Example 2: The demand function for a popular make of 12-speed bicycle is given by

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1 Sometimes, the unit price will not be given. Instead, product will be sold at market price, and you ll be given both supply and demand equations. In this case, we can find the equilibrium point (Section 4.) which will give us the equilibrium quantity and price. Example : The demand function for a popular make of 1-speed bicycle is given by ( ) where p is the unit price in dollars and x is the quantity demanded in p D x x units of a thousand. The supply function for the same product is given by ( ) where p is the unit price in dollars and x is the quantity p S x x x supplied in units of a thousand. Begin by entering the functions into GGB. a. Assume the market price is set at the equilibrium price. Find the equilibrium point. b. Determine the consumers surplus. Recall: Q E CS D( x) dx Q P First apply the formula. 0 E E Lesson 1 Other Applications of Integration 3

2 c. Determine the producers surplus. Recall: Q E PS QE PE S( x) dx 0 First apply the formula. Probability The study of probability deals with the likelihood of a certain outcome of an experiment. When you flip a coin, the probability that the coin lands with heads facing up is 1. When you roll a six-sided die and record the number that lands on the uppermost face, the probability that the die lands with the number 3 facing upwards is 1 6. You can also find the probability of something occurring over a continuous interval. Suppose you want to know how long a brand of light bulb lasts. Now the possible set of answers contains more than a discrete set of numbers (e.g., something other than 1,, 3, etc.). The light bulbs can last any positive length of time. We can find the probability that the light bulbs life span is on a given interval [a, b]. This problem is an example of a problem that involves a probability density function. This type of function can be used to determine the probability that an event occurs on a given interval [a, b]. All probability density functions must meet these three criteria: 1. f( x) 0 for all x. The area under the graph of f ( x ) is exactly The probability that an event occurs in an event [a, b] can be computed using the definite b integral f ( xdx ). a Lesson 1 Other Applications of Integration 4

3 0.00x Example 3: The function f( x) 0.00e gives the life span of a popular brand of light bulb, where x gives the lifespan in hours and f ( x ) is the probability density function. Find the probability that the lifespan is between 500 hours and 1000 hours. a. Set up the integral needed to answer the question. b. Enter the function into GGB and calculate the answer using GGB. Example 4: A company finds that the percent of its locations that experience a profit in the first 36 1 year of business has the probability density function Px ( ) x 1 x, 0 x I. What is the probability that more than 50% of the company s locations experienced a profit during the first year of business? a. Set up the integral needed to answer the question. b. Enter the function into GGB and calculate the answer using GGB. II. What is the probability that between 30% and 60% of the company s locations experienced a profit during the first year of business? a. Set up the integral needed to answer the question. b. Enter the function into GGB and calculate the answer using GGB. Lesson 1 Other Applications of Integration 5

4 Lesson Functions of Several Variables So far, we have looked at functions of a single variable. In this section, we will consider functions of more than one variable. You are already familiar with some examples of these. P( x, y) x y A ( P, i, t) P(1 i) These formulas are functions of several variables. We have just never called them that before. We will, for the most part, limit our discussion to functions of two variables. Functions of Two Variables Definition: A real valued function of two variables, f, consists of a set A of ordered pairs of real numbers (x, y) called the domain of the function, and a rule that associates with each ordered pair in the domain of f one and only one real number, denoted by z f ( x, y). You will need to learn two skills using functions of several variables: Evaluating at a given point and determining the domain. Example 1: Suppose f ( xy, ) 3xy 6 ln( xy) Compute f ( 1, 3). Enter the function as shown in GGB. t Example : The volume of a cylindrical tank with radius r and height h is given by the formula f (, rh) rh. Find the volume of a tank with radius 6 feet and height 0 feet. Enter the function as shown in GGB. Lesson Functions of Several Variables 1

5 Example 3: The monthly payment that amortizes a loan of A dollars in t years when the interest rate is r per year is given by Find the monthly payment for a mortgage of $50,000 that will be amortized over 5 years with an interest rate of 4.5% per year. Enter the function as shown in GGB. Example 4: Use the following table to answer the following question. Find P f( A, r, t) f(150,5.5,0), where P is the monthly payment, A is the amount financed in thousands of dollars, r is the interest rate, and t is the length of the loan in years. 3x Example 5: Find the domain of the function: f ( x, y). x 5y Lesson Functions of Several Variables

6 Example 6: Find the domain of the function: f ( xy, ) 16 x y. Example 7: Find the domain of the function: f ( x, y) x 3y Lesson Functions of Several Variables 3