Business Calculus Study Guide: Rates of Change, Marginal Cost, and Marginal Revenue

Q1. Finding Average Velocity and Velocity

Background

Topic: Average and Instantaneous Rate of Change (Velocity)

This question tests your understanding of how to compute the average velocity over a time interval and the instantaneous velocity at a specific time using a position function.

Key formulas:

• $v_{\text{ave}} = \frac{\Delta s}{\Delta t} = \frac{f(t + \Delta t) - f(t)}{\Delta t}$

• $v = \lim_{\Delta t \to 0} \frac{f(t + \Delta t) - f(t)}{\Delta t} = \frac{ds}{dt}$

Step-by-Step Guidance

1. Identify the position function: $s(t) = 3t^2 + 5$.

2. For part (a), calculate $s(10)$ and $s(10.1)$ to find the average velocity over $[10, 10.1]$.

3. Use the formula $v_{\text{ave}} = \frac{s(10.1) - s(10)}{10.1 - 10}$ to set up the calculation.

4. For part (b), find the instantaneous velocity at $t = 10$ by differentiating $s(t)$ to get $v(t)$.

5. Set up $v(10)$ using the derivative you found.

Try solving on your own before revealing the answer!

Final Answer:

(a) $v_{\text{ave}} = 60.3$ meters/second

(b) $v(10) = 60$ meters/second

The average velocity uses the difference quotient, while the instantaneous velocity uses the derivative.

Q2. Estimating Change in y Using dy/dx

Background

Topic: Linear Approximation and Rate of Change

This question tests your ability to estimate the change in a function using its derivative at a given point.

Key formula:

• $\Delta y \approx \frac{dy}{dx} \Delta x$

Step-by-Step Guidance

1. Identify the given derivative: $\frac{dy}{dx} = 8$ at $x = 3$.

2. Determine the change in $x$: $\Delta x = 3.5 - 3 = 0.5$.

3. Set up the estimation formula: $\Delta y \approx 8 \times 0.5$.

Try solving on your own before revealing the answer!

Final Answer:

$\Delta y \approx 4$

The estimated change in $y$ is 4 units when $x$ increases from 3 to 3.5.

Q3. Finding a Rate of Change

Background

Topic: Derivative and Rate of Change

This question tests your ability to find the rate of change of a function and evaluate it at specific points.

Key formula:

• $\frac{dy}{dx}$ is the rate of change of $y$ with respect to $x$.

Step-by-Step Guidance

1. Write the function: $y = x^4$.

2. Find the derivative: $\frac{dy}{dx}$.

3. Evaluate $\frac{dy}{dx}$ at $x = 2$ and $x = -1$.

Try solving on your own before revealing the answer!

Final Answer:

$\frac{dy}{dx} = 4x^3$; at $x = 2$, $\frac{dy}{dx} = 32$; at $x = -1$, $\frac{dy}{dx} = -4$.

These values represent the rate of change of $y$ at the specified points.

Q4. Rate of Change of Price with Respect to Quantity

Background

Topic: Marginal Analysis in Economics

This question tests your ability to find the rate of change (marginal change) of price with respect to quantity using a demand function.

Key formula:

• $\frac{dp}{dq}$ is the rate of change of price with respect to quantity.

Step-by-Step Guidance

1. Write the demand function: $p = 100 - q^2$.

2. Find the derivative $\frac{dp}{dq}$.

3. Evaluate $\frac{dp}{dq}$ at $q = 5$.

Try solving on your own before revealing the answer!

Final Answer:

$\frac{dp}{dq} = -2q$; at $q = 5$, $\frac{dp}{dq} = -10$

The price is decreasing at a rate of $10 per unit when$q = 5$.

Q5. Marginal Cost and Applications

Background

Topic: Marginal Cost in Economics

This question tests your understanding of how to compute marginal cost from a total-cost function and interpret its meaning.

Key formula:

• $\text{marginal cost} = \frac{dc}{dq}$

Step-by-Step Guidance

1. Write the cost function: $c = 0.1q^2 + 3$.

2. Find the derivative $\frac{dc}{dq}$.

3. Evaluate $\frac{dc}{dq}$ at $q = 4$.

Try solving on your own before revealing the answer!

Final Answer:

$\frac{dc}{dq} = 0.2q$; at $q = 4$, $\frac{dc}{dq} = 0.8$

The marginal cost at $q = 4$ is $0.80 per pound.

Q6. Average Cost per Unit

Background

Topic: Average Cost Calculation

This question tests your ability to compute the average cost per unit and relate it to total cost.

Key formula:

• $\bar{c} = \frac{c}{q}$

Step-by-Step Guidance

1. Identify total cost $c$ and number of units $q$.

2. Plug values into the formula $\bar{c} = \frac{c}{q}$.

3. Multiply both sides by $q$ to relate total cost and average cost.

Try solving on your own before revealing the answer!

Final Answer:

If $c = 100$ and $q = 20$, then $\bar{c} = 5$.

Total cost can be written as $c = q \bar{c}$.

