Q1. The value of a recently purchased car can be modeled by . Is the car appreciating or depreciating in value over time? What is the purchase price of the car? What is the value of the car 2 years after purchase? How is the car changing in value at 2 years?

Background

Topic: Exponential Functions & Derivatives in Business Calculus

This question tests your understanding of exponential decay models, initial value interpretation, and how to use derivatives to analyze rates of change.

Key Terms and Formulas:

• Exponential decay: , where means depreciation.

• Initial value: is the purchase price.

• Derivative: gives the rate of change of value at time .

Step-by-Step Guidance

1. Examine the exponent in to determine if the value is increasing or decreasing over time. What does a negative exponent indicate?

2. Find the purchase price by evaluating . Substitute into the formula.

3. Calculate the value of the car after 2 years by substituting into .

4. To find how the car is changing in value at 2 years, compute the derivative and evaluate it at .

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Q2. The total cost of producing tricycles is . What is the cost of producing 10 tricycles? Find the marginal cost function. What is the cost of producing the 11th tricycle?

Background

Topic: Cost Functions & Marginal Analysis

This question tests your ability to evaluate cost functions, find marginal cost using derivatives, and interpret marginal cost in a business context.

Key Terms and Formulas:

• Total cost function:

• Marginal cost: , the derivative of the cost function

• Cost of producing the 11th tricycle:

Step-by-Step Guidance

1. Evaluate by substituting into the cost function.

2. Find the marginal cost function by differentiating with respect to .

3. Calculate to find the cost of producing the 11th tricycle.

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Q3. The revenue function for a one-product firm is given by . Find the value of that results in maximum revenue.

Background

Topic: Optimization Using Calculus

This question tests your ability to find the maximum of a function using derivatives, a key skill in business calculus for maximizing profit or revenue.

Key Terms and Formulas:

• Revenue function:

• Critical points: Solve

• Maximum revenue: Check the value of at critical points

Step-by-Step Guidance

1. Find the derivative with respect to .

2. Set and solve for to find critical points.

3. Check the second derivative or use a sign chart to confirm which critical point gives the maximum revenue.

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Q4. The demand equation for beach umbrellas is , . Find the value of and the corresponding price that maximize the revenue.

Background

Topic: Revenue Maximization with Demand Functions

This question tests your ability to use demand equations to construct revenue functions and apply calculus to find maximum revenue.

Key Terms and Formulas:

• Demand equation:

• Revenue function:

• Maximum revenue: Find where

Step-by-Step Guidance

1. Write the revenue function:

2. Expand and simplify .

3. Find the derivative and set it equal to zero to find critical points.

4. Check which value of within the domain gives the maximum revenue.

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Q5. A seamstress is planning to sell prom dresses. If 60 prom dresses are offered for sale, she can charge $350 each. However, if she makes more than 60 prom dresses, she must lower the price by $5 for each prom dress in excess of the 60. How many prom dresses should the seamstress make to maximize her revenue?

Background

Topic: Revenue Optimization with Piecewise Pricing

This question tests your ability to construct a revenue function based on changing price and use calculus to maximize revenue.

Key Terms and Formulas:

• Revenue function:

• Piecewise price: for

• Maximum revenue: Find where

Step-by-Step Guidance

1. Write the price function for :

2. Construct the revenue function:

3. Expand and simplify .

4. Find the derivative and set it equal to zero to find the value of that maximizes revenue.

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Q6. A bookstore has an annual demand of 4,200 of a best-selling book. It costs xr$ for number of orders placed per year.

Background

Topic: Inventory Optimization (Economic Order Quantity)

This question tests your ability to use calculus to minimize total inventory costs using the EOQ model.

Key Terms and Formulas:

• Annual demand:

• Order quantity:

• Number of orders per year:

• EOQ formula:

• Where is order cost, is holding cost per unit per year

Step-by-Step Guidance

1. Write the total cost function:

2. Substitute the given values: , ,

3. Find the derivative and set it equal to zero to solve for the optimal .

Try solving on your own before revealing the answer!