holds water” • The graph of is concave down on any interval where and concave up on any interval where

8. Points of Inflection • A point on the curve where the concavity changes is called a point of inflection. • If the second derivative is zero for some x, we may be able to find a point of inflection. • It IS possible for the second derivative to be zero at a point that is NOT a point of inflection. • A point of inflection may occur where the second derivative fails to exist.

9. Relative Extrema • Let f(x) be defined on an interval, I, and let x0 be in I. • 1. If f(x) has a relative extremum at x = x0 then either f’(x)=0 or f is not differentiable at x = x0. • 2. Values at which the derivative is zero at x0 or at which f is not differentiable at x = x0 are called critical numbers. • 3. If f is defined on an open interval, its relative extrema occur at critical numbers. • NOTE: This does NOT mean that a critical number MUST yield a relative extremum.

10. This is what happens around the point x0: The First Derivative and Relative Extrema

11. Assume that f is twice differentiable at x0. If: The Second Derivative and Relative Extrema

12. An Example: • This first derivative is equal to zero at x=0, x=1 and x= -1. • These are the critical values. • Examine the sign of the derivative around these values:

13. Sign of the First Derivative:

14. Furthermore… • The function is decreasing from ( ) and on (0,1) because the derivative is negative on those intervals. • The function is increasing on (-1,0) and on ( 1, ) because the derivative is positive on those intervals. • We will examine the second derivative for what it can tell us. • The second derivative is:

15. The Second Derivative. • The second derivative is equal to zero at x = • Examine the sign of the second derivative around these points: +++++ ------- +++++

16. Concavity • The function is concave up in those areas where the second derivative is positive and concave down in that area where the second derivative is negative. • If you check the sign of the second derivative at the critical values, you will find that this reinforces what we said before about the relative max and min.

17. Inflection Points • You can tell where the function changes concavity by finding the inflection points. • Evaluate the function at those values where the second derivative is zero; that is, at x = • Take a look at the graph of the original function:

18. The Graph

19. Does It Check? • Check the intervals on which the function is increasing and decreasing. • Check the location of relative maxima and/or minima. • Check the concavity of the function. • The graph should match information determined from the derivatives.