Optimization and Graph Analysis in Calculus

Q3. A homeowner wants to build a rectangular planting bed with an area equal to 200 ft² and a width x between 1 and 25 ft. The bed will be behind the house, so no fence is along the back wall of the house.

Part (a): Build a function f(x) that gives the total length of the fence.

Background

Topic: Optimization and Function Construction

This question is about expressing a real-world constraint (fencing a rectangular area with one side against a wall) as a function, which is a common application of calculus in optimization problems.

Key Terms and Formulas:

• Let x = width of the bed (perpendicular to the house).

• Let y = length of the bed (parallel to the house, but not against the wall).

• Area constraint: $x \cdot y = 200$

• Total fence needed: Only three sides (two widths and one length) require fencing.

Step-by-Step Guidance

1. Express the area constraint as $x \cdot y = 200$ and solve for $y$ in terms of $x$.

2. Write the total length of fence needed as $f(x) = 2x + y$ (since the back side against the house does not need fencing).

3. Substitute your expression for $y$ from step 1 into the formula for $f(x)$ to get $f(x)$ in terms of $x$ only.

4. Remember to state the domain for $x$ (given as $1 \leq x \leq 25$).

Try solving on your own before revealing the answer!

Q3b. Find the minimum length of the fence.

Background

Topic: Optimization Using Calculus

This part asks you to minimize the function you found in part (a), which is a classic calculus optimization problem. You will use derivatives to find critical points and determine the minimum value within the given domain.

Key Terms and Formulas:

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

• Endpoints: Check $x = 1$ and $x = 25$ as possible candidates for the minimum.

Step-by-Step Guidance

1. Take the derivative $f'(x)$ of your function $f(x)$ from part (a).

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

3. Check the value of $f(x)$ at the critical point(s) and at the endpoints $x = 1$ and $x = 25$.

4. Compare these values to determine which gives the minimum fence length.

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Q6. Let $f(x) = 2x^3 + 3x^2 - 12x$.

Part (a): Compute $f'(x)$, determine the intervals for which $f$ is increasing or decreasing, and list (if any) extreme value(s) of $f$.

Background

Topic: First Derivative Test and Critical Points

This question tests your ability to find the derivative of a polynomial, use it to determine where the function is increasing or decreasing, and identify local maxima and minima (extreme values).

Key Terms and Formulas:

• First derivative: $f'(x)$ gives the slope of the tangent line to $f(x)$.

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

• Increasing: Where $f'(x) > 0$; Decreasing: Where $f'(x) < 0$.

Step-by-Step Guidance

1. Compute $f'(x)$ using the power rule.

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

3. Use a sign chart or test values in each interval determined by the critical points to see where $f'(x)$ is positive (increasing) or negative (decreasing).

4. Identify any local maxima or minima by checking the sign changes of $f'(x)$ at the critical points.

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Part (b): Compute $f''(x)$, determine the intervals for which $f$ is concave up or down, and list (if any) inflection points.

Background

Topic: Second Derivative Test and Concavity

This part asks you to analyze the concavity of the function and find inflection points using the second derivative.

Key Terms and Formulas:

• Second derivative: $f''(x)$ tells you about the concavity of $f(x)$.

• Concave up: Where $f''(x) > 0$; Concave down: Where $f''(x) < 0$.

• Inflection point: Where $f''(x)$ changes sign.

Step-by-Step Guidance

1. Compute $f''(x)$ by differentiating $f'(x)$.

2. Set $f''(x) = 0$ and solve for $x$ to find possible inflection points.

3. Test intervals around these points to determine where $f''(x)$ is positive (concave up) or negative (concave down).

4. List the $x$-values where the concavity changes as inflection points.

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Part (c): Choose the letter with the graph of $f(x)$.

Background

Topic: Graph Matching

This part asks you to match the function $f(x) = 2x^3 + 3x^2 - 12x$ to its graph based on the analysis from parts (a) and (b).

Step-by-Step Guidance

1. Recall the key features of $f(x)$: degree (cubic), end behavior, and the number and location of turning points and inflection points.

2. Compare these features to the graphs provided (A, B, C, D).

3. Look for the graph that matches the increasing/decreasing intervals, concavity, and intercepts you found earlier.

Try solving on your own before revealing the answer!