Applications of Exponential Functions

Transcription

1 Applications of Exponential Functions MATH 151 Calculus for Management J. Robert Buchanan Department of Mathematics Spring 2014

2 Objectives After this lesson we will be able to solve problems involving growth, decay, relative and absolute rate of change, and elastic demand.

3 Background It has been observed that in many fields (economics, biology, chemistry, etc.) that quantities change at a rate proportional to the quantity itself.

4 Background It has been observed that in many fields (economics, biology, chemistry, etc.) that quantities change at a rate proportional to the quantity itself. We can express this relationship with the equation dy dt = k y where y is the quantity, and k is a constant.

5 Exponential Growth Theorem (Law of Exponential Growth) If y is a positive quantity whose rate of change with respect to time is proportional (with proportionality constant k > 0) to the quantity present at time t, then dy dt = k y and y = c e kt where c is the initial value and k is the growth rate.

6 Illustration y = c e kt with c > 0 and k >

7 Example A termite colony grows exponentially at a rate of 4% per day. Initially there are 750 termites in the colony. 1 Find an exponential function for the number of termites in the colony after t days. 2 How many termites are in the colony after 7 days? 3 How many days will it take for the colony to double in size?

8 Example A termite colony grows exponentially at a rate of 4% per day. Initially there are 750 termites in the colony. 1 Find an exponential function for the number of termites in the colony after t days. y = 750e 0.04t 2 How many termites are in the colony after 7 days? 3 How many days will it take for the colony to double in size?

9 Example A termite colony grows exponentially at a rate of 4% per day. Initially there are 750 termites in the colony. 1 Find an exponential function for the number of termites in the colony after t days. y = 750e 0.04t 2 How many termites are in the colony after 7 days? y(7) = 750e 0.04(7) = How many days will it take for the colony to double in size?

10 Example A termite colony grows exponentially at a rate of 4% per day. Initially there are 750 termites in the colony. 1 Find an exponential function for the number of termites in the colony after t days. y = 750e 0.04t 2 How many termites are in the colony after 7 days? y(7) = 750e 0.04(7) = How many days will it take for the colony to double in size? t = ln 2 = days 0.04

11 Exponential Decay Theorem (Law of Exponential Decay) If y is a positive quantity whose rate of change with respect to time is proportional (with proportionality constant k < 0) to the quantity present at time t, then dy dt = k y and y = c e kt where c is the initial value and k is the decay rate.

12 Illustration y = ce kt with c > 0 and k >

13 Application: Radioactive Decay Radioactive decay is an example of exponential decay. Each radioactive element has a different decay constant k related to its half life, t 1/2.

14 Application: Radioactive Decay Radioactive decay is an example of exponential decay. Each radioactive element has a different decay constant k related to its half life, t 1/2. y = ce kt 1 2 c = ce kt 1/2 1 2 = e kt 1/2 ln 1 2 = ln e kt 1/2 ln 2 1 = kt 1/2 ln 2 = kt 1/2 ln 2 k = t 1/2

15 Example The element 137 Cs has a decay constant of k = yr 1. 1 What is the half-life of 137 Cs? 2 How much of 1000 grams of 137 Cs will remain after 10 years?

16 Example The element 137 Cs has a decay constant of k = yr 1. 1 What is the half-life of 137 Cs? t 1/2 = ln 2 = years How much of 1000 grams of 137 Cs will remain after 10 years?

17 Example The element 137 Cs has a decay constant of k = yr 1. 1 What is the half-life of 137 Cs? t 1/2 = ln 2 = years How much of 1000 grams of 137 Cs will remain after 10 years? y = 1000e 0.023(10) = grams

18 Limited Growth Most populations are limited in size by environmental factors such as availability of food, availability of shelter, predators, etc. The mathematical model for limited growth is y = c(1 e kt ) where c and k are positive constants.

19 Illustration y = c(1 e kt ) c 1 e k t

20 Example (1 of 3) The Department of Forestry has begun a reclamation project in an area where most of the trees were destroyed by fire. It is estimated that the number of trees in t years will be P = e 0.025t. What is the initial number of trees?

21 Example (1 of 3) The Department of Forestry has begun a reclamation project in an area where most of the trees were destroyed by fire. It is estimated that the number of trees in t years will be P = e 0.025t. What is the initial number of trees? P(0) = e 0.025(0) = = 125

22 Example (2 of 3) P = e 0.025t. What will be the population of trees after 15 years?

23 Example (2 of 3) P = e 0.025t. What will be the population of trees after 15 years? P(15) = e 0.025(15) =

24 Example (3 of 3) P = e 0.025t. When will the population of trees reach 500?

25 Example (3 of 3) P = e 0.025t. When will the population of trees reach 500? 500 = e 0.025t 150 = 525e 0.025t = e 0.025t ln = 0.025t t = ln = years

26 Absolute Rate of Change (1 of 2) Remarks: We know that the derivative of a function gives the functions instantaneous rate of change.

