AP Calculus AB Rates of Change in Context and Straight-Line Motion Quiz: 10 Questions with Explanations

A At 4 seconds, the object is 6 feet to the left of the originB At 4 seconds, the object speeds up at 6 feet per second squaredC At 4 seconds, the object moves left at 6 feet per secondD At 4 seconds, the object has moved 6 feet in total

The derivative of position is velocity. The negative sign gives direction (left) and the units are feet per second.

A \(0\)B \(-4\)C \(12\)D \(4\)

v = 4t^3 – 8t and a = 12t^2 – 8, so a(1) = 4. The value -4 is the velocity at t = 1.

A it changes direction from right to leftB it changes direction from left to rightC it keeps moving leftD it speeds up

Velocity changes from negative to positive at t = 2, so the particle turns from left to right.

A it is increasingB it is constantC it is decreasingD it cannot be determined

v and a have opposite signs, so the particle is slowing down.

A \(7\)B \(-7\)C \(0\)D \(\frac{1}{7}\)

Speed is the absolute value of velocity, so it is 7.

A \(24\text{ ft/s}\)B \(7\text{ ft/s}\)C \(9\text{ ft/s}\)D \(12\text{ ft/s}\)

(27 – 3)/(3 – 1) = 12. Forgetting to divide by the elapsed time gives 24.

A \(6\pi\)B \(9\pi\)C \(3\pi\)D \(2\pi\)

A′ = 2 pi r, so A′(3) = 6 pi. The value 9 pi is A(3).

A \(t=2\)B \(t=0\)C \(t=4\)D \(t=1\)

v = 0 at t = 2 and v changes from positive to negative, so the particle turns around at t = 2.

A pencils per dollarB dollars per pencilC dollarsD pencils

A derivative has units of output per input, here dollars per pencil.

A \(2t-4\)B \(2\)C \(0\)D \(t^{2}\)

v = 2t – 4 and a = v′ = 2, a constant. The expression 2t – 4 is the velocity.