Business Calculus Study Guidance: Continuity, Cost, Revenue, and Marginal Analysis

Q1. For the function $f(x)$ defined piecewise, graph $f(x)$ and determine where it is not continuous.

Background

Topic: Continuity of Piecewise Functions

This question tests your understanding of continuity, especially for piecewise-defined functions. You need to identify points where the function is not continuous by analyzing the graph and the definition of $f(x)$.

Key Terms and Formulas:

• Continuity: A function $f(x)$ is continuous at $x=a$ if $\lim_{x \to a} f(x) = f(a)$.

• Piecewise Function: A function defined by different expressions for different intervals of $x$.

Step-by-Step Guidance

1. Identify the intervals and expressions for $f(x)$ from the piecewise definition.

2. Graph each segment of $f(x)$ according to its formula and domain.

3. Check the endpoints where the formula changes (e.g., $x = 1$, $x = 4$) for possible discontinuities.

4. For each endpoint, compare the left-hand and right-hand limits to the function value at that point.

Try solving on your own before revealing the answer!

Final Answer:

$f(x)$ is continuous everywhere except at $x = 1$ and $x = 4$.

At these points, the left and right limits do not match the function value, so discontinuities occur.

Q2. Given the total cost of producing $x$ items is $C(x) = 5000 + 12x$ and the revenue is $R(x) = 17x$, answer the following:

Background

Topic: Cost, Revenue, and Marginal Analysis

This question tests your ability to apply calculus concepts to business scenarios, specifically cost and revenue functions, marginal cost, and profit analysis.

Key Terms and Formulas:

• Total Cost ($C(x)$): The total expense incurred in producing $x$ items.

• Revenue ($R(x)$): The total income from selling $x$ items.

• Marginal Cost: $C'(x)$, the derivative of the cost function, representing the cost of producing one more item.

• Marginal Revenue: $R'(x)$, the derivative of the revenue function, representing the revenue from selling one more item.

• Profit Function: $P(x) = R(x) - C(x)$

Step-by-Step Guidance

1. Write down the cost and revenue functions: $C(x) = 5000 + 12x$, $R(x) = 17x$.

2. To find the cost of producing the 401st item, calculate $C(401) - C(400)$.

3. Find the marginal cost by differentiating $C(x)$: $C'(x) = \frac{d}{dx}(5000 + 12x)$.

4. Find the marginal revenue by differentiating $R(x)$: $R'(x) = \frac{d}{dx}(17x)$.

5. Set up the profit function: $P(x) = R(x) - C(x)$, and find the profit for selling the 401st item.

Try solving on your own before revealing the answer!

Final Answer:

The profit of selling the 401st item is approximately $5$.

This is found by calculating the difference between marginal revenue and marginal cost for the 401st item.