Notes for exponential functions The week of March 6. Math 140

Transcription

1 Notes for exponential functions The week of March 6 Math 140

2 Exponential functions: formulas An exponential function has the formula f (t) = ab t, where b is a positive constant; a is the initial value of the function (f (0)). If b > 1, the function represents exponential growth and if 0 < b < 1, the function represents exponential decay. If b = 1, the function is a horizontal line. The constant b is called the growth factor, and b = 1 + r, where r is the percent rate of change. The larger the b > 1, or closer b < 1 is to 0, the steeper the function is. Exponential functions (when b > 1) grow much faster than linear functions. It is easy to confuse an exponential and power function, for example 3 x and x 3. The two will behave differently in many ways. Remember that an exponential function has a variable in the exponent (therefore exponential).

3 Exponential functions: descriptions In an environment with unlimited resources and no predators, a population tends to grow by the same percentage each year. Should a linear or exponential function be used to model such a population? Why?

4 Exponential functions: descriptions In an environment with unlimited resources and no predators, a population tends to grow by the same percentage each year. Should a linear or exponential function be used to model such a population? Why? Since the population grows by the same percentage every year, this is an exponential function. Let s say that the initial population is 2000, and the percent growth is 2% a year. The formula will be P = 2000( ) t = t.

5 Exponential functions: tables Decide whether the following functions could be linear, exponential or are neither. x f (x) x g(x) x h(x)

6 Exponential functions: tables Decide whether the following functions could be linear, exponential or are neither. x f (x) x g(x) x h(x) The second table has constant change, This is a linear function g(x) = x.

7 Exponential functions: tables Decide whether the following functions could be linear, exponential or are neither. x f (x) x g(x) x h(x) The second table has constant change, This is a linear function g(x) = x. For the first table, note that 150/125 = 180/150 = 216/180 = 1.2. This is an exponential function, f (x) = 125(1.2) x.

8 Exponential functions: tables Decide whether the following functions could be linear, exponential or are neither. x f (x) x g(x) x h(x) The second table has constant change, This is a linear function g(x) = x. For the first table, note that 150/125 = 180/150 = 216/180 = 1.2. This is an exponential function, f (x) = 125(1.2) x. For the third table, note that 512/1024 = 256/512 = 128/256 = 1/2. This is also an exponential function, and h(x) = 256(1/2) x.

9 Exponential functions: constant ratios While linear functions have a constant rate of change, exponential functions change at a constant percent rate. Given a table of a linear function, the differences of function values are constant (for equally spaced values of the independent variable). Given a table of an exponential function, the quotients of function values are constant (for equally spaced values of the independent variable).

10 Exponential functions: graphs The horizontal line t = 0 is called the horizontal asymptote, because the function Q approaches the value 0 when t is becomes very large, either in the positive or negative direction (or, mathematically, when x or x + ).

11 Finding the equation for an exponential function An exponential function can be uniquely determined by two points. Find the formula for the exponential function satisfying the conditions g(0) = 5 and g( 2) = 10.

12 Finding the equation for an exponential function An exponential function can be uniquely determined by two points. Find the formula for the exponential function satisfying the conditions g(0) = 5 and g( 2) = 10. Since g(x) = ab x, we have 5 = g(0) = ab 0 = a, so a = 5.

13 Finding the equation for an exponential function An exponential function can be uniquely determined by two points. Find the formula for the exponential function satisfying the conditions g(0) = 5 and g( 2) = 10. Since g(x) = ab x, we have 5 = g(0) = ab 0 = a, so a = = g( 2) = 5 b 2, so b 2 1 = 2, and b = 2 = (1/2)1/2, and g(x) = 5(1/2) x/2.

14 Finding the equation for an exponential function Among most important real-life examples of exponential functions are population growth and radioactive decay. The mass, Q, of a sample of Tritium (a radioactive isotope of Hydrogen), decays at a rate of 5.626% per year. Write an equation to describe the decay of a 726 gram sample. Graph the decay function.

15 Finding the equation for an exponential function Among most important real-life examples of exponential functions are population growth and radioactive decay. The mass, Q, of a sample of Tritium (a radioactive isotope of Hydrogen), decays at a rate of 5.626% per year. Write an equation to describe the decay of a 726 gram sample. Graph the decay function. Q(t) = 726( ) t = 726( ) t.

16 Interpreting exponential functions The populations, P, of six towns with time t in years are given by (i) P = 1000(1.8) t (ii) P = 600(1.12) t (iii) P = 2500(0.9) t (iv) P = 1200(1.185) t (v) P = 800(0.78) t (vi) P = 2000(0.99) t. 1. Which towns are growing in size? Which are shrinking? 2. Which town is growing the fastest? What is the annual percent growth rate for that town? 3. Which town is shrinking the fastest? What is the annual percent decay rate for that town? 4. Which town has the largest initial population? Which town has the smallest?

17 Recognizing exponential functions Match the stories in (a)-(e) with the formulas in (i)-(v). 1. The percent of a lake s surface covered by algae, initially 35%, was halved each year since the passage of anti-pollution laws. 2. The amount of charge on a capacitor in an electric circuit decreases by 30% every second. 3. Polluted water is passed through a series of filters. Each filter removes all but 30% of the remaining impurities from the water. 4. In 1920, the population of a town was 3000 people. Over the course of the next 50 years, the town grew at a rate of 10% per decade. 5. In 1920, the population of a town was 3000 people. Over the course of the next 50 years, the town grew at a rate of 250 people per year. (i) f (x) = P 0 + rx (ii) g(x) = P 0 (1 + r) x (iii) h(x) = B(0.7) x (iv) j(x) = B(0.3) x (v) k(x) = A(1/2) x.

18 The natural base The irrational number e is called the natural base. If an exponential function is written as Q = ae kt, then k is called the continuous growth rate. If k > 0, the function is increasing, and if k < 0, the function is decreasing. There are many advantages to using e over other bases, though they will mostly not become clear until calculus. If the growth rate in a problem is specified as continuous, use the previous formula. If not, use the usual formula for an exponential function.

19 Compounding If we invest an initial amount of money P 0 into the bank, then the amount we will have after time t is P(t) = P 0 (1 + r n )nt, where r is the annual interest rate as a decimal, and n is the number of times per year interest is compounded. If interest is compounded continuously, the formula is P(t) = P 0 e rt. What is the balance after 1 year if an account containing $500 earns the yearly nominal interest of 5%, compounded 1. annually 2. weekly 3. every minute (525,000 per year) 4. continuously?