Unit 9 Practice Test

Parametric Equations, Polar Coordinates, and Vector-Valued Functions

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⏱️ Practice Test Instructions

⏰ Time Remaining: 45:00
📝 Instructions: This practice test contains 15 multiple-choice questions and 6 free-response questions. You have 45 minutes to complete the test. Use the timer to simulate real exam conditions. Check your answers as you go, and calculate your final score at the end.

📝 Multiple Choice Questions (15 questions)

1
Find dy/dx for the parametric equations x = t², y = t³ - 2t
2
Convert the polar equation r = 4sin(θ) to rectangular coordinates
3
Find the velocity vector for r(t) = ⟨t², 3t, sin(t)⟩
4
Find the area enclosed by the polar curve r = 2 + cos(θ)
5
Find the arc length of the parametric curve x = t², y = t³ from t = 0 to t = 2
6
Find the second derivative d²y/dx² for x = cos(t), y = sin(t)
Note: Questions 7-15 follow the same format. Complete all multiple choice questions before proceeding to free response.

✏️ Free Response Questions (6 questions)

1
Find the area enclosed by the polar curve r = 3sin(2θ)

Solution:

1) The curve r = 3sin(2θ) is a 4-petal rose

2) One petal occurs when 0 ≤ θ ≤ π/2

3) Area of one petal: A = (1/2)∫0π/2 [3sin(2θ)]² dθ

4) A = (9/2)∫0π/2 sin²(2θ) dθ

5) Using sin²(u) = (1-cos(2u))/2: A = (9/4)∫0π/2 (1-cos(4θ)) dθ

6) A = (9/4)[θ - sin(4θ)/4]0π/2 = (9/4)(π/2) = 9π/8

7) Total area = 4 × (9π/8) = 9π/2

Answer: 9π/2

2
Find the arc length of the parametric curve x = eᵗcos(t), y = eᵗsin(t) from t = 0 to t = π

Solution:

1) dx/dt = eᵗcos(t) - eᵗsin(t) = eᵗ(cos(t) - sin(t))

2) dy/dt = eᵗsin(t) + eᵗcos(t) = eᵗ(sin(t) + cos(t))

3) (dx/dt)² + (dy/dt)² = e²ᵗ[(cos(t) - sin(t))² + (sin(t) + cos(t))²]

4) = e²ᵗ[cos²(t) - 2cos(t)sin(t) + sin²(t) + sin²(t) + 2sin(t)cos(t) + cos²(t)]

5) = e²ᵗ[2cos²(t) + 2sin²(t)] = 2e²ᵗ

6) L = ∫0π √(2e²ᵗ) dt = √2 ∫0π eᵗ dt

7) L = √2[eᵗ]0π = √2(e^π - 1)

Answer: √2(e^π - 1)

Note: Questions 3-6 follow the same format. Complete all free response questions before calculating your final score.

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🎯 Study Recommendations: Review any topics you missed and retake this practice test. Focus on the areas where you struggled, and consider reviewing Unit 10: Infinite Sequences and Series next.