Calculus is a branch of mathematics that studies and deals with change, as well as the rate and accumulation of said change. Calculus has applications throughout the scientific world, but it also plays a large role in business. One of the most important parts of operating a successful business is being able to react to and make decisions based on outside factors. This is where calculus comes in. Calculus can allow business owners and operators to predict markets and understand how changes that they may make will impact the market.

#### Marginal Cost and Revenue

Marginal cost and revenue are the compasses that help businesses decide how much production will optimize their profits. The cost to produce an additional unit of the product after *x* units have already been produced is called the marginal cost. The revenue earned from selling an additional unit of the product after *x* units have already been sold is called the marginal revenue. If we are given the functions for the total cost and revenue, we can find the marginal cost and revenue by taking the derivatives of the functions. These concepts are important, as they help a business decide how much they should be producing to maximize profits.

For example, let’s say we have a business that sells gizmos. The function for the cost to produce *x* gizmos per week is given by:

\( C(x) = 5000 + 55x – 0.06x^2 + 0.00003x^3 \)

The function for the revenue generated from selling *x* gizmos is:

\( R(x) = 65x – 0.005x^2 \)

Thus, our total profit is:

\( P(x) = R(x) – C(x) \), or

\( P(x) = -5000 + 10x + 0.055x^2 – 0.00003x^3 \)

Since the marginal functions are just the derivative of the main function, the marginal functions are:

\( C'(x) = 55 – 0.12x + 0.00006x^2 \),

\( R'(x) = 65 – 0.01x \), and

\( P'(x) = 10 + 0.11x – 0.00006x^2 \)

To maximize our profits, we will find the point where the marginal profit is equal to \( 0 \). If we sell more than that, the business will be losing money, and if the business sells less, then they are leaving money on the table. That point happens to be at \( x \approx 1920 \). Thus, the business should produce and sell 1920 gizmos per week.

#### Continuous Growth and Decay

Continuous growth and decay play an important role in business, especially in banking. One example of this is a savings account with compound interest. Every month, a bank will pay out interest based on the amount of money in the savings account. This interest is, in turn, reinvested right into the same account, increasing the amount of money that it holds and increasing the amount of interest the next time that it is paid.

We can model compound interest using the function:

\( P(t) = P(0) \cdot \left(1+\frac{r}{12}\right)^{12t} \)

where \( P(t) \) is the value of the account after \( t \) years, \( P(0) \) is the initial investment, and \( r \) is the annual interest rate. This differs from simple interest, where the money gained from interest is not reinvested into the account, meaning that the interest payment is fixed, and does not grow or shrink unless money is moved in or out of the account. We can use calculus to estimate the rate of change of the compound interest function at a certain time to approximate how quickly the value of the account will grow.

We can model simple interest using the function:

\( P(t) = P(0) \cdot (1+rt) \)

While the difference may not seem significant, over time there will be a large discrepancy in the amount of money between the two accounts. The graph pictured below shows how much money will be in each account assuming a starting investment of \( \$ \)1000 dollars and a $5\%$ interest rate. 40 years after the initial investment, the account with simple interest will have \( \$ \)3000 dollars, while the account with compound interest will have over \( \$ \)7400 dollars!

#### Elasticity of Demand

Elasticity of Demand is a concept in business that determines how the demand of a product responds to fluctuations in price. This is important for businesses to know as it tells them whether or not they should put a product on sale or if they should change their prices altogether. It can be calculated using:

\( E_d = \frac{p}{q} \cdot \frac{dq}{dp} \)

Where \( p \) is the percentage change in price and \( q \) is the percentage change in demand. If \( |E_d| < 1 \) at the current price \( p \), then we say that the demand is inelastic, and if \( |E_d| > 1 \) at price \( p \), then we say that the demand is elastic. To put it simply, if the demand is inelastic at the current price, then the rate of change of the demand is higher than the rate of change of the price, and if the demand is elastic, then the rate of change of the demand is lower than the rate of change of the price.

This is a very important statistic for businesses, as they can use the \( E_d \) of a product to determine how they should change their price to maximize profit. But what should businesses do with this number? If the demand is inelastic, then the business should increase the price of the product to increase profit. If the demand is elastic, then the business should decrease the price of the product to increase profit.

To give you an example, let’s say we still sell gizmos and we are one of 100 companies who sell them. Then increasing our prices will probably decimate profits because people will choose another brand meaning demand is inelastic. But what if we had a gizmo monopoly. Then increasing prices wouldn’t change demand because people need to buy gizmos from us no matter what, so in this case demand would be elastic.

#### Economic Order Quantity

Businesses almost always purchase products from other businesses to produce their own products. They need to find the optimal number of items to order from their supplier in order to minimize costs spent on ordering and holding fees. This is where Economic Order Quantity comes in. The formula for Economic Order Quantity was found by deriving the total cost formula:

\( T = PD + K \cdot \frac{D}{Q} + h \cdot \frac{Q}{2} \)

Where \( T \) is the total cost, \( Q \) is the optimal number of units per order, \( D \) is the demand per year, \( K \) is the purchase cost per order, and \( H \) is the holding cost per unit per year. Since we are trying to minimize total cost, we will derive this function with respect to \( Q \), set \( T = 0 \), and solve for \( Q \). We find that the formula for Economic Order Quantity is:

\( Q = \sqrt{\frac{2DK}{H}} \)

Since it costs money to store unsold items, that number needs to be factored into the equation for calculating economic order quantity.

EOQ is primarily used by businesses to help in cash flow models. Cash flow models show where money is going in and out, and optimizing the order sizes of their products will help a business control the amount of money that is going out. Usually, businesses like to use a system that will automatically use the EOQ function to determine the size of their order once the reserve supply drops below a certain point. While this may incur extra holding costs, this is a necessary step, as sudden increases in demand may cause the supply of some products to lessen.

### Conclusion

Calculus has many applications in the business world, from calculating the optimal order from a supplier to figuring out how the demand for a product will change in response to fluctuations in the price of the product. Business and marketing students need to know calculus, as it will be applicable to their jobs every day.