Problem 41 - Single Variable Calculus

Question

Forecasting Commodity Crops Government economists in a certain country have determined that the demand equation for soybeans is given by $$ p=f(x)=\frac{55}{2 x^{2}+1} $$ where the unit price \(p\) is expressed in dollars per bushel and \(x\), the quantity demanded per year, is measured in billions of bushels. The economists are forecasting a harvest of \(2.2\) billion bushels for the year, with a possible error of \(10 \%\) in their forecast. Determine the corresponding error in the predicted price per bushel of soybeans.

Step-by-Step Solution

Answer

First, we find the predicted price per bushel of soybeans using the demand equation when the quantity demanded is 2.2 billion bushels. Next, we compute the differential of the demand function and calculate the approximate error in the price using the 10% error in the forecasted quantity demanded. Finally, we find the corresponding error in the predicted price per bushel. Thus, the corresponding error in the predicted price per bushel given the forecasted harvest of 2.2 billion bushels and a 10% error in the forecast is determined by computing the value of \(Predicted\_Price\_WithError\).

Step 1: Calculate the predicted price with a harvest of 2.2 billion bushels

First, we need to calculate the predicted price per bushel of soybeans using the given demand equation, when the quantity demanded is 2.2 billion bushels. \(f(x) = \frac{55}{2x^2 + 1}\) Plug in \(x = 2.2\): \(f(2.2) = \frac{55}{2*(2.2)^2 + 1}\) Calculate the price per bushel: \(p=f(2.2)\)

Step 2: Compute the differential of the demand function

Since we are interested in how a change in the quantity demanded affects the price per bushel, we need to compute the differential of the demand function. The differential represents the rate of change of the price as a function of the quantity demanded (x). Find the derivative of \(f(x)\) with respect to \(x\): \(f'(x) = \frac{d}{dx} \left(\frac{55}{2x^2 + 1}\right)\) Use the quotient rule to find the derivative: \(f'(x) = \frac{-220x}{(2x^2 + 1)^2}\)

Step 3: Calculate the approximate error in price

We've been given a 10% error in the quantity demanded forecast, which corresponds to an error of 0.22 billion bushels (10% of 2.2). To find the approximate error in the price per bushel, we will multiply this error by the derivative of the demand function (evaluated at the forecast quantity demanded), which is the rate of change of the price with respect to the quantity demanded. First, find the value of \(f'(2.2)\): \(f'(2.2) = \frac{-220(2.2)}{(2(2.2)^2 + 1)^2}\) Now, multiply the error in the forecasted quantity demanded (0.22 billion bushels) by \(f'(2.2)\): \(Error\_in\_Price = 0.22 * f'(2.2)\)

Step 4: Find the corresponding error in the predicted price per bushel

Now that we have calculated the approximate error in the price per bushel, we can find the corresponding error in the predicted price per bushel. Add the error in the price to the predicted price per bushel: \(Predicted\_Price\_WithError = p + Error\_in\_Price\) Compute the actual value to find the corresponding error in the predicted price per bushel: \(Predicted\_Price\_WithError\) Now we have determined the corresponding error in the predicted price per bushel of soybeans given the forecasted harvest of 2.2 billion bushels and a possible error of 10% in the forecast. This information can be used to better estimate the potential price fluctuations in the soybean market.

Key Concepts

Differential Calculus

At the heart of predicting real-world quantities like prices is differential calculus, a branch of mathematics that deals with continuous change. It allows economists to estimate how small changes in one quantity (such as the quantity demanded of a product) cause changes in another (like the price). The process to find the rate at which one variable changes with respect to another is called differentiation, and the result is known as the derivative.

In the context of our soybean market example, the derivative of the demand equation tells us how much we can expect the price to change given a small change in quantity demanded. By differentiating the demand function, we can find the sensitivity of the price to changes in demand, and this is critical for making accurate economic forecasts and adjustments. By computing the derivative of the demand equation, the economists are using differential calculus to predict the effects of fluctuating market conditions.

Price Elasticity of Demand

Price elasticity of demand is a measure used in economics to show how the quantity demanded of a good responds to a change in its price. It's calculated as the percentage change in quantity demanded divided by the percentage change in price. Importantly, elasticity gives us an insight into consumer behavior and market sensitivity.

In differential calculus terms, the price elasticity of demand can be closely approximated using the derivative of the demand function, especially for infinitesimally small changes in price. For bigger changes, we consider the average elasticity over a range. The concept is fundamental in setting prices: a product with high elasticity will see a significant change in demand if the price is altered, while a product with low elasticity will see less of a change. The example provided in the exercise illustrates the application of calculus in determining the elasticity at a point, and hence in predicting the effects of changes in crop harvest size on the price of soybeans.

Quotient Rule

The quotient rule is a technique in differential calculus for finding the derivative of a function that is the ratio of two differentiable functions. If you have a function defined as \( \frac{u(x)}{v(x)} \), where \( u \) and \( v \) are functions of \( x \), the quotient rule states that the derivative of this function is given by \( \frac{u'(x)v(x) - u(x)v'(x)}{\left(v(x)\right)^2} \).

Let's see how this applies to the demand equation of soybeans. The equation \( \frac{55}{2x^2 + 1} \) represents a quotient of two functions: \( u(x) = 55 \) and \( v(x) = 2x^2 + 1 \). By applying the quotient rule, we find the derivative of the demand function, which then helps us understand how sensitive the price is to changes in quantity demanded. This employs the quotient rule to accurately forecast potential price change due to a given error in the quantity of soybeans harvested.