Math232 section 1.2 Linear functions and its applications

Math232 section 1.2 Linear functions and its applications Zheng Chen@suno zchen@suno

Def. of functions • A function involves two variables say, x and y, when x changes, y will change with x and satisfy the following conditions: for a any given x value, there is only one y value corresponding to this value. X is called as independent variable, y dependent variable. denoted as y=f(x), f is the name for this function, or the rule (about how to decide y value from a given x value). Read f(2) as f of 2, which means the function value when x =2. Ex 1, f(x)=3.3 x is a function, f(2)=6.6 Ex 2 . g(x)=1.5 for any real values x.

Cont.of functions • Linear function: a function with the form y=ax +b, where a and b are constants. y=-2x+3 a linear function;

Applications of linear functions • Now we know for a linear function, linear means the graph is a line. • Ex 1 A practical problem: someone makes copies of a music cd to earn money. To start this business, he needs to buy original cd, license, cd burner, after making 1st copy, he can make more. essentially, he will spend same money to make 2nd one and third one. the cost function is like the following: C=500+0.8x, when x is the number of copies, $500 fixed fee before production

Def. of a cost function • gerenrally, Cost Function C(x) = a x + b (cost is expenditure) x = # items produced, C = C(x) = cost of x items, a = unit cost, b = fixed cost.

Another related function • Revenue function: R(x) = k x (Revenue is income) where x = # of items sold, R = R(x) = Revenue, k = unit sell price • Ex 2 selling the copies, R(x)=5.8x price of one copy=$5.8, x number of copies

What does the difference mean? • The difference means the profit from production and selling Profit Function P(x) = R(x)- C(x) Break Even Analysis: Break Even when P(x) = 0 (same as R(x) = C(x)) Break Even means no loss and no profit

Break even • Ex 3, C(x)=500+0.8 R(x)=5.8x • Break even when 500+0.8x=5.8x Solving this equation, we have x=100

Geometric meaning • The above process is same as to • solve a system • Now the solution is the intersection point of two lines with equation: and the intersection point is called equilibrium pt.

Hw ex1) Let C(x) = 220 x + 8000 = manufacter's cost to produce x bicycles. (a) Fixed cost = C(0) (b) Unit cost = cost for 1 additional item = C(1) - C(0)  (c) Find the cost to produce 40 bicycles. (d) How many bicycles was produced if the cost was $30,000 ex2) Let C(x) = 35 x + 4400 = cost to produce x copies of a certain book. Find: (a) fixed cost and unit cost (b) cost to produce 1000 books (c) How many books are produced if the total cost was $14900 zchen@suno

hw • Ex3, Let R(x) = 420x where x = # bicycles sold, R = Revenue • Ex4, Let R(x) = 85x where x = # of books sol

hw • Ex5, Let C(x) = 220 x + 8000 = manufacter's cost to produce x bicycles. • Let R(x) = 420x where x = # bicycles sold, R = Revenue • (a) Find profit function. • (b) Find break even point. • (c) Profit is $3000. Find x. • (d) Profit is greater than $5000. • (e) Loss is $3000. • (f) Loss is greater than $2000

hw • Ex6, Let C(x) = 35 x + 4400 = cost to produce x copies of a certain book. • Let R(x) = 85x where x = # of books sold • (a) Find profit function (b) Find break even point.

Depreciation (or Appreciation) • Straight Line Depreciation (or Appreciation) B(t) = m t + n B(t) = (book) value of an item after t years since new. n = value of new item, m = annual appreciation value (depreciation when m < 0 and |m| =depreciation amount) Ex7, Let B(t) = -3000 t + 24000 be the book value of a certain car

Math232 section 1.2 Linear functions and its applications