Exponential Functions and Models Lesson 3.1

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

Contrast 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? 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 YearOption AOption 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

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 1100 * 0.05 = $5.00$ * 0.05 = $5.25$ * 0.05 = $5.51$ View completed table

Compounded Interest Completed table

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

Compound Interest Consider an amount A 0 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 A n

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 * 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 hourAmount remaining 1100 – 0.15 * 100 = – 0.15 * 85 = 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 * B t Consider letting B = e k Then by substitution y = A * (e k ) 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 b t k can be found by solving e k = b Then Q = a e k*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 e k*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*B t We know B = e k Change to the form Q = A*e k*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 = e k