Q1. You need to construct a bookcase with a frame that holds 3 interior shelves and encloses an area of 18 square feet. The wood used for the frame costs $3 per foot and the wood used for the shelves costs $2 per foot. Find the dimensions that minimize the total cost of building material. Justify your answer with the first or second derivative test.

Background

Topic: Optimization (Applications of Derivatives)

This problem tests your ability to set up and solve an optimization problem involving cost, using calculus to find minimum values and justify them with derivative tests.

Key Terms and Formulas:

• Critical Number: A value where the derivative is zero or undefined, and a possible location for extrema.

• First Derivative Test: Determines if a critical point is a minimum or maximum by analyzing sign changes of the derivative.

• Second Derivative Test: Uses the sign of the second derivative at a critical point to determine concavity and classify extrema.

• Area Constraint: (if is base, is height)

• Cost Function: Express total cost in terms of and using the given costs per foot.

Step-by-Step Guidance

1. Let be the base and be the height of the bookcase. Write the area constraint: .

2. Express the total cost in terms of and . The frame consists of the perimeter (2 sides and top/bottom), and there are 3 interior shelves. Assign the correct cost per foot to each part.

3. Use the area constraint to solve for in terms of (or vice versa), and substitute into your cost function so that is a function of a single variable.

4. Take the derivative of with respect to , set it equal to zero, and solve for the critical number(s).

5. Set up the second derivative or use the first derivative test to justify that your critical number gives a minimum cost.

Try solving on your own before revealing the answer!

Q2. An open-top cylindrical can has volume cm. Find the minimal surface area of the can.

Background

Topic: Optimization (Applications of Derivatives)

This question asks you to minimize the surface area of a cylinder with a fixed volume, a classic calculus optimization problem.

Key Terms and Formulas:

• Volume of Cylinder:

• Surface Area (open-top):

• Constraint:

Step-by-Step Guidance

1. Write the constraint equation: .

2. Solve for in terms of using the constraint, and substitute into the surface area formula so is a function of $r$ only.

3. Take the derivative of with respect to , set it equal to zero, and solve for the critical value(s) of $r$.

4. Use the second derivative test or analyze the sign of the first derivative to confirm that this value of gives a minimum surface area.

Try solving on your own before revealing the answer!

Q3. A closed box is to be made out of 108 square feet of material. Its base must be a rectangle whose length is twice the width. Find the maximum volume that can be enclosed.

Background

Topic: Optimization (Applications of Derivatives)

This problem involves maximizing the volume of a box with a fixed surface area and a base with a specific length-to-width ratio.

Key Terms and Formulas:

• Let width = , length = , height = .

• Surface Area:

• Volume:

Step-by-Step Guidance

1. Write the surface area constraint in terms of and , and solve for $h$ in terms of $x$.

2. Substitute into the volume formula to express as a function of only.

3. Take the derivative of with respect to , set it equal to zero, and solve for the critical value(s) of $x$.

4. Use the second derivative test or analyze the sign of the first derivative to confirm that this value of gives a maximum volume.

Try solving on your own before revealing the answer!

Q4. A rectangle is to be inscribed in the region between the x-axis, the y-axis, and the curve in the first quadrant. Find the maximal area the rectangle can have, and the dimensions at which that area is achieved.

Background

Topic: Optimization (Applications of Derivatives)

This problem asks you to maximize the area of a rectangle inscribed under a parabola in the first quadrant.

Key Terms and Formulas:

• Rectangle's base: (along x-axis)

• Rectangle's height:

• Area:

Step-by-Step Guidance

1. Express the area as a function of using the given curve for the height.

2. Write .

3. Take the derivative of with respect to , set it equal to zero, and solve for the critical value(s) of $x$ in the first quadrant.

4. Use the second derivative test or analyze the sign of the first derivative to confirm that this value of gives a maximum area.

Try solving on your own before revealing the answer!

Q5. Find the derivative of , and then find the equation of the line tangent to the curve at .

Background

Topic: Implicit Differentiation and Tangent Lines

This question tests your ability to use implicit differentiation to find and then use the point-slope form to write the equation of a tangent line.

Key Terms and Formulas:

• Implicit Differentiation: Differentiate both sides of the equation with respect to , treating as a function of $x$.

• Tangent Line: , where is the slope at the point .

Step-by-Step Guidance

1. Differentiate both sides of with respect to , applying the product rule where necessary.

2. Collect all terms involving on one side and factor to solve for $\frac{dy}{dx}$.

3. Substitute and into your expression for to find the slope at the given point.

4. Use the point-slope form to write the equation of the tangent line at .

Try solving on your own before revealing the answer!