Growth and Decay: Integral Exponents Section 5-1 Growth and Decay: Integral Exponents Objective: To define and apply integral exponents

Introduction Exponential Growth and Exponential Decay These are functions where we have variables for exponents Ex) Ex) Let’s look at an EXAMPLE: Suppose that the cost of a hamburger has been increasing at the rate of 9% per year. That means each year the cost is 1.09 times the cost of the previous year. Or Current Cost = Cost of Previous Year + 0.09(Cost of Previous Year) Suppose that a hamburger currently costs $4. Let’s look at some projected future costs by using a table Initial Cost Time (years from now) Cost (dollars) 1 2 3 t 4 4(1.09) 4(1.09)2 4(1.09)3 4(1.09)t X 1.09 X 1.09 X 1.09 X 1.09

Exponential Growth and Exponential Decay Initial Cost Cost at time t Time (years from now) Cost (dollars) 1 2 3 t 4 4(1.09) 4(1.09)2 4(1.09)3 4(1.09)t X 1.09 X 1.09 X 1.09 X 1.09 This table suggests that cost is a function of time t. We can not only predict cost in the future, but we could predict costs in the past. When t > 0, the function gives future costs When t < 0, the function gives past costs a) 5 years from now b) 5 years ago The cost will be about $6.15 The cost was about $2.60

Exponential Growth and Exponential Decay The previous example (cost of hamburgers) was an example of exponential growth. Let’s look at an example of Exponential Decay: Say Arielle bought her graphing calculator for $70, and its value depreciates (decreases) by 9% each year. How much will the calculator be worth after t years? Current Cost = Cost of Previous Year - 0.09(Cost of Previous Year) OR Each year we multiply the previous year’s cost by 0.91 Current Cost = 0.91(Cost of Previous Year) Let’s briefly compare the two situations: Hamburger cost: $4 now Calculator Cost: $70 now Cost increasing at 9% each year Cost decreasing at 9% each year

Exponential Growth and Exponential Decay Hamburger cost: $4 now Calculator Cost: $70 now Cost increasing at 9% each year Cost decreasing at 9% each year Let’s look at the graphs of these functions Exponential Growth Exponential Decay Growth and decay can be modeled by: If r > 0 (1+r > 1)  Exponential Growth If 0 > r > -1 (0 > 1+r > 1)  Exp. Decay

Exponential Growth and Exponential Decay R is between zero and negative one (0 > -0.15 > -1 So Exponential Decay Let’s use this formula in an example: Suppose that a radioactive isotope decays so that the radioactivity present decreases by 15% per day. If 40 kg are present now, find the amount present: a) 6 days from now b) 6 days ago There will be about 15.1 kg left after 6 days There was about 106.1 kg 6 days ago.

Review of Laws of Exponents

Apply the of Laws of Exponents Example: Distribute the exponents Solution Distribute the exponents Subtract the exponents

Apply the of Laws of Exponents Example: Get rid of the negative exponents Solution Get a common base Get rid of the negative exponents

Where do we see Exponential Growth? Wikipedia

Homework P173-174: 1-21(odd) Day 2: 23-45 (odd)