1. Clicker Question 1 What is the derivative function f '(x ) of the function ? (Hint: Algebra first, calculus second!) A. 12x 2 – (5/2)x -1/2 B. 12x 2 – 10x -1/2 C. 8x – (5/2)x -1/2 D. 8x – (5/2)x -3/2 E. 8x + (5/2)x -3/2

2. Derivatives: Exponential Functions & Units (11/18/11) The units of any derivative function are output units per input unit. This makes sense since it is measuring rate of change (on a graph, slope ). For example, if a function f (t ) is measuring position of an object (in feet) at a given time t (in seconds), the units of the derivative function are what?

3. Clicker Question 2 What is the instantaneous rate of change of f (x ) = x 5 at the point x = 2? A. 32 output units per input unit B. 80 output units per input unit C. 10 output units per input unit D. 40 output units per input unit E. 60 output units per input unit

4. Derivatives of 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.

5. The Derivative of the General Exponential Function If f (x ) = a x is the general exponential function, then f '(x ) = a x (ln(a )). In particular, 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. Its value always equals its rate of change (i.e., its height equals its slope)!

6. Clicker Question 3 What (to one decimal place) is the slope of tangent line to the curve f (x ) = 2 x at the point (3, 8)? A. 12.0 B. 5.5 C. 8.0 D. 10.2 E. 3.7

7. Assignment for Monday Read the rest of Section 3.1 and finish any exercises assigned there last time. Find the equation of the tangent line to the curve y = x 3 + 3 x at the point (2, 17). Test corrections will be due at 4:45 on Tuesday. Office hours will start a 2 instead of 3:15. Remember that corrections are done on your own (or with help from me) and should be done on separate sheets.