Business Calculus Exam 1 Review: Step-by-Step Guidance

Q1. Perform the indicated operation:

\[ \left( \frac{1}{9}y^2 - \frac{1}{3}y + \frac{2}{9} \right) + \left( 5y^2 + \frac{5}{6}y - \frac{1}{9} \right) \]

Background

Topic: Polynomial Addition

This question tests your ability to add polynomials by combining like terms (terms with the same variable and exponent).

Key Terms and Formulas:

• Like terms: Terms that have the same variable raised to the same power.

• Combining like terms: Add or subtract the coefficients of like terms.

Step-by-Step Guidance

1. Write both polynomials in a single expression, grouping like terms together:

   \[ \left( \frac{1}{9}y^2 - \frac{1}{3}y + \frac{2}{9} \right) + \left( 5y^2 + \frac{5}{6}y - \frac{1}{9} \right) \]

2. Combine the $y^2$ terms: $\frac{1}{9}y^2 + 5y^2$.

3. Combine the $y$ terms: $-\frac{1}{3}y + \frac{5}{6}y$.

4. Combine the constant terms: $\frac{2}{9} - \frac{1}{9}$.

5. Simplify each group by finding common denominators where necessary, but do not compute the final values yet.

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Q2. Determine whether the statement is true or false, and explain why:

The function $f(x) = x^3 + x^6 + x^8$ is an example of a cubic polynomial.

Background

Topic: Polynomial Degree Classification

This question tests your understanding of how to classify polynomials by their degree (the highest exponent of the variable).

Key Terms and Formulas:

• Degree of a polynomial: The highest exponent of the variable in the polynomial.

• Cubic polynomial: A polynomial whose highest degree term is $x^3$.

Step-by-Step Guidance

1. Identify the degree of each term in the polynomial: $x^3$, $x^6$, $x^8$.

2. Determine the highest degree among all terms.

3. Recall the definition of a cubic polynomial and compare it to the degree found in step 2.

4. Decide if the polynomial fits the definition of a cubic polynomial based on its highest degree.

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Q3. Multiply the following rational expressions. Write the answer in lowest terms:

\[ \frac{15w^2}{7} \cdot \frac{21}{55w} \]

Background

Topic: Multiplying Rational Expressions

This question tests your ability to multiply rational expressions and simplify the result to lowest terms.

Key Terms and Formulas:

• Rational expression: A fraction where the numerator and/or denominator are polynomials.

• Simplifying: Cancel common factors in the numerator and denominator.

Step-by-Step Guidance

1. Multiply the numerators together: $15w^2 \times 21$.

2. Multiply the denominators together: $7 \times 55w$.

3. Write the product as a single fraction.

4. Factor and cancel any common factors between the numerator and denominator.

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Q4. Simplify the given expression. Assume all variables represent positive real numbers. Write answers with only positive exponents:

\[ \frac{(18k^2)^2}{6k^8} \]

Background

Topic: Exponents and Simplifying Rational Expressions

This question tests your ability to apply exponent rules and simplify rational expressions.

Key Terms and Formulas:

• Power of a product: $(ab)^n = a^n b^n$

• Power of a power: $(a^m)^n = a^{mn}$

• Quotient rule: $\frac{a^m}{a^n} = a^{m-n}$

Step-by-Step Guidance

1. Expand the numerator using the power of a product rule: $(18k^2)^2 = 18^2 \cdot (k^2)^2$.

2. Simplify the numerator: $18^2 = 324$ and $(k^2)^2 = k^4$.

3. Write the expression as $\frac{324k^4}{6k^8}$.

4. Simplify the coefficients and apply the quotient rule for exponents to the $k$ terms.

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Q5. The sales of a small company were $24,000 in its second year of operation and $54,000 in its fifth year. Let y represent sales in the xth year of operation. Assume that the data can be approximated by a straight line.

• (a) Find the slope of the sales line, and give an equation for the line in the form $y = mx + b$.

• (b) Use your answer from part (a) to find out how many years must pass before the sales surpass $90,000.

Background

Topic: Linear Equations and Slope

This question tests your ability to find the equation of a line given two points and use it to solve for a specific value.

Key Terms and Formulas:

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

• Equation of a line: $y = mx + b$

Step-by-Step Guidance

1. Identify the two points: $(2, 24000)$ and $(5, 54000)$.

2. Calculate the slope $m$ using the slope formula.

3. Substitute one point and the slope into $y = mx + b$ to solve for $b$.

4. Write the equation of the line.

5. Set $y = 90000$ and solve for $x$ to find when sales surpass $90,000$.

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Q6. Joanne sells silk-screened T-shirts at community festivals and craft fairs. Her marginal cost to produce one T-shirt is $2.50. Her total cost to produce 40 T-shirts is $150, and she sells them for $6 each.

• a. Find the linear cost function for Joanne's T-shirt production.

• b. How many T-shirts must she produce and sell in order to break even?

• c. How many T-shirts must she produce and sell to make a profit of $700?

Background

Topic: Linear Cost, Revenue, and Profit Functions

This question tests your ability to construct and analyze cost and revenue functions, and solve for break-even and profit conditions.

Key Terms and Formulas:

• Cost function: $C(x) = mx + b$ where $m$ is the marginal cost and $b$ is the fixed cost.

• Revenue function: $R(x) = p \cdot x$ where $p$ is the price per item.

• Break-even point: $C(x) = R(x)$

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

Step-by-Step Guidance

1. Use the marginal cost and total cost information to write the cost function $C(x)$.

2. Set $C(x) = R(x)$ to solve for the break-even quantity.

3. Set $P(x) = 700$ and solve for $x$ to find the quantity needed for a $700 profit.

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Q7. Give the domain and the range of the function in the graph.

Background

Topic: Domain and Range of Functions

This question tests your ability to interpret a graph and determine the set of possible input (domain) and output (range) values.

Key Terms and Formulas:

• Domain: All possible $x$-values for which the function is defined.

• Range: All possible $y$-values the function can take.

Step-by-Step Guidance

1. Look at the graph and identify the leftmost and rightmost points to determine the domain.

2. Identify the lowest and highest points on the graph to determine the range.

3. Express your answers in interval notation.

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Q8. A charter flight charges in the form of $R = 6000 + 80x$ per person for each unsold seat on the plane. The plane holds 100 passengers. Let x represent the number of unsold seats. Complete parts (a) through (c).

• a. Find an expression for the total revenue received for the flight, $R(x)$.

• b. Choose the correct graph of the function $R(x)$.

• c. Find the number of unsold seats that will produce the maximum revenue and the maximum revenue itself.

Background

Topic: Revenue Functions and Maximization

This question tests your ability to construct a revenue function and analyze it to find maximum values.

Key Terms and Formulas:

• Revenue function: $R(x) = ( ext{price per ticket}) \times ( ext{number of tickets sold})$

• Maximizing revenue: Find the vertex of the quadratic function $R(x)$ if applicable.

Step-by-Step Guidance

1. Express the number of tickets sold as $100 - x$ (since $x$ is the number of unsold seats).

2. Write the revenue function $R(x)$ in terms of $x$.

3. Expand and simplify the expression for $R(x)$.

4. Identify the vertex of the quadratic function to find the value of $x$ that maximizes revenue.

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