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AP Calculus AB Unit 3A: Chain Rule and Composite Functions — practice questions — page 4

Explore 96 free Unit 3A: Chain Rule and Composite Functions questions in AP Calculus AB Mathematics. Open each question or exercise to read its content, reference answers and available explanation.

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Unit 3A: Chain Rule and Composite Functions questions and exercises with answers

Showing questions 91–96 of 96.

General solution of cos θ = 0:Read the answer and worked solution →If sin θ = 12/13 (acute), find tan(θ/2):Read the answer and worked solution →In triangle with sides 7, 8, 9, find cos of the angle opposite to side 9 (use cosine rule):Read the answer and worked solution →General solution of sin θ = sin α:Read the answer and worked solution →If tan θ + cot θ = 2, find sin 2θ:Read the answer and worked solution →In which quadrant is sin θ positive and cos θ negative?Read the answer and worked solution →

Worked practice questions

General solution of cos θ = 0:

  1. θ = π/2 + nπ
  2. θ = nπ
  3. θ = π/4 + nπ
  4. θ = 0 + 2nπ
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Correct answer: A. θ = π/2 + nπ

cos θ = 0 at θ = π/2 and 3π/2 (i.e., 90° and 270°). **General:** θ = π/2 + nπ where n ∈ ℤ. (Every π apart.)

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If sin θ = 12/13 (acute), find tan(θ/2):

  1. 5/12
  2. Cannot determine
  3. 1/5
  4. 2/3
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Correct answer: C. 1/5

For acute θ with sinθ = 12/13: cosθ = 5/13 (positive in Q1). tan(θ/2) = sinθ/(1 + cosθ) = (12/13)/(1 + 5/13) = (12/13)/(18/13) = 12/18 = 2/3. Hmm answer A is 1/5 my calc gives 2/3. Let me recheck. tan(θ/2) = (1 - cosθ)/sinθ = (1 - 5/13)/(12/13) = (8/13)/(12/13) = 8/12 = 2/3. So actually 2/3 is correct, which matches option C, not A. Setting answer to C.

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In triangle with sides 7, 8, 9, find cos of the angle opposite to side 9 (use cosine rule):

  1. 11/16
  2. 1/2
  3. 2/7
  4. 1/7
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Correct answer: C. 2/7

**Cosine rule:** c² = a² + b² - 2ab cosC. Here c = 9, a = 7, b = 8. 81 = 49 + 64 - 2(7)(8)cosC = 113 - 112 cosC. **So 112 cosC = 113 - 81 = 32, cosC = 32/112 = 2/7.**

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More Mathematics topics

Unit 3B: Implicit Differentiation and Inverse FunctionsUnit 1A3: Function Composition and InversesUnit 1A1: Relations, Functions and GraphsUnit 1A2: Trigonometric Functions & Inverse TrigUnit 2: Differentiation and Fundamental Derivative RulesUnit 6A: Indefinite Integrals & Fundamental TheoremUnit 1B3: Asymptotic Behavior and Infinite LimitsUnit 1B1: Limit Concepts and Graphical Limits