Problem 58 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

DEMAND FuNCTIONS The demand function for the Luminar desk lamp is given by $$ p=f(x)=-0.1 x^{2}-0.4 x+35 $$ where \(x\) is the quantity demanded in thousands and \(p\) is the unit price in dollars. a. Find \(f^{\prime}(x)\). b. What is the rate of change of the unit price when the quantity demanded is 10,000 units \((x=10)\) ? What is the unit price at that level of demand?

Step-by-Step Solution

Answer

The first derivative of the demand function is \(f'(x) = -0.2x - 0.4\). The rate of change of the unit price when the quantity demanded is 10,000 units is -2.4 dollars per thousand units, and the unit price at that level of demand is 21 dollars.

Step 1: Find the first derivative of the demand function

To find the first derivative of the demand function \(f(x) = -0.1x^2 - 0.4x + 35\), use the power rule, which states that the derivative of a function \(ax^n\) with respect to \(x\) is \(anx^{n-1},\) where a and n are constants. So, we have: \(f^\prime (x) = \frac{d}{dx} (-0.1x^2 - 0.4x + 35) \) Applying the power rule, \(f^\prime(x) = -0.2x - 0.4\)

Step 2: Evaluate the rate of change of the unit price for x = 10

To find the rate of change of the unit price when the quantity demanded is 10,000 \((x = 10)\), substitute \(x = 10\) into the first derivative, \(f^\prime(x) = -0.2x - 0.4\): \(f^\prime(10) = -0.2(10) - 0.4\) \(f^\prime(10) = -2 - 0.4\) \(f^\prime(10) = -2.4\) So, the rate of change of the unit price when the quantity demanded is 10,000 units is -2.4 dollars per thousand units.

Step 3: Find the unit price for x = 10

To find the unit price when the quantity demanded is 10,000 \((x = 10)\), substitute \(x = 10\) into the demand function, \(f(x) = -0.1x^2 - 0.4x + 35\): \(f(10) = -0.1(10)^2 - 0.4(10) + 35\) \(f(10) = -0.1(100) - 4 + 35\) \(f(10) = -10 + 31\) \(f(10) = 21\) So, the unit price when the quantity demanded is 10,000 units is 21 dollars.

Key Concepts

Demand Function

A demand function illustrates the relationship between the quantity of a good that consumers are willing and able to purchase and the good's price. In this exercise, the demand function for the Luminar desk lamp is defined by the equation \[ p=f(x)=-0.1 x^{2}-0.4 x+35 \] where \( x \) is the quantity of lamps demanded in thousands, and \( p \) is the price in dollars. By analyzing this function, we can predict how price fluctuates as demand changes, which is invaluable for setting competitive pricing strategies.

• The coefficients \(-0.1\) and \(-0.4\) determine the rate at which the price changes as the quantity varies.
• The constant \(35\) serves as the base price without any price reduction due to demand.

This quadratic function shape implies prices drop off more swiftly at higher quantities than they do at lower quantities, which is typical in pricing models aimed to stimulate demand.

Rate of Change

The concept of the rate of change offers insight into how the demand function's price component reacts when the demand shifts. In simple terms, it tells us how much the price will increase or decrease for each additional unit of demand.

• In mathematical terms, this is derived from the derivative of the demand function. It shows how sensitive the price is in response to changes in demand.
• For example, when you find that the rate of change is \(-2.4\), it tells us the price will decrease by \($2.4\) per thousand units increase in demand.
• This can help businesses understand when a high demand-driven price drop calls for strategic adjustments to maintain profitability.

Such calculations are key to crafting dynamic pricing models that adapt to changes in consumer purchasing behavior.

First Derivative

The first derivative of a function is a fundamental calculus tool that explains the rate at which that function's values change. If you remember calculus basics, finding the first derivative is essential in understanding how steeply a function rises or falls.
In this exercise, the first derivative of the demand function \(f(x)=-0.1x^2 - 0.4x + 35\) is computed using differentiation rules. The process involves:

• Using the power rule to differentiate each term of the function individually.
• Finding \(f'(x) = -0.2x - 0.4\), which represents the slope of the original demand function at any point \(x\).

By evaluating \(f'(x)\) at specific \(x\) values, one can gain practical insights into the behavior and trends of the function over its domain.

Power Rule

The power rule is a basic yet powerful tool in calculus for finding derivatives, especially when dealing with polynomial functions like the one in this exercise. The rule states: \[ \frac{d}{dx}(a x^n) = a n x^{n-1} \] where \(a\) is the coefficient and \(n\) is the exponent. This rule simplifies the calculation process by providing a straightforward way to find the rate at which a function's value changes.

• In our demand function, applying the power rule to the term \(-0.1x^2\) results in \(-0.2x\).
• The linear term \(-0.4x\) becomes \(-0.4\) after applying the rule, since the derivative of \(x\) is 1.
• The constant \(35\) vanishes as its derivative is 0.

Understanding and applying the power rule is crucial for determining how variables interact, which is especially valuable in economics-related functions where demand and supply need to be analyzed perceptively.