Business Calculus Study Guidance: Limits, Derivatives, Tangents, Optimization, and Integrals

Q1. Decide whether the limit exists. If it exists, find its value: Find $\lim_{x \to 0} \frac{x}{|x|}$

Background

Topic: Limits and Piecewise Functions

This question tests your understanding of limits involving absolute value functions and how the behavior differs from the left and right sides of a point.

Key Terms and Formulas:

• Limit: $\lim_{x \to a} f(x)$ is the value $f(x)$ approaches as $x$ gets close to $a$.

• Absolute Value: $|x|$ is $x$ if $x \geq 0$, and $-x$ if $x < 0$.

Step-by-Step Guidance

1. Consider the function $\frac{x}{|x|}$ for $x > 0$ and $x < 0$ separately.

2. For $x > 0$, $|x| = x$, so $\frac{x}{|x|} = 1$.

3. For $x < 0$, $|x| = -x$, so $\frac{x}{|x|} = -1$.

4. Analyze the left-hand and right-hand limits as $x \to 0$.

Try solving on your own before revealing the answer!

Q2. Use the graph to determine whether each statement is true or false: $\lim_{x \to 2} f(x) = 1$

Background

Topic: Limits from Graphs

This question tests your ability to interpret a graph and determine the value a function approaches as $x$ approaches a specific point.

Key Terms and Formulas:

• Limit from a graph: Look at the $y$-value the function approaches as $x$ gets close to the target value from both sides.

Step-by-Step Guidance

1. Examine the graph near $x = 2$ and observe the $y$-values as $x$ approaches $2$ from the left and right.

2. Check if the function approaches the same value from both sides.

3. Compare the value the function approaches to $1$.

Try solving on your own before revealing the answer!

Q3. Determine the continuity of the function at the given points: $f(x) = x^2$ for $x \neq 0$, $f(0) = 1$

Background

Topic: Continuity of Piecewise Functions

This question tests your understanding of continuity at a point, especially for piecewise-defined functions.

Key Terms and Formulas:

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

• Piecewise function: Defined differently for different intervals of $x$.

Step-by-Step Guidance

1. Find $\lim_{x \to 0} f(x)$ for $x \neq 0$ (i.e., $\lim_{x \to 0} x^2$).

2. Compare this limit to $f(0)$, which is $1$.

3. Determine if the function is continuous at $x = 0$ by checking if the limit equals the function value.

Try solving on your own before revealing the answer!

Q4. Find the indicated derivative: $\frac{d}{dx}(x^3 + 3x^2 + x + 1)$

Background

Topic: Basic Differentiation

This question tests your ability to apply the power rule and sum rule for derivatives.

Key Terms and Formulas:

• Power Rule: $\frac{d}{dx} x^n = n x^{n-1}$

• Sum Rule: $\frac{d}{dx}(f(x) + g(x)) = f'(x) + g'(x)$

Step-by-Step Guidance

1. Apply the power rule to each term: $x^3$, $3x^2$, $x$, and $1$.

2. Differentiate each term separately.

3. Combine the results to form the derivative.

Try solving on your own before revealing the answer!

Q5. Find the equation of the line tangent to the graph of the function at the indicated point: $y = x^2$ at $x = 2$

Background

Topic: Tangent Lines and Derivatives

This question tests your ability to find the slope of a tangent line using derivatives and write the equation of the tangent line.

Key Terms and Formulas:

• Tangent line: A line that touches the curve at one point and has the same slope as the curve at that point.

• Slope at $x = a$: $m = f'(a)$

• Point-slope form: $y - y_1 = m(x - x_1)$

Step-by-Step Guidance

1. Find $f'(x)$ for $y = x^2$ using the power rule.

2. Evaluate $f'(2)$ to get the slope at $x = 2$.

3. Find the point on the curve at $x = 2$ ($y = 2^2$).

4. Write the equation of the tangent line using the point-slope form.

Try solving on your own before revealing the answer!

Q6. Find the number of units that must be produced and sold in order to yield the maximum profit, given the following equations for revenue and cost: $R(x) = 40x$, $C(x) = 0.01x^2 + 14x + 80$

Background

Topic: Optimization in Business Calculus

This question tests your ability to maximize profit by finding the critical points of a profit function.

Key Terms and Formulas:

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

• Critical points: Where $P'(x) = 0$

Step-by-Step Guidance

1. Write the profit function: $P(x) = 40x - (0.01x^2 + 14x + 80)$

2. Simplify the profit function.

3. Find $P'(x)$ by differentiating the profit function.

4. Set $P'(x) = 0$ and solve for $x$ to find the critical point.

Try solving on your own before revealing the answer!

Q7. Evaluate the integral: $\int 3x^2 + 7x - 5 dx$

Background

Topic: Basic Integration

This question tests your ability to apply the power rule for integration to polynomials.

Key Terms and Formulas:

• Power Rule for Integration: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$

• Constant of integration: $C$

Step-by-Step Guidance

1. Integrate each term separately using the power rule.

2. Combine the results and add the constant of integration.

Try solving on your own before revealing the answer!

Q8. Find the antiderivative $F(x)$ of the function $f(x)$ below satisfying the initial condition $F(0) = 8$: $f(x) = 3x^2 - 5x + 7$

Background

Topic: Initial Value Problems in Integration

This question tests your ability to find an antiderivative and use an initial condition to solve for the constant.

Key Terms and Formulas:

• Antiderivative: $F(x)$ such that $F'(x) = f(x)$

• Initial condition: $F(0) = 8$

Step-by-Step Guidance

1. Integrate $f(x)$ term by term to find $F(x)$.

2. Add the constant $C$ to the antiderivative.

3. Use the initial condition $F(0) = 8$ to solve for $C$.

Try solving on your own before revealing the answer!