Calculus has many real-world uses and applications in the physical sciences, computer science, economics, business, and medicine. I will briefly refer to some of these uses and applications in the real estate industry.

Let’s start by using some calculation examples in speculative real estate development (ie new home construction). Logically, a new home builder wants to make a profit after completing each home in a new home community. This builder will also need to be able to (hopefully) maintain a positive cash flow during the construction process of each house or each phase of the house development. There are many factors that go into calculating a profit. For example, we already know that the profit formula is: *P = R – C*that is, the gain (*P*) is equal to income (*R.*) minus the cost (*against*). Although this main formula is very simple, there are many variables that can influence this formula. For example, low cost (*against*), there are many different cost variables, such as the cost of construction materials, labor costs, real estate maintenance costs prior to purchase, utility costs, and premium costs. insurance during the construction phase. These are some of the many costs to factor into the formula mentioned above. Low income (*R.*), variables such as the home’s base sale price, additional improvements or additions to the home (security system, surround sound system, granite countertops, etc.) could be included. Just plugging all these different variables into itself can be a daunting task. However, this is further complicated if the rate of change is non-linear, which requires us to adjust our calculations because the rate of change of one or all of these variables is in the shape of a curve (i.e., exponential rate of change). ? This is an area where calculus comes into play.

Let’s say last month we sold 50 homes with an average sales price of $500,000. Without taking other factors into account, our revenues (*R.*) is the price ($500,000) per x (50 homes sold) which equals $25,000,000. Let’s assume that the total cost to build the 50 houses was $23,500,000; so the gain (*P*) is 25,000,000 – $23,500,000 which equals $1,500,000. Now, knowing these numbers, your boss has asked you to maximize profit for the next month. How do you do this? What price can you put?

As a simple example of this, let’s first calculate the marginal gain in terms of *X* to build a house in a new residential community. We know that the income (*R.*) is equal to the demand equation (*p*) times units sold (*X*). We write the equation as

*R = pixels*.

Suppose we have determined that the demand equation for selling a house in this community is

*p* = $1,000,000 – *X*/10.

With $1,000,000 you know you won’t sell any houses. Now, the cost equation (*against*) is

$300,000 + $18,000*X* ($175,000 in fixed costs of materials and $10,000 per house sold + $125,000 in fixed costs of labor and $8,000 per house).

From this we can calculate the marginal benefit in terms of *X* (units sold), then use the marginal profit to calculate the price we should charge to maximize profit. So the income is

*R.* = *pixels* = ($1,000,000 – *X*/10) * (*X*) = $1,000,000*X* – *x^2*/10.

Therefore, the profit is

*P* = *CR* = ($1,000,000*X* – *x^2*/10) – ($300,000 + $18,000*X*) = 982,000x – (*x^2*/10) – $300,000.

From this we can calculate the marginal benefit by taking the derivative of the benefit

*dp/dx* = 982,000 – (*X*/5)

To calculate the maximum profit, we set the marginal profit equal to zero and solve

982,000 – (*X*/5) = 0

*X* = 4910000.

we plug *X* Go back to the demand function and get the following:

*p* = $1,000,000 – (4910000)/10 = $509,000.

So the price we need to set to get the maximum profit for each house we sell should be $509,000. The following month, he sells 50 more houses under the new pricing structure, earning a net profit increase of $450,000 from the previous month. Great job!

Now, for the next month, your boss is asking you, the community developer, to find a way to reduce home construction costs. You already know that the cost equation (*against*) was:

$300,000 + $18,000*X* ($175,000 in fixed costs of materials and $10,000 per house sold + $125,000 in fixed costs of labor and $8,000 per house).

After astute negotiations with his construction vendors, he was able to reduce fixed material costs to $150,000 and $9,000 per house, and reduce labor costs to $110,000 and $7,000 per house. As a result, his cost equation (*against*) has changed to

*against* = $260,000 + $16,000*X*.

Due to these changes, you will need to recalculate the base benefit

*P* = *CR* = ($1,000,000*X* – *x^2*/10) – ($260,000 + $16,000*X*) = 984,000*X* – (*x^2*/10) – $260,000.

From this we can calculate the new marginal gain by taking the derivative of the new calculated gain

*dp/dx* = 984,000 – (*X*/5).

To calculate the maximum profit, we set the marginal profit equal to zero and solve

984,000 – (*X*/5) = 0

*X* = 4920000.

we plug *X* Go back to the demand function and get the following:

*p* = $1,000,000 – (4920000)/10 = $508,000.

So the price we need to set to get the new maximum profit for each house we sell should be $508,000. Now, even though we lowered the sales price from $509,000 to $508,000, and we still sell 50 units as in the previous two months, our profit has still increased because we cut costs to the tune of $140,000. We can find out by calculating the difference between the first *P = R – C* and the second *P = R – C* which contains the new cost equation.

1st *P* = *CR* = ($1,000,000*X* – *x^2*/10) – ($300,000 + $18,000*X*) = 982,000*X* – (*x^2*/10) – $300,000 = 48,799,750

2nd *P* = *CR* = ($1,000,000*X* – *x^2*/10) – ($260,000 + $16,000*X*) = 984,000*X* – (*x^2*/10) – $260,000 = 48,939,750

Taking the second profit minus the first profit, you can see a difference (increase) of $140,000 in profit. Thus, by reducing home construction costs, you can make the company even more profitable.

Let’s recap. By simply applying the demand function, marginal profit, and maximum profit from the calculation, and nothing else, he was able to help his company increase its monthly profit from the ABC Home Community project by hundreds of thousands of dollars. Through a little negotiation with his construction suppliers and labor leaders, he was able to reduce his costs and through a simple readjustment of the cost equation (*against*), you could quickly see that by cutting costs, you increased profit once again, even after adjusting your maximum profit by lowering the selling price by $1,000 per unit. This is an example of the wonder of calculus when applied to real world problems.