Business Calculus Final Exam Study Guide – Step-by-Step Guidance

Q1. Section R.4: Find the break-even value (e.g., #59e and 61c)

Background

Topic: Break-Even Analysis in Business Applications

This question tests your understanding of how to determine the break-even point, which is the value (often in units or dollars) where total revenue equals total cost, resulting in zero profit or loss.

Key Terms and Formulas:

• Break-even point: The value where total revenue equals total cost.

• Total Revenue (TR): $TR = p \times x$ (where $p$ is price per unit, $x$ is number of units sold)

• Total Cost (TC): $TC = F + V \times x$ (where $F$ is fixed cost, $V$ is variable cost per unit, $x$ is number of units)

• Break-even equation: $TR = TC$

Step-by-Step Guidance

1. Write the expressions for total revenue and total cost based on the information given in the problem.

2. Set the total revenue equal to the total cost to form the break-even equation: $TR = TC$.

3. Solve the equation for $x$ (the number of units or value at break-even).

4. Check that your solution makes sense in the context of the problem (e.g., $x$ should be positive).

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Q2. Section 1.1: Find the limit by using a graph (e.g., #21-40)

Background

Topic: Limits and Graphical Analysis

This question tests your ability to estimate the value a function approaches as $x$ approaches a certain value, using the graph of the function.

Key Terms and Concepts:

• Limit: The value that $f(x)$ approaches as $x$ approaches a specific value $a$.

• Graphical approach: Observe the $y$-value the function approaches from both sides of $x = a$.

Step-by-Step Guidance

1. Locate the point $x = a$ on the $x$-axis of the graph.

2. Trace the graph as $x$ approaches $a$ from the left ($x \to a^-$) and from the right ($x \to a^+$).

3. Observe the $y$-value the function is approaching from both sides.

4. If the left and right limits are the same, that is the limit. If not, the limit does not exist.

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Q3. Section 1.2: Find the limit by direct substitution (e.g., #13-18)

Background

Topic: Evaluating Limits Algebraically

This question tests your ability to find limits by directly substituting the value of $x$ into the function, provided the function is continuous at that point.

Key Terms and Formulas:

• Direct substitution: Plug the value of $x$ into $f(x)$ to find the limit, if $f$ is continuous at that point.

Step-by-Step Guidance

1. Identify the value $x$ is approaching (e.g., $x \to a$).

2. Substitute $x = a$ directly into the function $f(x)$.

3. Simplify the expression to see if you get a real number.

4. If you get an indeterminate form (like $0/0$), further algebraic manipulation may be needed.

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Q4. Section 1.2: Find the indeterminate form limit by factoring (e.g., #19-24)

Background

Topic: Limits and Algebraic Manipulation

This question tests your ability to resolve indeterminate forms (like $0/0$) by factoring and simplifying before evaluating the limit.

Key Terms and Formulas:

• Indeterminate form: An expression like $0/0$ or $\infty/\infty$ when substituting directly.

• Factoring: Rewrite numerator and denominator to cancel common factors.

Step-by-Step Guidance

1. Substitute the value into the function to check if you get an indeterminate form.

2. If you get $0/0$, factor the numerator and denominator to find and cancel common factors.

3. After simplifying, substitute the value again to find the limit.

4. Be careful to only cancel factors, not terms.

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Q5. Section 1.2: Determine continuity of the function (e.g., #47-51, 57e-g)

Background

Topic: Continuity of Functions

This question tests your understanding of when a function is continuous at a point or on an interval.

Key Terms and Concepts:

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

• Types of discontinuity: Removable, jump, infinite.

Step-by-Step Guidance

1. Check if $f(a)$ is defined.

2. Find $\lim_{x \to a} f(x)$ from both sides.

3. Compare the limit to $f(a)$. If they are equal, the function is continuous at $a$.

4. If not, identify the type of discontinuity.

Try solving on your own before revealing the answer!