Relative Rates of Growth Section 8.3 Relative Rates of Growth

Section 8.3 Relative Rates of Growth Learning Target: I can compare rates of growth (grows faster/slower/at the same rate)

Comparing Rates of Growth Definition: Faster, Slower, Same-rate Growth as x Let f(x) and g(x) be positive for x sufficiently large. 1. f grows faster than g (and g grows slower than f) as x if , or equivalently, if 2. f and g grow at the same rate as x if

Example 1: (Grows Faster) Shows that the function ex grows faster than x2 as x . Therefore, ex grows faster than x2.

Exploration Show that ax, a>1, grows faster than x2 as x Show that 3x grows faster than 2x as x If a > b > 1, show that ax grows faster than bx as x Therefore, ax grows faster than x2. Therefore, 3x grows faster than 2x. Therefore, ax grows faster than bx. Because a/b > 1

Example 2: (Grows slower) Show that ln x grows slower than: x as x x2 as x Therefore, ln x grows slower than x. Therefore, ln x grows slower than x2.

Example 3: (Grows at the Same Rate) Let a and b be numbers greater than 1. Show that logax and logbx grow at the same rate as x . Therefore, they grow at the same rate.

Definition Transitivity of Growing Rates – If f grows at the same rate as g as x and g grows at the same rate as h as x , then f grows at the same rate as h as x **Note: Growing at the same rate is a transitive relation.

Example 4: Put the following functions in order from SLOWEST to FASTEST: Therefore, 2x grows faster than x2. Therefore, 2x grows faster than (ln2)x. Therefore, ex grows faster than (ln2)x.

Example 4, continued Therefore, (ln2)x grows faster than x2. Therefore, ex grows faster than 2x.

8.3 Homework #3, 6, 9, 12, 17, 24, 25, 27, 30, 46 - 51

Quiz Day  After you complete your quiz, please complete pages 12 – 14… Integral Refreshers. I will be checking off 8.3 and the Integral Refreshers WS tomorrow in class.