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

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 1–30 of 96.

The value of $i^3$ is:Read the answer and worked solution →For the complex number $z = 2 + 5i$, the imaginary part $\operatorname{Im}(z)$ is:Read the answer and worked solution →The complex conjugate of $3 - 4i$ is:Read the answer and worked solution →The modulus $|3 + 4i|$ equals:Read the answer and worked solution →The multiplicative inverse of $i$ (i.e. $1/i$) is:Read the answer and worked solution →The value of $i^{10}$ is:Read the answer and worked solution →The value of $(1 + i)^2$ is:Read the answer and worked solution →The roots of the equation $x^2 + 1 = 0$ are:Read the answer and worked solution →The modulus $|1 - i|$ equals:Read the answer and worked solution →The product $(2 + 3i)(2 - 3i)$ equals:Read the answer and worked solution →The value of $i + i^2 + i^3 + i^4$ is:Read the answer and worked solution →If $z = a + ib$ is non-zero, its multiplicative inverse $z^{-1}$ is:Read the answer and worked solution →A student differentiates f(x) = x^5 using the power rule. The derivative is:Read the answer and worked solution →A cost function is the constant f(x) = 7. Its derivative is:Read the answer and worked solution →For f(x) = 3x^2, a student pulls the 3 out and differentiates the rest. The derivative is:Read the answer and worked solution →If f(x) = x^3 + x, then f'(x) equals:Read the answer and worked solution →A function is written as one function multiplied by another. The rule to reach for is the:Read the answer and worked solution →A student guesses the derivative of a product is simply the product of the derivatives. Is that right?Read the answer and worked solution →A function appears as one function divided by another. The rule to use is the:Read the answer and worked solution →A student meets a composition, with one function sitting inside another. The rule for that is the:Read the answer and worked solution →Differentiating f(x) = (2x + 1)^3 with the chain rule gives:Read the answer and worked solution →A physics student needs the derivative of sin x. It is:Read the answer and worked solution →The same student now needs the derivative of cos x. It is:Read the answer and worked solution →A function is known to be differentiable at a point. What follows about continuity there?Read the answer and worked solution →Another function is continuous at a point. Must it also be differentiable there?Read the answer and worked solution →A student differentiates the derivative one more time. The result is the:Read the answer and worked solution →Third and fourth derivatives, and those beyond, are known together as:Read the answer and worked solution →Velocity is the rate of change of position. The rate of change of velocity is:Read the answer and worked solution →An equation cannot be solved neatly for y, so a student differentiates both sides as they stand. That method is:Read the answer and worked solution →Knowing the derivative of a function, a student can also compute the derivative of its:Read the answer and worked solution →

Worked practice questions

The value of $i^3$ is:

  1. $i$
  2. $-1$
  3. $-i$
  4. $1$
Show answer and solution

Correct answer: C. $-i$

We have i³ = i²·i. That is i³ = (−1)i. So i³ = −i.

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For the complex number $z = 2 + 5i$, the imaginary part $\operatorname{Im}(z)$ is:

  1. 5
  2. $5i$
  3. $-5$
  4. 2
Show answer and solution

Correct answer: A. 5

For z = a + ib, the imaginary part is b; here Im(z) = 5.

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The complex conjugate of $3 - 4i$ is:

  1. $-3 - 4i$
  2. $4 - 3i$
  3. $3 + 4i$
  4. $-3 + 4i$
Show answer and solution

Correct answer: C. $3 + 4i$

The conjugate of a + ib is a − ib, so conj(3 − 4i) = 3 + 4i.

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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