Unit 7 Practice Test

Differential Equations - 15 Multiple Choice + 6 Free Response Questions

⏱️ Test Timer
00:00

Recommended time: 45 minutes

📋 Test Instructions

💡 Important: This practice test covers all topics from Unit 7: Differential Equations. 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
Solve the separable differential equation: dy/dx = 2xy
2
What is the general solution to dy/dx = y?
3
Use Euler's method with h = 0.5 to approximate y(1) for dy/dx = x + y, y(0) = 1
4
A population grows exponentially. If it doubles every 4 hours, what is the growth constant k?
5
Solve dy/dx = x/y with initial condition y(0) = 2
6
What is the half-life of a radioactive substance that decays according to dN/dt = -0.1N?
7
For the slope field of dy/dx = x - y, what is the slope at the point (1,1)?
8
Solve dy/dx = 3x²y with y(0) = 1
9
A bacteria culture grows at a rate proportional to its size. If it triples in 6 hours, what is the growth rate constant?
10
Use Euler's method with h = 0.25 to approximate y(0.5) for dy/dx = 2y, y(0) = 1
11
Solve dy/dx = y/x with y(1) = 2
12
For the logistic equation dy/dt = 0.1y(100 - y), what is the carrying capacity?
13
Solve dy/dx = e^x with y(0) = 0
14
A radioactive substance has a half-life of 5 years. What percentage remains after 15 years?
15
For the slope field of dy/dx = y - x, what is the slope at (0,0)?

✍️ Section B: Free Response (6 questions)

1
Solve the separable differential equation dy/dx = x²y with initial condition y(0) = 1.

Solution:

1) Separate variables: dy/y = x² dx

2) Integrate both sides: ∫dy/y = ∫x² dx

3) ln|y| = x³/3 + C

4) y = e^(x³/3 + C) = Ce^(x³/3)

5) Use y(0) = 1: 1 = Ce^0 = C, so C = 1

Answer: y = e^(x³/3)

2
Use Euler's method with step size h = 0.5 to approximate y(1) for dy/dx = x + y, y(0) = 1.

Solution:

1) Start: (x₀, y₀) = (0, 1)

2) Step 1: x₁ = 0.5, y₁ = 1 + 0.5(0 + 1) = 1 + 0.5 = 1.5

3) Step 2: x₂ = 1.0, y₂ = 1.5 + 0.5(0.5 + 1.5) = 1.5 + 1.0 = 2.5

Answer: y(1) ≈ 2.5

3
A bacteria culture grows exponentially. If it doubles every 3 hours and starts with 100 bacteria, find the population after 9 hours.

Solution:

1) P(t) = P₀e^(kt) where P₀ = 100

2) P(3) = 200 = 100e^(3k), so e^(3k) = 2

3) k = ln(2)/3

4) P(9) = 100e^(9·ln(2)/3) = 100e^(3ln(2)) = 100·2³ = 800

Answer: 800 bacteria

4
Solve dy/dx = y/x with initial condition y(1) = 3.

Solution:

1) Separate variables: dy/y = dx/x

2) Integrate both sides: ∫dy/y = ∫dx/x

3) ln|y| = ln|x| + C

4) y = e^(ln|x| + C) = Cx

5) Use y(1) = 3: 3 = C(1), so C = 3

Answer: y = 3x

5
A radioactive substance decays at a rate proportional to its mass. If 25% remains after 10 years, find the half-life.

Solution:

1) M(t) = M₀e^(-kt) where M₀ is initial mass

2) M(10) = 0.25M₀ = M₀e^(-10k), so e^(-10k) = 0.25

3) -10k = ln(0.25) = ln(1/4) = -ln(4), so k = ln(4)/10

4) For half-life: 0.5M₀ = M₀e^(-kt), so t = ln(2)/k = ln(2)/(ln(4)/10) = 10ln(2)/ln(4) = 5 years

Answer: 5 years

6
For the logistic equation dy/dt = 0.2y(50 - y), find the carrying capacity and describe the long-term behavior.

Solution:

1) The logistic equation is dy/dt = ky(L - y) where L is carrying capacity

2) Comparing: 0.2y(50 - y) = ky(L - y), so k = 0.2 and L = 50

3) Carrying capacity = 50

4) Long-term behavior: As t → ∞, y → 50 (approaches carrying capacity)

Answer: Carrying capacity = 50, y approaches 50 as t → ∞

📊 Test Results

📚 Study Recommendations

If you scored 80%+:
  • Great job! You have a solid foundation in differential equations
  • Focus on the few topics you missed
  • Move on to Unit 8: Applications of Integration
  • Practice more challenging differential equation problems
If you scored below 80%:
  • Review separable differential equations
  • Practice Euler's method calculations
  • Focus on exponential growth and decay
  • Use the main Unit 7 page for review