Skip to content

Quiz: Higher-Order Derivatives and Motion

Test your understanding of higher-order derivatives and motion with these review questions.

1. The second derivative f''(x) represents:

  1. The slope of f(x)
  2. The rate of change of the rate of change
  3. The area under f(x)
  4. The average rate of change
Show Answer

The correct answer is B. The second derivative is the derivative of the derivative, representing how the rate of change itself is changing. In physics, this corresponds to acceleration.

Concept Tested: Second Derivative

2. If position is s(t), then velocity v(t) equals:

  1. s''(t)
  2. s'(t)
  3. ∫s(t)dt
  4. s(t)/t
Show Answer

The correct answer is B. Velocity is the derivative of position: v(t) = s'(t). This gives the instantaneous rate of change of position with respect to time.

Concept Tested: Velocity from Position

3. Acceleration a(t) is defined as:

  1. The first derivative of position
  2. The second derivative of position (or first derivative of velocity)
  3. The integral of position
  4. Position divided by time squared
Show Answer

The correct answer is B. Acceleration is the derivative of velocity, which makes it the second derivative of position: a(t) = v'(t) = s''(t).

Concept Tested: Acceleration

4. If s(t) = t³ − 6t², what is the acceleration at t = 2?

  1. −12
  2. 0
  3. 6
  4. 12
Show Answer

The correct answer is B. First, v(t) = s'(t) = 3t² − 12t. Then a(t) = v'(t) = 6t − 12. At t = 2: a(2) = 6(2) − 12 = 0.

Concept Tested: Position Velocity Acceleration

5. The third derivative is sometimes called:

  1. Snap
  2. Crackle
  3. Jerk
  4. Pop
Show Answer

The correct answer is C. The third derivative of position (derivative of acceleration) is called jerk. It measures how quickly acceleration is changing.

Concept Tested: Jerk

6. What is the difference between speed and velocity?

  1. They are the same thing
  2. Speed is the absolute value of velocity
  3. Velocity is the absolute value of speed
  4. Speed is the derivative of velocity
Show Answer

The correct answer is B. Speed is the magnitude (absolute value) of velocity: speed = |v(t)|. Velocity can be positive or negative (indicating direction), but speed is always non-negative.

Concept Tested: Speed vs Velocity

7. An object is speeding up when:

  1. Velocity is positive
  2. Acceleration is positive
  3. Velocity and acceleration have the same sign
  4. Velocity and acceleration have opposite signs
Show Answer

The correct answer is C. An object speeds up when velocity and acceleration have the same sign (both positive or both negative). When they have opposite signs, the object is slowing down.

Concept Tested: Speeding Up Slowing Down

8. If f(x) = sin(x), what is f'''(x)?

  1. sin(x)
  2. cos(x)
  3. −sin(x)
  4. −cos(x)
Show Answer

The correct answer is D. Starting with sin(x): f'(x) = cos(x), f''(x) = −sin(x), f'''(x) = −cos(x). The pattern continues with period 4.

Concept Tested: Higher Trig Derivatives

9. The notation d²y/dx² represents:

  1. (dy/dx)²
  2. The second derivative
  3. d(y²)/dx
  4. Twice the first derivative
Show Answer

The correct answer is B. The notation d²y/dx² is Leibniz notation for the second derivative. It means d/dx(dy/dx), the derivative of the derivative.

Concept Tested: Second Derivative Notation

10. A particle moves along a line with position s(t) = t⁴ − 4t³. When is the particle at rest?

  1. t = 0 only
  2. t = 3 only
  3. t = 0 and t = 3
  4. t = 0, t = 1, and t = 3
Show Answer

The correct answer is C. At rest means velocity = 0. v(t) = s'(t) = 4t³ − 12t² = 4t²(t − 3). Setting v(t) = 0: t = 0 or t = 3.

Concept Tested: Direction of Motion