Business Calculus Assignment Guidance: Limits, Extrema, Optimization, and Applications

Q1(a). Find the limit: $\lim\limits_{x \to 0^-} \frac{\ln(1 + 2x)}{x^2}$

Background

Topic: Limits involving logarithmic functions and indeterminate forms.

This question tests your ability to evaluate limits, especially as $x$ approaches a value from the left, and to recognize and handle indeterminate forms using algebraic manipulation or L'Hôpital's Rule.

Key Terms and Formulas

• Indeterminate Form: An expression like $\frac{0}{0}$ or $\frac{\infty}{\infty}$, which requires further analysis.

• L'Hôpital's Rule: If $\lim\limits_{x \to a} \frac{f(x)}{g(x)}$ yields $\frac{0}{0}$ or $\frac{\infty}{\infty}$, then $\lim\limits_{x \to a} \frac{f(x)}{g(x)} = \lim\limits_{x \to a} \frac{f'(x)}{g'(x)}$ (if the latter limit exists).

• Derivative of $\ln(u)$: $\frac{d}{dx} \ln(u) = \frac{u'}{u}$

Step-by-Step Guidance

1. First, substitute $x = 0$ into the numerator and denominator to check if the limit is an indeterminate form.

2. Recognize that as $x \to 0^-$, $1 + 2x \to 1$ from below, so $\ln(1 + 2x) \to 0$ and $x^2 \to 0$ (from above).

3. Since both numerator and denominator approach $0$, you have a$\frac{0}{0}$ indeterminate form. This suggests L'Hôpital's Rule can be applied.

4. Differentiate the numerator and denominator with respect to $x$:

   • Numerator: $\frac{d}{dx} \ln(1 + 2x) = \frac{2}{1 + 2x}$

   • Denominator: $\frac{d}{dx} x^2 = 2x$

5. Set up the new limit using the derivatives:

   $\lim\limits_{x \to 0^-} \frac{2}{(1 + 2x) \cdot 2x}$

Try solving on your own before revealing the answer!

Q1(b). Find the limit: $\lim\limits_{x \to -2} \frac{x^2 + 2x}{x^2 + x - 2}$

Background

Topic: Limits involving rational functions and factoring.

This question tests your ability to evaluate limits of rational functions, especially when direct substitution gives an indeterminate form.

Key Terms and Formulas

• Factoring: Expressing polynomials as products of their factors to simplify expressions.

• Indeterminate Form: $\frac{0}{0}$, which may be resolved by simplification.

Step-by-Step Guidance

1. Substitute $x = -2$ into the numerator and denominator to check if you get $\frac{0}{0}$.

2. Factor both the numerator and denominator:

   • Numerator: $x^2 + 2x = x(x + 2)$

   • Denominator: $x^2 + x - 2 = (x + 2)(x - 1)$

3. Simplify the expression by canceling common factors, if possible.

4. After simplification, substitute $x = -2$ into the remaining expression to evaluate the limit.

Try solving on your own before revealing the answer!

Q1(c). Find the limit: $\lim\limits_{x \to \infty} \frac{x^2}{e^{3x}}$

Background

Topic: Limits at infinity involving exponential and polynomial functions.

This question tests your understanding of how exponential functions grow compared to polynomial functions as $x$ approaches infinity.

Key Terms and Formulas

• Exponential Growth: $e^{kx}$ grows much faster than any polynomial as $x \to \infty$.

• L'Hôpital's Rule: Useful for $\frac{\infty}{\infty}$ forms.

Step-by-Step Guidance

1. As $x \to \infty$, both numerator and denominator approach infinity, so you have an $\frac{\infty}{\infty}$ indeterminate form.

2. Apply L'Hôpital's Rule: Differentiate numerator and denominator with respect to $x$.

3. First derivatives: Numerator $= 2x$, Denominator $= 3e^{3x}$.

4. Set up the new limit: $\lim\limits_{x \to \infty} \frac{2x}{3e^{3x}}$ and consider applying L'Hôpital's Rule again if needed.

Try solving on your own before revealing the answer!

Q2(a). Find the local and absolute extrema of $f(x) = 2x^3 - 3x^2 - 12x + 24$ on $[-3, 4]$

Background

Topic: Finding extrema (maximum and minimum values) of a function on a closed interval.

