A final optimization Example: Alice wants to get to the bus stop as - - PDF document

Mt020.02 Slide 1 on 4/5/00

raj

A final optimization Example: Alice wants to get to the bus stop as quickly as possible. The bus stop is across a grassy park, 2000 feet west and 600 feet north of her starting position. Alice can walk west along the edge of the park

• n the sidewalk at a speed of 6 feet per
• second. She can also travel through the grass

in the park, but only at a rate of 4 feet per second (the park is a favorite place to walk dogs so she must walk with care). What path will get her to the bus stop fastest? Solution:

Mt020.02 Slide 2 on 4/5/00

raj

We introduce a new family of functions today which will occupy our attention until the end of the semester: the exponential functions. Suppose that there were a kind of bacteria in which each cell split into two carbon copies of the parent just like clockwork each week. Assume that we start with two cells of this kind

• f bacteria and chart the colony size as the

weeks go by: start of week 1 2 3 4 colony size 2 4 8 16 A clever observer will note that this is a function with C(w) = colony size after w weeks given by C(w) = 2w. This is an example of an exponential function where the base (2) is a

Mt020.02 Slide 3 on 4/5/00

raj

constant (>0) and the exponent (x) is a variable. If each bacteria cell had split into 4 babies the function describing the colony size would be altered to become B(w) = 4w. Examples of exponential functions: y = 7x, p = (.3)t, h = πs, z = (3/2)v There is one function in this family of exponential functions for each base, a > 0. Notice that the exponent, x, need not be a whole number. Indeed if f(x) = 2x, then f(3/2) = 2(3/2) = 23 = 2 2, and f(.3) = f(3/10) = 23

10

= 1.231 .

Mt020.02 Slide 4 on 4/5/00

raj

Graphing these exponential functions is educational (these have 0 < a < 1): and these have 1 < a < ∞ : Properties of Exponential Expressions: ax ay = a(x+y) (ab)x = ax bx ( ax / ay ) = a(x-y) ( a / b )x = ax / bx ( ax )y = a(xy)

Mt020.02 Slide 5 on 4/5/00

raj

This is the kind of practice problems we can expect on Friday’s homework to practice our manipulation of exponents: Simplify: y(-3/2) y(5/3) = Simplify: x 2n−2y 2n x 5n+1y −n      

1 3

Mt020.02 Slide 6 on 4/5/00

raj

Solve for x: 3(3x-4) = 35. Solve for x: 32x – (12)3x + 27 = 0 Solve for x: 8x = 1 32      

x−2