Exponential Functions and Models Lesson 5.3

Exponential Functions Contrast Linear Functions Change at a constant rate Rate of change (slope) is a constant Exponential Functions Change at a changing rate Change at a constant percent rate

Definition An exponential function Note the variable is in the exponent The base is a C is the coefficient, also considered the initial value (when x = 0)

Explore Exponentials Given f(1) = 3, for each unit increase in x, the output is multiplied by 1.5 Determine the exponential function x f(x) 1 0.75 2 3 4

Explore Exponentials Graph these exponentials What do you think the coefficient C and the base a do to the appearance of the graphs?

Contrast Linear vs. Exponential Suppose you have a choice of two different jobs at graduation Start at $30,000 with a 6% per year increase Start at $40,000 with $1200 per year raise Which should you choose? One is linear growth One is exponential growth

Which Job? How do we get each next value for Option A? Year Option A Option B 1 $30,000 $40,000 2 $31,800 $41,200 3 $33,708 $42,400 4 $35,730 $43,600 5 $37,874 $44,800 6 $40,147 $46,000 7 $42,556 $47,200 8 $45,109 $48,400 9 $47,815 $49,600 10 $50,684 $50,800 11 $53,725 $52,000 12 $56,949 $53,200 13 $60,366 $54,400 14 $63,988 $55,600 How do we get each next value for Option A? When is Option A better? When is Option B better? Rate of increase a constant $1200 Rate of increase changing Percent of increase is a constant Ratio of successive years is 1.06

Amount of interest earned Example Consider a savings account with compounded yearly income You have $100 in the account You receive 5% annual interest At end of year Amount of interest earned New balance in account 1 100 * 0.05 = $5.00 $105.00 2 105 * 0.05 = $5.25 $110.25 3 110.25 * 0.05 = $5.51 $115.76 4   5 View completed table

Compounded Interest Completed table

Table of results from calculator Compounded Interest Table of results from calculator Set Y= screen y1(x)=100*1.05^x Choose Table (♦ Y) Graph of results

Assignment A Lesson 5.3A Page 415 Exercises 1 – 57 EOO

Compound Interest Consider an amount A0 of money deposited in an account Pays annual rate of interest r percent Compounded m times per year Stays in the account n years Then the resulting balance An

Exponential Modeling Population growth often modeled by exponential function Half life of radioactive materials modeled by exponential function

Growth Factor Recall formula new balance = old balance + 0.05 * old balance Another way of writing the formula new balance = 1.05 * old balance Why equivalent? Growth factor: 1 + interest rate as a fraction

Decreasing Exponentials Consider a medication Patient takes 100 mg Once it is taken, body filters medication out over period of time Suppose it removes 15% of what is present in the blood stream every hour At end of hour Amount remaining 1 100 – 0.15 * 100 = 85 2 85 – 0.15 * 85 = 72.25 3 4 5 Fill in the rest of the table What is the growth factor?

Decreasing Exponentials Completed chart Graph Growth Factor = 0.85 Note: when growth factor < 1, exponential is a decreasing function

Solving Exponential Equations Graphically For our medication example when does the amount of medication amount to less than 5 mg Graph the function for 0 < t < 25 Use the graph to determine when

General Formula All exponential functions have the general format: Where A = initial value B = growth rate t = number of time periods

Typical Exponential Graphs When B > 1 When B < 1

Using e As the Base We have used y = A * Bt Consider letting B = ek Then by substitution y = A * (ek)t Recall B = (1 + r) (the growth factor) It turns out that

Continuous Growth The constant k is called the continuous percent growth rate For Q = a bt k can be found by solving ek = b Then Q = a ek*t For positive a if k > 0 then Q is an increasing function if k < 0 then Q is a decreasing function

Continuous Growth For Q = a ek*t Assume a > 0 k > 0 k < 0

Continuous Growth For the function what is the continuous growth rate? The growth rate is the coefficient of t Growth rate = 0.4 or 40% Graph the function (predict what it looks like)

Converting Between Forms Change to the form Q = A*Bt We know B = ek Change to the form Q = A*ek*t We know k = ln B (Why?)

Continuous Growth Rates May be a better mathematical model for some situations Bacteria growth Decrease of medicine in the bloodstream Population growth of a large group

Example A population grows from its initial level of 22,000 people and grows at a continuous growth rate of 7.1% per year. What is the formula P(t), the population in year t? P(t) = 22000*e.071t By what percent does the population increase each year (What is the yearly growth rate)? Use b = ek

Assignment B Lesson 5.3B Page 417 Exercises 65 – 85 odd