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

Mt020.02 Slide 1 on 4/5/00 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 on 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: raj

Mt020.02 Slide 2 on 4/5/00 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 of bacteria and chart the colony size as the weeks go by: start of 1 2 3 4 week colony 2 4 8 16 size A clever observer will note that this is a function with C(w) = colony size after w weeks given by C(w) = 2 w . This is an example of an exponential function where the base (2) is a raj

Mt020.02 Slide 3 on 4/5/00 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) = 4 w . Examples of exponential functions: y = 7 x , p = (.3) t , h = π s , 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) = 2 x , then f(3/2) = 2 (3/2) = 2 3 = 2 2, and f(.3) = f(3/10) = = 1.231 . 10 2 3 raj

Mt020.02 Slide 4 on 4/5/00 Graphing these exponential functions is educational (these have 0 < a < 1): and these have 1 < a < ∞ : Properties of Exponential Expressions: a x a y = a (x+y) (ab) x = a x b x ( a x / a y ) = a (x-y) ( a / b ) x = a x / b x ( a x ) y = a (xy) raj

Mt020.02 Slide 5 on 4/5/00 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) = 1   Simplify: x 2 n − 2 y 2 n 3   x 5 n + 1 y − n   raj

Mt020.02 Slide 6 on 4/5/00 Solve for x: 3 (3x-4) = 3 5 . Solve for x: 3 2x – (12)3 x + 27 = 0 x − 2   Solve for x: 8 x = 1     32 raj