This question tests your ability to find critical points, evaluate endpoints, and determine local and absolute extrema.

Key Terms and Formulas

• Critical Points: Values of $x$ where $f'(x) = 0$ or $f'(x)$ is undefined.

• Endpoints: The values at $x = -3$ and $x = 4$ must also be checked for absolute extrema.

• First Derivative: $f'(x)$ gives the slope; set $f'(x) = 0$ to find critical points.

Step-by-Step Guidance

1. Find the first derivative: $f'(x) = 6x^2 - 6x - 12$.

2. Solve $f'(x) = 0$ to find critical points inside $[-3, 4]$.

3. Evaluate $f(x)$ at each critical point and at the endpoints $x = -3$ and $x = 4$.

4. Compare all these values to determine which are local and which are absolute extrema.

Try solving on your own before revealing the answer!

Q2(b). Find the local and absolute extrema of $f(x) = x^3 - 6x^2$ on $[0, \infty)$

Background

Topic: Extrema on a half-infinite interval.

This question tests your ability to find and classify critical points and analyze behavior as $x \to \infty$.

Key Terms and Formulas

• Critical Points: Where $f'(x) = 0$ or undefined.

• First Derivative: $f'(x)$

Step-by-Step Guidance

1. Find the first derivative: $f'(x) = 3x^2 - 12x$.

2. Solve $f'(x) = 0$ to find critical points in $[0, \infty)$.

3. Evaluate $f(x)$ at $x = 0$ and at the critical points.

4. Analyze the behavior of $f(x)$ as $x \to \infty$ to determine if there is an absolute maximum or minimum.

Try solving on your own before revealing the answer!

Q2(c). Find the local and absolute extrema of $f(x) = \frac{x^2}{x^2 + 1}$ on $(-\infty, \infty)$

Background

Topic: Extrema of rational functions on the entire real line.

This question tests your ability to find critical points and analyze limits at infinity.

Key Terms and Formulas

• Critical Points: Where $f'(x) = 0$ or undefined.

• First Derivative: Use the quotient rule: $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$

Step-by-Step Guidance

1. Let $u = x^2$, $v = x^2 + 1$. Compute $u'$ and $v'$.

2. Apply the quotient rule to find $f'(x)$.

3. Set $f'(x) = 0$ and solve for $x$ to find critical points.

4. Evaluate $f(x)$ at the critical points and analyze the limits as $x \to \pm\infty$.

Try solving on your own before revealing the answer!

Q2(d). Find the local and absolute extrema of $f(x) = 4x \ln(x) - 7x$ on $(0, \infty)$

Background

Topic: Extrema of functions involving logarithms on an open interval.

This question tests your ability to use derivatives to find critical points and analyze behavior as $x \to 0^+$ and $x \to \infty$.

Key Terms and Formulas

• Product Rule: $\frac{d}{dx}[u \cdot v] = u'v + uv'$

• Derivative of $\ln(x)$: $\frac{1}{x}$

Step-by-Step Guidance

1. Find the first derivative using the product rule:

   $f'(x) = 4 \ln(x) + 4 - 7$

2. Simplify $f'(x)$ and set it equal to zero to find critical points.

3. Solve for $x$ to find where extrema may occur.

4. Analyze the behavior of $f(x)$ as $x \to 0^+$ and $x \to \infty$ to determine absolute extrema.

Try solving on your own before revealing the answer!

Q3. The concentration $C(t) = \frac{0.16t}{t^2 + 4t + 4}$ (mg/cm³) of a drug in a patient's bloodstream, where $t$ is hours after the drug is taken. When is the concentration maximum, and what is the maximum concentration?

Background

Topic: Optimization of a rational function in an applied context.

This question tests your ability to find the maximum value of a function modeling a real-world scenario using calculus.

Key Terms and Formulas

• Critical Points: Where $C'(t) = 0$ or undefined.

• Quotient Rule: $\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}$

Step-by-Step Guidance

1. Let $u = 0.16t$, $v = t^2 + 4t + 4$. Compute $u'$ and $v'$.

2. Apply the quotient rule to find $C'(t)$.

3. Set $C'(t) = 0$ and solve for $t$ to find when the concentration is maximum.

4. Substitute the value of $t$ back into $C(t)$ to find the maximum concentration.

Try solving on your own before revealing the answer!

