Linear Functions and Mathematical Modeling Section 1.3 Linear Functions and Mathematical Modeling

Mathematical Modeling Formulating real world problems into the language of mathematics. Ex. The monthly payment, M, necessary to repay a home loan of P dollars, at a rate of r % per year (compounded monthly), for t years, can be found using

Function A rule that assigns to each value of x one and only one value of y. is a function. Ex. We write f (x) , read “f of x”, in place of y to show the dependency of y on x . So and NOTE: It is not f times x

Linear Function A linear function can be expressed in the form m and b are constants Can be used for Simple Depreciation Linear Supply and Demand Functions Linear Cost, Revenue, and Profit Functions

Simple Depreciation Ex. A computer with original value $2000 is linearly depreciated to a value of $200 after 4 years. Find an equation for the value, V, of the computer at the end of year t.

Cost, Revenue, and Profit Functions Ex. A shirt producer has a fixed monthly cost of $3600. If each shirt has a cost of $3 and sells for $12 find: a. The cost function Cost: C(x) = 3x + 3600 where x is the number of shirts produced. b. The revenue function Revenue: R(x) = 12x where x is the number of shirts sold. c. The profit from 900 shirts Profit: P(x) = Revenue – Cost = 12x – (3x + 3600) = 9x – 3600 P(900) = 9(900) – 3600 = $4500

Linear Demand Ex. The quantity demanded of a particular game is 5000 games when the unit price is $6. At $10 per unit the quantity demanded drops to 3400 games. Find a demand equation relating the price p, and the quantity demanded, x (in units of 100).