Unit
Applications of the Derivative
Cornell University
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
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Cornell University
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
As we will see later, derivatives can have economic applications. But first, we must learn the terminology used when discussing basic economic principles. For example, the total cost c(x) is fixed cost plus variable cost. Fixed cost is given, variable cost depends on the number of items sold. Revenue, r(x), is the price multiplied by the number of items made. Profit is the revenue minus cost, or p(x) = r(x) - c(x). The break even point is the number of items that must be produced in order to make a profit, or when r(x) = c(x).
Let's explore the ideas of cost, revenue, and profit in a problem. Your company manufactures magic brooms. Your fixed costs are 300,000 Galleons per annum for rent; utilities; elf salary, clothing, and benefits; dividends for your goblin shareholders. Each broom costs 55 Galleons to make, and is sold for 175 Galleons.
So we have a bunch of parts here. A; find the total cost c(x) as a function of the number x of brooms produced. Let's do that first.
So total cost c(x) is fixed cost plus variable cost. Let me write that down. So this is part a. C(x), fixed cost, plus variable cost. Now we're told that the fixed cost is 300,000 Galleons. The variable cost is going to depend on the number of brooms sold, but it's 55 Galleons for each broom, times the number of brooms sold. So this is my function; 300,000 plus 55x.
B. Let's take a look. Find the total revenue function r(x). Revenue, r(x), is price times the number manufactured. So price times x. Now the price of the brooms is 175. 175 Galleons times x, that's your revenue function. This is the amount of money coming into your company.
Part c. Find the total profit function p(x). Profit is revenue minus cost. So p(x) is r(x) minus c(x). So that's going to be the revenue function 175x minus this function. So minus 300,000 minus 55x. So this can be simplified a little bit. 175x minus 55x is 120x minus 300,000. That's your profit function.
Part d; how many brooms must be produced for your company to make a profit? We're looking for the break-even point. The break-even point is where revenue equals cost or where profit is 0. So set the profit equal to 0 to find the break-even point 120x minus 300,000 equals 0. So 120x equals 300,000. Divide both sided by 120, and you get x equals 300,000 over 120. That's 2500. 2500 brooms have to be manufactured in order to break even. That means a profit of 0. So to make a profit, you have to manufacture more than 2500. More than 2500 brooms.
So cost, revenue, profit, and the break-even point. The break even is the point where the profit equals 0. Pass that, and you'll earn a profit.