Problem 2 - Active Calculus

Question

The cost, \(C\) (in dollars) to produce \(g\) gallons of ice cream can be expressed as \(C=f(g)\). (a) In the expression \(f(300)=350\), what are the units of \(300 ?\) [Choose: ? | dollars | gallons | dollars*gallons | dollars/gallon | gallons/dollar] what are the units of \(350 ?\) [Choose: ? | dollars | gallons | dollars*gallons | dollars/gallon | gallons/dollar] (b) In the expression \(f^{\prime}(300)=1.2,\) what are the units of 300? [Choose:? | dollars | gallons | dollars*gallons | dollars/gallon | gallons/dollar] what are the units of 1.2? [Choose: ? | dollars | gallons | dollars*gallons | dollars/gallon | gallons/dollar] (Be sure that you can carefully put into words the meanings of each of these statement in terms of ice cream and money.)

Step-by-Step Solution

Answer

300 and 300 are in gallons. 350 is in dollars. 1.2 is in dollars per gallon.

Step 1: Determine Units for 300 in f(300)=350

The function represents the cost as a function of the number of gallons of ice cream produced. Here, the input to the function is 300. Thus, 300 represents the amount of ice cream, measured in gallons.

Step 2: Determine Units for 350 in f(300)=350

The output of the function, f(300), is 350. This output represents the cost to produce the 300 gallons of ice cream. Thus, 350 is measured in dollars.

Step 3: Determine Units for 300 in f'(300)=1.2

In the derivative function, the input value is again 300. This input remains the same as in the original function, representing the amount of ice cream produced, measured in gallons.

Step 4: Determine Units for 1.2 in f'(300)=1.2

The derivative f'(g) represents how the cost changes with respect to the number of gallons. Thus, f'(300)=1.2 means that the cost changes by 1.2 dollars for each additional gallon produced. Therefore, the unit for 1.2 is dollars per gallon.

Key Concepts

Function Notation

Understanding function notation is key when interpreting problems involving functions. A function, denoted as \(f(g)\) in our example, shows a relationship between two sets of numbers. Here, \(f\) represents the cost of producing ice cream, and \(g\) represents the number of gallons of ice cream produced. Thus, \(f(g)\) means the cost (in dollars) to produce \(g\) gallons of ice cream. When we see expressions like \(f(300) = 350\), it tells us that producing 300 gallons of ice cream costs 350 dollars.

• \(g\) is the input, representing gallons of ice cream.
• \(f(g)\) is the output, representing the cost in dollars.

It's important to note the input/output relationship and the respective units associated with these numbers for clear interpretation.

Derivative Interpretation

Derivatives help us understand how one quantity changes as another quantity changes. In this context, \(f'(g)\) represents the rate at which the cost changes as the production of ice cream changes.

When we see \(f'(300) = 1.2\), it tells us the rate of change of the cost when 300 gallons are being produced. Specifically, the cost increases by 1.2 dollars for each additional gallon of ice cream produced.

• \(f'(g)\) is the derivative of the function \(f(g)\)
• The input \(300\) represents the current production level in gallons
• The output, 1.2, gives the rate of cost change, in dollars per gallon

This understanding is crucial for decision-making and cost estimation in production scenarios.

Unit Analysis

Unit analysis involves ensuring that we correctly interpret the units associated with each quantity in a problem. This keeps calculations and interpretations consistent and meaningful.

For the function \(f(300) = 350\):

• The input value \(300\) is in gallons (ice cream produced)
• The output value \(350\) is in dollars (cost to produce 300 gallons)

For the derivative \(f'(300) = 1.2\):

• The input value \(300\) remains in gallons
• The output value \(1.2\) means dollars per gallon (change in cost per gallon increase)

Being precise with units helps avoid errors and ensures you understand what each part of the equation represents in real-world terms. It makes the problems more relatable and easier to solve.