Business Calculus Study Guide: Linear Functions, Slope, and Cost Applications

Q1. Find the slope of the line containing the points (-2, 6) and (-4, 9).

Background

Topic: Slope of a Line

This question tests your understanding of how to calculate the slope between two points on a line, a foundational concept in calculus and linear modeling.

Key Terms and Formulas

• Slope (m): The measure of the steepness of a line, calculated as the change in y divided by the change in x between two points.

Formula for slope between points $(x_1, y_1)$ and $(x_2, y_2)$:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Step-by-Step Guidance

1. Label your points: Let $(-2, 6)$ be $(x_1, y_1)$ and $(-4, 9)$ be $(x_2, y_2)$.

2. Substitute the values into the slope formula:

   $m = \frac{9 - 6}{-4 - (-2)}$

3. Simplify the numerator and denominator separately.

Try solving on your own before revealing the answer!

Q2. Find the slope and the y-intercept of the graph of $y = mx + b$ (general linear equation).

Background

Topic: Linear Equations in Slope-Intercept Form

This question checks your ability to identify the slope and y-intercept from the standard form of a linear equation.

Key Terms and Formulas

• Slope (m): The coefficient of $x$ in the equation $y = mx + b$.

• Y-intercept (b): The constant term, representing where the line crosses the y-axis.

General form: $y = mx + b$

Step-by-Step Guidance

1. Identify the coefficient of $x$ in the equation; this is your slope ($m$).

2. Identify the constant term; this is your y-intercept ($b$).

3. Express the y-intercept as an ordered pair: $(0, b)$.

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Q3. Find an equation of the line with slope 3 containing the point (-1, -5).

Background

Topic: Point-Slope Form of a Line

This question tests your ability to write the equation of a line given a point and a slope, using the point-slope form.

Key Terms and Formulas

• Point-Slope Form: $y - y_1 = m(x - x_1)$

• Where $(x_1, y_1)$ is a point on the line and $m$ is the slope.

Step-by-Step Guidance

1. Identify the given point $(-1, -5)$ as $(x_1, y_1)$ and the slope $m = 3$.

2. Substitute these values into the point-slope formula:

   $y - (-5) = 3(x - (-1))$

3. Simplify the equation to standard or slope-intercept form as needed.

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Q4. Find the equation of the line containing the points (-1, 4) and (3, -6). Then determine the line’s y-intercept and express it as an ordered pair.

Background

Topic: Equation of a Line from Two Points

This question tests your ability to find the equation of a line given two points, and to determine the y-intercept.

Key Terms and Formulas

• Slope (m): $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Point-Slope Form: $y - y_1 = m(x - x_1)$

• Slope-Intercept Form: $y = mx + b$

Step-by-Step Guidance

1. Label the points: $(-1, 4)$ as $(x_1, y_1)$ and $(3, -6)$ as $(x_2, y_2)$.

2. Calculate the slope:

   $m = \frac{-6 - 4}{3 - (-1)}$

3. Use the point-slope form with one of the points and the calculated slope.

4. Rearrange to slope-intercept form to find $b$ (the y-intercept).

5. Express the y-intercept as an ordered pair: $(0, b)$.

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Q5. Raggs, Ltd., a clothing firm, has fixed costs of $10,000$ per month. To produce $x$ units of a suit, it costs $20$per suit in addition to the fixed costs. The total cost$C(x)$of producing$x$suits in a month is given by$C(x) = 20x + 10,000$.

Background

Topic: Cost Functions in Business Applications

This question tests your understanding of fixed and variable costs, and how to model total cost as a linear function of production quantity.

Key Terms and Formulas

• Fixed Costs: Costs that do not change with the level of production (e.g., rent, maintenance).

• Variable Costs: Costs that change with the number of units produced (e.g., materials, labor).

• Total Cost Function: $C(x) = ( ext{variable cost per unit}) \times x + ( ext{fixed cost})$

Step-by-Step Guidance

1. Identify the fixed cost ($10,000$) and the variable cost per unit ($20$).

2. Write the total cost function: $C(x) = 20x + 10,000$.

3. For part (A): To graph the variable cost, fixed cost, and total cost functions, consider:

   • Variable cost: $20x$ (starts at 0, increases with $x$)

   • Fixed cost: $10,000$ (horizontal line)

   • Total cost: $20x + 10,000$ (starts at $10,000$ when $x=0$)

4. For part (B): To find the total cost for $x = 100$ and $x = 400$, substitute these values into $C(x)$:

   $C(100) = 20 \times 100 + 10,000$

   $C(400) = 20 \times 400 + 10,000$

5. Simplify the expressions for each value of $x$ to find the total cost.

Try solving on your own before revealing the answer!