Q7. Marginal Cost from Average-Cost Equation

Background

Topic: Marginal Cost from Average Cost

This question tests your ability to find the marginal cost function from an average-cost equation and evaluate it at a specific quantity.

Key formula:

• $\bar{c} = 0.0001q^2 - 0.02q + 5 + \frac{5000}{q}$

• Marginal cost: $\frac{d\bar{c}}{dq}$

Step-by-Step Guidance

1. Write the average-cost function.

2. Differentiate $\bar{c}$ with respect to $q$ to find the marginal-cost function.

3. Evaluate the marginal-cost function at $q = 50$.

Try solving on your own before revealing the answer!

Final Answer:

Marginal-cost function: $0.0002q - 0.02 - \frac{5000}{q^2}$; at $q = 50$, marginal cost is $-1.98$.

This value represents the rate of change of average cost at $q = 50$ units.

Q8. Marginal Revenue

Background

Topic: Marginal Revenue in Economics

This question tests your understanding of how to compute marginal revenue from a total-revenue function.

Key formula:

• $\text{marginal revenue} = \frac{dr}{dq}$

Step-by-Step Guidance

1. Write the total-revenue function: $r = 2q$.

2. Find the derivative $\frac{dr}{dq}$.

Try solving on your own before revealing the answer!

Final Answer:

$\frac{dr}{dq} = 2$

The marginal revenue is $2 per unit$ sold.

Q9. Income–Education: Rate of Change of Income

Background

Topic: Rate of Change in Applied Context

This question tests your ability to find the rate of change of income with respect to years of education and evaluate it at a specific value.

Key formula:

• $y = 6x^{9/4} + 5900$

• Rate of change: $\frac{dy}{dx}$

Step-by-Step Guidance

1. Write the income function.

2. Find the derivative $\frac{dy}{dx}$.

3. Evaluate $\frac{dy}{dx}$ at $x = 16$.

Try solving on your own before revealing the answer!

Final Answer:

$\frac{dy}{dx} = 6 \cdot \frac{9}{4} x^{5/4}$; at $x = 16$, $\frac{dy}{dx} = 216$

The rate of change of income at 16 years of education is $216$ dollars per year.

Q10. Marginal-Cost Function for Cost Functions

Background

Topic: Marginal Cost for Cost Functions

This question tests your ability to find the marginal-cost function and evaluate it at given values of $q$.

Key formula:

• $\frac{dc}{dq}$

Step-by-Step Guidance

1. Write the cost function $c$ for each problem.

2. Find the derivative $\frac{dc}{dq}$.

3. Evaluate the marginal-cost function at the specified values of $q$.

Try solving on your own before revealing the answer!

Final Answer:

For each cost function, differentiate with respect to $q$ and substitute the given values to find the marginal cost.

Q11. Marginal-Cost Function for Average Cost Functions

Background

Topic: Marginal Cost from Average Cost

This question tests your ability to find the marginal-cost function from an average-cost function and evaluate it at indicated values.

Key formula:

• $\frac{d\bar{c}}{dq}$

Step-by-Step Guidance

1. Write the average-cost function $\bar{c}$ for each problem.

2. Find the derivative $\frac{d\bar{c}}{dq}$.

3. Evaluate the marginal-cost function at the specified values of $q$.

Try solving on your own before revealing the answer!

Final Answer:

For each average-cost function, differentiate with respect to $q$ and substitute the given values to find the marginal cost.

Q12. Cost Function: Rate of Change and Average Rate of Change

Background

Topic: Rate of Change and Average Rate of Change for Cost Functions

This question tests your ability to find the instantaneous and average rate of change for a cost function.

Key formula:

• $c = 0.4q^2 + 4q + 5$

• Instantaneous rate: $\frac{dc}{dq}$

• Average rate: $\frac{c(3) - c(2)}{3 - 2}$

Step-by-Step Guidance

1. Write the cost function.

2. Find the derivative $\frac{dc}{dq}$ and evaluate at $q = 2$.

3. Calculate $c(2)$ and $c(3)$, then set up the average rate formula.

Try solving on your own before revealing the answer!

Final Answer:

Instantaneous rate at $q = 2$: $\frac{dc}{dq} = 0.8(2) + 4 = 5.6$

Average rate over $[2, 3]$: $\frac{c(3) - c(2)}{1} = 7.2$

Q13. Approximating Total Cost Using Marginal and Average Cost

Background

Topic: Cost Approximation Using Marginal and Average Cost

This question tests your ability to use marginal and average cost to estimate total cost for a change in production.

Key formula:

• Total cost for $q$ units: $c = q \bar{c}$

• Marginal cost approximates the cost of one additional unit.

Step-by-Step Guidance

1. Identify the average cost and marginal cost at $q = 20$.

2. Calculate total cost for 20 bikes: $c = 20 \times 200$.

3. Add the marginal cost for the 21st bike to estimate total cost for 21 bikes.

Try solving on your own before revealing the answer!

Final Answer:

Total cost for 21 bikes is approximately $4150$.

This uses the average cost for the first 20 bikes and the marginal cost for the 21st bike.