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AP Calculus AB Unit 7: Differential Equations and Modeling — practice questions — page 3

Explore 65 free Unit 7: Differential Equations and Modeling 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 7: Differential Equations and Modeling questions and exercises with answers

Showing questions 61–65 of 65.

Value of arctan(1):Read the answer and worked solution →Simplify arctan(sin x / (1+cos x)) for x ∈ (-π, π):Read the answer and worked solution →arccos(x) + arccos(-x) =Read the answer and worked solution →A matrix with all elements zero is called:Read the answer and worked solution →Mode of {1, 2, 2, 3, 4, 4, 4, 5}:Read the answer and worked solution →

Worked practice questions

Value of arctan(1):

  1. π/4
  2. 0
  3. π/2
  4. π
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Correct answer: A. π/4

tan(π/4) = 1, so arctan(1) = π/4.

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Simplify arctan(sin x / (1+cos x)) for x ∈ (-π, π):

  1. x²
  2. x/2 (use half-angle: sin x/(1+cos x) = tan(x/2))
  3. x
  4. arctan(x)
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Correct answer: B. x/2 (use half-angle: sin x/(1+cos x) = tan(x/2))

**Half-angle:** sin x = 2 sin(x/2) cos(x/2), 1 + cos x = 2 cos²(x/2). Ratio = sin(x/2)/cos(x/2) = tan(x/2). For x/2 ∈ (-π/2, π/2), arctan(tan(x/2)) = x/2.

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arccos(x) + arccos(-x) =

  1. x
  2. π (since arccos(-x) = π - arccos(x))
  3. 0
  4. 2 arccos(x)
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Correct answer: B. π (since arccos(-x) = π - arccos(x))

arccos(-x) = π - arccos(x) (reflection identity). Sum = arccos(x) + π - arccos(x) = π.

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More Mathematics topics

Unit 3B: Implicit Differentiation and Inverse FunctionsUnit 3A: Chain Rule and Composite 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 Limits