1. Clicker Question 1 At what point do the graphs of f (x ) = 2 x and g (x ) = 10 x cross? A. (1, 0) B. (0, 1) C. (0, 0) D. (2, 10) E. (10, 2)

2. Clicker Question 2 What is the value of P (t ) = 100(5 t ) when t = -2 ? A. 4 B. -2 C. -4 D. 2500 E. 100

3. About that number e …. (9/30/11) What is this mysterious number e anyway? Well, it is the natural limit of compounding, in the following sense: If you invest $1 for 1 year at 100% interest compounded annually, you make $1(1+1) 1 = $2.00 If you compound twice, you make $1(1 + ½) 2 = $2.25

4. The story of e, continued Compound quarterly: $1(1+1/4) 4 = $2.44 Compound monthly: $1(1+1/12) 12 = $2.61 Compound daily: $1(1+1/365) 365 = $2.71 Definition: e = limit (1+1/n) n as n gets big. So, compound continuously, make $e !!! We will see that e is “natural” in various ways as this course goes on.

5. New Functions from Old: Inverse Functions Inverse functions: If a function can be reversed (i.e., the output becomes the input and the input becomes the output), the reverse function is called the inverse of the original. For example: the inverse of f (x) = x 3 is the cube root function. A function is (directly) invertible only if its graph hits every horizontal line at most once (it is “one-to-one”).

6. Inverses - Examples f (x ) = x 2 is not directly invertible. (Why?) But if we restrict its domain so that it is one-to-one, then we can invert it on that domain. (To what?) In general to invert a given formula: Write in the form y = f (x ) Solve for x. Exchange x and y in your formula Examples….

7. Clicker Question 3 Find a formula (y as a function of x ) for the inverse of y = (2x +1) 3. A. y = (2x +1) 1/3 B. y = (2x +1) -3 C. y = (x 1/3 – 2) + 1 D. y = (x 1/3 – 1) / 2 E. y = 2x 1/3 - 1

8. Inverse Trig Functions The sin and cos functions cannot be directly inverted. (Why?) But restricting the domains appropriately (How? Look at the graphs!), we obtain the inverse, or arc, trig functions, arcsin, arccos, arctan, etc. These functions take in a number and put out an angle (in radians!). So, e.g., arcsin(x ) means “the angle whose sin is x.”

9. Clicker Question 4 What is the arcsin(-  2 / 2)? A. 45 ° B. -  / 4 C.  / 4 D. - 45 ° E. 0

10. Assignment for Monday Read Section 1.6 pages 58-62 and 67-69. In Section 1.6 do Exercises 1, 2, 3-13 odd, 21, 63-71 odd. (Suggestion on 69: draw a right triangle and label its sides so that one of the acute angles has sin equal to x. Using Pythagoras, what is the cos of that angle? Same technique on 71.)