27 Absolute Rate of Change (1 of 2) Remarks: We know that the derivative of a function gives the functions instantaneous rate of change. Suppose that a pair of new shoes costs f (t) (where cost is measured in dollars and t measures time in years).

28 Absolute Rate of Change (1 of 2) Remarks: We know that the derivative of a function gives the functions instantaneous rate of change. Suppose that a pair of new shoes costs f (t) (where cost is measured in dollars and t measures time in years). If f (0) = $200 and f (0) = $50/year then estimate the absolute difference in price between the same shoes now and in one year.

29 Absolute Rate of Change (2 of 2) Suppose that a new car costs g(t) (where cost is measured in dollars and t measures time in years).

30 Absolute Rate of Change (2 of 2) Suppose that a new car costs g(t) (where cost is measured in dollars and t measures time in years). If g(0) = $25, 000 and f (0) = $50/year then estimate the absolute difference in price between the same car now and in one year.

31 Absolute Rate of Change (2 of 2) Suppose that a new car costs g(t) (where cost is measured in dollars and t measures time in years). If g(0) = $25, 000 and f (0) = $50/year then estimate the absolute difference in price between the same car now and in one year. For the example of the shoes and the car, the absolute rate of change was the same.

32 Absolute Rate of Change (2 of 2) Suppose that a new car costs g(t) (where cost is measured in dollars and t measures time in years). If g(0) = $25, 000 and f (0) = $50/year then estimate the absolute difference in price between the same car now and in one year. For the example of the shoes and the car, the absolute rate of change was the same. A more meaningful comparison may be between the changes as fractions of the item being purchased.

33 Relative Rate of Change Definition If y = f (t) then the rate of change is f (t) and the relative rate of change is f (t) f (t) = d [ln f (t)] dt provided f (t) > 0.

34 Example Suppose the gross domestic product t years from now is predicted to be G(t) = 8.2e t trillion dollars. Find the relative change in GDP 25 years from now.

35 Example Suppose the gross domestic product t years from now is predicted to be G(t) = 8.2e t trillion dollars. Find the relative change in GDP 25 years from now. d [ ] ln 8.2e t dt = d [ ln ln e dt = d [ ln ] t dt t ] = 1 2 t 1/2 = 1 2 (25) 1/2 = 0.10 (when t = 25) Economists would call this a 10% change in GDP.

36 Price Elasticity of Demand (1 of 2) When the price of a product changes, consumers often change the amount of product they buy.

37 Price Elasticity of Demand (1 of 2) When the price of a product changes, consumers often change the amount of product they buy. If a decrease in price causes a greater demand, then demand for the product is said to be elastic (example: beer).

38 Price Elasticity of Demand (1 of 2) When the price of a product changes, consumers often change the amount of product they buy. If a decrease in price causes a greater demand, then demand for the product is said to be elastic (example: beer). If a decrease in price does not cause a greater demand, then demand for the product is said to be inelastic (example: postage stamps).

39 Price Elasticity of Demand (2 of 2) Definition If the demand function p = D(x) is differentiable, then the price elasticity of demand is E = 1 x D(x) D (x). For a given price, the demand is elastic if E > 1, the demand is inelastic if E < 1, and the demand has unit elasticity if E = 1.

40 Graphs of Revenue and Elasticity x x 12

41 Elasticity and Marginal Revenue R(x) = x p = x D(x) R (x) = D(x) + xd (x) = D(x) D(x) ( E R (x) = D(x) 1 1 ) E If E > 1 then R (x) > 0 (elastic demand, increasing revenue). If 0 < E < 1 then R (x) < 0 (inelastic demand, decreasing revenue). If E = 1 then R (x) = 0 (unit elasticity, maximum revenue).

42 Example The demand for wool blankets is given by p = 33 2 x dollars, where x is in thousands of units. 1 What is the price per unit if 25 blankets are sold? 2 Find the elasticity of demand. 3 Find the level of sales that maximizes the revenue.

43 Solution 1 p = = 23

44 Solution 1 p = = 23 2 E = 1 D(x) x D (x) = 33 2 x

45 Solution 1 p = = 23 2 E = 1 D(x) x D (x) = 33 2 x 3 Revenue is maximized when E = 1

46 Solution 1 p = = 23 2 E = 1 D(x) x D (x) = 33 2 x 3 Revenue is maximized when E = 1 33 x 2 = 1 33 x = 3 x = 11 x = 121