Q4. A parcel delivery service will deliver a package only if the length plus girth does not exceed 108 inches. Find the dimensions of a rectangular box with square ends that satisfies this restriction and has maximum volume. What is the maximum volume?

Background

Topic: Constrained optimization (maximizing volume under a constraint).

This question tests your ability to set up and solve an optimization problem with a geometric constraint.

Key Terms and Formulas

• Volume of a box: $V = l \cdot w^2$ (for length $l$ and square ends of width $w$)

• Girth: The perimeter around the box's cross-section (for square ends, $4w$)

• Constraint: $l + 4w \leq 108$

Step-by-Step Guidance

1. Let $w$ be the width of the square end and $l$ the length. The constraint is $l + 4w = 108$ (for maximum volume, use equality).

2. Express $l$ in terms of $w$: $l = 108 - 4w$.

3. Write the volume as a function of $w$: $V(w) = (108 - 4w)w^2$.

4. Find $V'(w)$ and set it to zero to find the value of $w$ that maximizes the volume.

5. Substitute this value of $w$ back into $l = 108 - 4w$ to find the corresponding length.

Try solving on your own before revealing the answer!

Q5. A fence is to be built to enclose a rectangular area of 800 sq ft. Three sides cost $6/ft, the fourth side costs $18/ft. Find the rectangle dimensions for the most economical fence.

Background

Topic: Optimization with cost constraints.

This question tests your ability to minimize cost given area and variable costs for different sides.

Key Terms and Formulas

• Area constraint: $xy = 800$ (let $x$ and $y$ be the rectangle's sides)

• Cost function: $C = 18x + 6(2y + x)$ (if $x$ is the side with higher cost)

Step-by-Step Guidance

1. Let $x$ be the length of the side with $18/ft$ cost, $y$ the other side.

2. Write the area constraint: $xy = 800$.

3. Express the cost function: $C = 18x + 6x + 12y = 24x + 12y$.

4. Use the area constraint to express $y$ in terms of $x$: $y = \frac{800}{x}$.

5. Substitute $y$ into the cost function to get $C(x)$, then find $C'(x)$ and set it to zero to find the minimum cost dimensions.

Try solving on your own before revealing the answer!

Q6(a). A company sells $x$ smartphones per week. Price-demand: $p = 500 - 0.4x$, Cost: $C(x) = 20000 + 20x$. What price and quantity maximize weekly revenue? What is the maximum revenue?

Background

Topic: Revenue maximization using price-demand functions.

This question tests your ability to express revenue as a function of quantity, find its maximum, and interpret the result in a business context.

Key Terms and Formulas

• Revenue function: $R(x) = p \cdot x$

• Price-demand equation: $p = 500 - 0.4x$

Step-by-Step Guidance

1. Express revenue as a function of $x$: $R(x) = x(500 - 0.4x)$.

2. Expand and simplify $R(x)$.

3. Find $R'(x)$ and set it to zero to find the value of $x$ that maximizes revenue.

4. Substitute this $x$ back into the price-demand equation to find the price.

5. Substitute $x$ into $R(x)$ to find the maximum revenue.

Try solving on your own before revealing the answer!

Q6(b). What is the maximum weekly profit? What price and quantity maximize profit?

Background

Topic: Profit maximization using cost and revenue functions.

This question tests your ability to set up and maximize the profit function, and interpret the results in a business context.

Key Terms and Formulas

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

• Revenue function: $R(x) = x(500 - 0.4x)$

• Cost function: $C(x) = 20000 + 20x$

Step-by-Step Guidance

1. Write the profit function: $P(x) = R(x) - C(x)$.

2. Substitute the expressions for $R(x)$ and $C(x)$ to get $P(x)$ in terms of $x$.

3. Find $P'(x)$ and set it to zero to find the value of $x$ that maximizes profit.

4. Substitute this $x$ into the price-demand equation to find the price.

5. Substitute $x$ into $P(x)$ to find the maximum profit.

Try solving on your own before revealing the answer!