Clicker Question 1 The function f (x ) is graphed on the board. If the derivative function f '(x ) were graphed, where would it intersect the x – axis? A. -3, -1, and +1 B. -2 only C. -2 and 0 D. 0 and 2 E. It will not intersect the x - axis

Derivatives of Power & Exponential Functions. Sums & Constant Multipliers. (2/18/09) First, power functions: What is the derivative of f (x ) = x 2 ? What about f (x ) = x 4 ? What about f (x ) = x -1 ? What about f (x ) = x 1/2 ? Do we see a pattern?? (Mathematicians look for patterns, then try to prove that those patterns always hold.) A proved result is called a theorem.

The Power Rule Theorem. If f (x ) = x r is any power function (i.e., r is any fixed real number), then f '(x ) = r x r – 1. We can prove this theorem pretty easily in the case that r is a positive whole number, using Pascal’s Triangle. We can extend the proof further (to negative r, fractional r, etc.) as we go on.

Clicker Question 2 According to the Power Rule, what is the derivative of f (x ) = x -2/3 ? A. -2/3 x 1/3 B. 2/3 x -4/3 C. -2/3 x -5/3 D. x -2/3 E. -3/2 x -5/3

Experimenting with Exponential Functions WARNING: The derivative of a x is NOT x a x-1 !! a x is not a power function, so don’t use the Power Rule! Let’s calculate the derivative of the exponential function f (x ) = a x (where a is a fixed positive real number) directly from the definition. We are left wondering what is. This is the rate of change (i.e., slope) of a x at x = 0. Look familiar??

The Derivative of the General Exponential Function Theorem. If f (x ) = a x is the general exponential function, then f '(x ) = a x (ln(a )). Corollary. If f (x ) = e x, then f '(x ) = e x. This is one reason why f (x ) = e x is called the natural exponential function.

Clicker Question 3 What is the slope of tangent line to the curve f (x ) = e x at the point (3, e 3 )? A. 3 B. e 3 C. e x D. 3e 2 E. ln(3)

Sums, Differences, and Constant Multipliers. Theorem. The derivative of the sum (or difference) of two functions is just the sum (or difference) of their derivatives. That is, “derivatives can be found by working term-by-term.” Theorem. The derivative of a constant multiplier times a function is just the constant multiplier times the derivative. That is, “when finding derivatives, constant multipliers just get carried along.”

For Friday… and Beyond Thursday’s Office Hour 4-5 (Feb 19 only). For Friday, read Section 3.1 and do Exercises 3-33 odd and 51. Also check over Homework #1 and have any questions ready. Monday’s class will be an optional Q & A session in Harder 203. Wednesday Feb 25 is Test #1.