Unit 4 Practice Test

Contextual Applications of Differentiation - 15 Multiple Choice + 6 Free Response Questions

← Back to Unit 4
⏱️ Test Timer
00:00

Recommended time: 45 minutes

📋 Test Instructions

💡 Important: This practice test covers all topics from Unit 4: Contextual Applications of Differentiation. Take your time and show all your work for free-response questions.
  • Multiple Choice: Select the best answer for each question
  • Free Response: Show all steps and justify your reasoning
  • Time Limit: 45 minutes recommended (AP exam timing)
  • Calculator: Graphing calculator allowed for some questions

📝 Section A: Multiple Choice (15 questions)

1
If s(t) represents the position of a particle at time t, what does s'(t) represent?
2
A spherical balloon is being inflated at a rate of 10 cm³/s. How fast is the radius increasing when the radius is 5 cm?
3
A ladder 10 feet long rests against a vertical wall. If the bottom slides away at 1 ft/s, how fast is the top sliding when the bottom is 6 feet from the wall?
4
Find the critical points of f(x) = x³ - 3x² + 2 on the interval [-1, 3].
5
A rectangular box with a square base has volume 1000 cm³. What dimensions minimize the surface area?
6
If f(x) = x³ - 6x² + 9x + 1, which of the following is true about x = 1?
7
A conical tank is being filled with water at 2 m³/min. If the height is twice the radius, how fast is the water level rising when the height is 4 m?
8
For f(x) = x⁴ - 8x² + 16, find the absolute maximum on [0, 3].
9
A particle moves along the x-axis with position s(t) = t³ - 6t² + 9t. When is the particle moving to the right?
10
Find the value of c guaranteed by the Mean Value Theorem for f(x) = x² on [1, 4].
11
A rectangle is inscribed in a semicircle of radius 2. What is the maximum area of such a rectangle?
12
If the temperature T(t) of an object changes at a rate of T'(t) = -0.1t + 2 °C/min, what is the temperature change from t = 0 to t = 10?
13
A snowball melts so that its surface area decreases at a rate of 1 cm²/min. How fast is the radius decreasing when the radius is 3 cm?
14
Find the absolute minimum value of f(x) = x + 1/x on (0, ∞).
15
Two cars start from the same point. One travels north at 60 mph and the other travels east at 80 mph. How fast is the distance between them increasing after 2 hours?

✍️ Section B: Free Response (6 questions)

1
A spherical balloon is being inflated at a rate of 20 cm³/s. How fast is the radius increasing when the radius is 4 cm?

Solution:

1) We want dr/dt when r = 4

2) Volume of sphere: V = (4/3)πr³

3) Differentiate with respect to t: dV/dt = 4πr² · dr/dt

4) Substitute known values: 20 = 4π(4)² · dr/dt

5) Simplify: 20 = 64π · dr/dt

6) Solve for dr/dt: dr/dt = 20/(64π) = 5/(16π) cm/s

Answer: 5/(16π) cm/s

2
Find the dimensions of a rectangle with perimeter 20 cm that has maximum area.

Solution:

1) Let x = length, y = width

2) Perimeter: 2x + 2y = 20, so y = 10 - x

3) Area: A = xy = x(10 - x) = 10x - x²

4) Find critical points: A' = 10 - 2x = 0

5) Solve: x = 5, so y = 10 - 5 = 5

6) Check second derivative: A'' = -2 < 0, so maximum

7) Maximum area occurs when rectangle is a square

Answer: 5 cm × 5 cm

3
A ladder 13 feet long rests against a vertical wall. If the bottom slides away at 2 ft/s, how fast is the top sliding when the bottom is 5 feet from the wall?

Solution:

1) We want dy/dt when x = 5

2) From Pythagorean theorem: x² + y² = 169

3) When x = 5: 25 + y² = 169, so y = 12

4) Differentiate: 2x(dx/dt) + 2y(dy/dt) = 0

5) Substitute: 2(5)(2) + 2(12)(dy/dt) = 0

6) Simplify: 20 + 24(dy/dt) = 0

7) Solve: dy/dt = -20/24 = -5/6 ft/s

8) The top is sliding down at 5/6 ft/s

Answer: -5/6 ft/s (sliding down)

4
Find all critical points of f(x) = x⁴ - 8x² + 16 and classify each as a local maximum, local minimum, or neither.

Solution:

1) Find f'(x) = 4x³ - 16x = 4x(x² - 4) = 4x(x - 2)(x + 2)

2) Set f'(x) = 0: x = 0, x = 2, x = -2

3) Find f''(x) = 12x² - 16

4) Test each critical point:

- f''(-2) = 12(4) - 16 = 32 > 0 → local minimum

- f''(0) = -16 < 0 → local maximum

- f''(2) = 12(4) - 16 = 32 > 0 → local minimum

Answer: x = -2 (local min), x = 0 (local max), x = 2 (local min)

5
A rectangular box with a square base and open top has volume 32 cm³. Find the dimensions that minimize the surface area.

Solution:

1) Let x = side of square base, h = height

2) Volume: V = x²h = 32, so h = 32/x²

3) Surface area: S = x² + 4xh = x² + 4x(32/x²) = x² + 128/x

4) Find critical points: S' = 2x - 128/x² = 0

5) Solve: 2x = 128/x², so 2x³ = 128, x³ = 64, x = 4

6) Find h: h = 32/4² = 32/16 = 2

7) Check second derivative: S'' = 2 + 256/x³ > 0 for x > 0, so minimum

Answer: 4 cm × 4 cm × 2 cm

6
Two cars start from the same point. One travels north at 50 mph and the other travels east at 30 mph. How fast is the distance between them increasing after 3 hours?

Solution:

1) Let x = distance east, y = distance north, z = distance between cars

2) From Pythagorean theorem: z² = x² + y²

3) After 3 hours: x = 30(3) = 90, y = 50(3) = 150

4) So z = √(90² + 150²) = √(8100 + 22500) = √30600 = 30√34

5) Differentiate: 2z(dz/dt) = 2x(dx/dt) + 2y(dy/dt)

6) Substitute: 2(30√34)(dz/dt) = 2(90)(30) + 2(150)(50)

7) Simplify: 60√34(dz/dt) = 5400 + 15000 = 20400

8) Solve: dz/dt = 20400/(60√34) = 340/√34 mph

Answer: 340/√34 mph

📊 Test Results

📚 Study Recommendations

If you scored 80%+:
  • Great job! You have a solid foundation
  • Focus on the few topics you missed
  • Move on to Unit 5: Analytical Applications
  • Practice more challenging problems
If you scored below 80%:
  • Review the concepts you struggled with
  • Practice more related rates problems
  • Focus on optimization techniques
  • Use the main Unit 4 page for review