Q: Which of the following best describes the instantaneous rate of change of a function $$f at$$ $$x_0$$?

The limit $$\displaystyle \lim_{h \to 0} \frac{f(x_0$ + h) - f(x_0$)}{h}$$, provided the limit exists.

Question 1

Which of the following best describes the instantaneous rate of change of a function $$f at$$ $$x_0$$?

Accepted Answer

The limit $$\displaystyle \lim_{h 	o 0} \frac{f(x_0$ + h) - f(x_0$)}{h}$$, provided the limit exists.

Question 2

Given the position function $$s(t) = f(t)$$ for a moving object, which expression gives the instantaneous velocity at time $$t$$?

Accepted Answer

$$\displaystyle \frac{ds}{dt}$$

Question 3

A car’s position is given by $$s(t) = 3t^2$ + 2t + 1. $$What is the car’s acceleration at $$t = 2$$ seconds?

Accepted Answer

$$6$$

Question 4

If the velocity of a particle is negative, what does this indicate about the particle’s motion?

Accepted Answer

The particle is moving backward.

Question 5

Which of the following correctly defines acceleration in terms of derivatives?

Accepted Answer

$$a(t) = \displaystyle \frac{d^2$ s}{dt^2$}$$

Question 6

A rock is thrown upward on the moon with initial velocity $$24 m/s. $$Its height is $$s(t) = 24t - 0.8t^2$. How long does it take to reach its highest point?

Accepted Answer

$$15$$ seconds

Question 7

Given $$v(t) = \frac{ds}{dt}$$, when is the particle speeding up?

Accepted Answer

When $$v(t)$$ and $$a(t)$$ have the same sign.

Question 8

If $$s(t) = t^3$ - 6t^2$ + 9t$$, at what time does the particle change direction?

Accepted Answer

$$t = 1$$ and $$t = 3$$

Question 9

Which of the following statements is true about the jerk $$j(t) of a $$moving object?

Accepted Answer

It is the derivative of acceleration with respect to time.

Question 10

A particle’s velocity is given by $$v(t) = 4 - 2t. At $$what time does the particle come to rest?

Accepted Answer

$$t = 2$$

Question 11

If a particle’s velocity graph crosses the $$t-$$axis, what does this indicate?

Accepted Answer

The particle changes direction.

Question 12

Suppose $$s(t) = 5t^2$ - 2t + 3. $$What is the average velocity on the interval $$[1, 3]$$?

Accepted Answer

$$16$$

Question 13

A car’s velocity is $$v(t) = 10 - t^2$. When is the acceleration positive?

Accepted Answer

Never, since $$a(t) = -2t is $$negative for $$t \u003e 0$$

Question 14

If $$s(t) is $$the position of a particle, which of the following is the correct interpretation of $$\displaystyle \frac{ds}{dt}$$?

Accepted Answer

The particle’s velocity.

Question 15

A particle moves along a line so that $$s(t) = 2t^3$ - 9t^2$ + 12t. $$What is the acceleration at $$t = 2$$?

Accepted Answer

$$0$$

Question 16

Which of the following best describes when a particle is slowing down?

Accepted Answer

When velocity and acceleration have opposite signs.

Section 3.4: The Derivative as a Rate of Change – Motion, Velocity, and Acceleration

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