2nd Derivatives, Differentiability and Products/Quotients (2/27/09) The second derivative f '' of a function f measures the rate of change of the rate of change of f . On a graph, f '' measures concavity. If f '' is positive, then f is concave up; if f '' is negative, then f is concave down, and if f '' is 0, the curve has no concavity at that point (not bending! E.g., if f is linear). If the concavity changes from up to down (or vice-versa) at a point a (so f '' goes from + to -), a is called an inflection point of f.

2nd Derivative - Units The units of the second derivative, in general, are output units per input unit per input unit. Example: f (x) = x 2 + 3x + 2, so f ''(x) = 2 . This means that the slope of f everywhere is changing at a rate of 2 vertical units per horizontal unit per horizontal unit. Example: If position s (in feet) is a function time t (in seconds), then s ''(t ) is acceleration, measured in feet/sec/sec.

Clicker Question 1 What is the second derivative of f (x ) = 2x 3 + 3x ? A. 12x + 3x B. 6x 2 + 3x (ln(3)) C. 12x + x (x – 1) 3x – 2 D. 12x + 3x (ln(3))2 E. T.G.I.F.

Clicker Question 2 If the position (in feet) of an object at time t (in seconds) is given by s (t ) = 5t 2 + 4t + 1 , what is the acceleration of the object at time t = 3 seconds? A. 10 feet/sec/sec B. 10 feet/sec C. 30 feet/sec/sec D. 34 feet/sec E. 34 feet/sec/sec

Concepts of Continuity and Differentiability A function f is continuous at a point if its graph does not break apart at that point. A function f is differentiable at a point if it has a well-defined derivative at the point, which means that the graph of f will neither break nor have a sharp bend at that point. If you look closely at f near that point, it will look like a straight line, i.e., it looks locally linear at every differentiable point.

Relationship of Continuity and Differentiability If f is differentiable at a point, then it is definitely continuous there. Put the opposite way, if f is discontinuous at a point, then it cannot possibly have a derivative there. On the other hand, the absolute value function f (x) = |x | at x = 0 is continuous but is not differentiable. Likewise for the more complicated function f (x) = x sin(1/x) at x = 0 (not locally linear!).

The Product and Quotient Rules for finding derivatives The Product Rule: (f g )'(x) = f (x) g '(x) + g (x) f '(x) In words, the derivative of a product is the first times the derivative of the second plus the second times the derivative of the first. The Quotient Rule: (f / g )'(x) = (g (x)f '(x)f (x)g '(x))/g 2(x) In words, the derivative of a quotient is bottom times derivative of top minus top times derivative of bottom over the bottom squared.

Clicker Question 3 What is the derivative of f (x) = x ex ? A. ex B. x 2 ex – 1 + ex C. ex (x + 1) D. x ex E. x ex – 1

Assignment On Monday we will have Lab #4 on 2nd derivatives and non-differentiable functions. For Wednesday, read Section 3.2 and do Exercises 1-15 odd, 27, 29, 31 and 45. Test #1 will be returned on Wednesday.