Question 1

Suppose an object moves along a line with position $$s(t). If$$ $$v(t) = s'(t)$$ and $$a(t) = v'(t)$$, which statement is true about the object's motion when $$v(t)$$ and $$a(t)$$ have opposite signs?

Accepted Answer

The object is decelerating.

Question 2

Given the position function $$s(t) = 4t^2$ - 9t$$, what is the velocity and acceleration of the object at $$t = 3$$?

Accepted Answer

$$v(3) = 15$$, $$a(3) = 8$$

Question 3

A particle has position $$s(t) = t^3$ - 6t^2$ + 9t (in $$feet) at time $$t (in $$seconds). At what times is the particle at rest?

Accepted Answer

$$t = 1 or$$ $$t = 3$$

Question 4

Given $$s(t) = 12t - 3t^2$, find the velocity and acceleration at t = 1$.

Accepted Answer

$$v(1) = 6 ft/s$$, $$a(1) = -6 ft/s$^2$$

Question 5

An object is dropped from a tower. Its height in feet after $$t$$ seconds is $$s(t) = 1024 - 16t^2$. How long does it take to hit the ground?

Accepted Answer

$$t = 8 s$$

Question 6

For the function $$s(t) = t^3$ - 6t^2$ + 9t$$, what is the change in position of the particle over the first 5 seconds?

Accepted Answer

$$20 ft$$

Question 7

Given $$s(t) = 12t - 3t^2$, on what interval is the speed of the particle increasing?

Accepted Answer

$$t \u003e 2$$

Question 8

An object is thrown vertically upward from the ground at a velocity of $$42 m/s$$, reaching a height $$s(t) = -0.7t^2$ + 42t$$ meters. When does the object reach its highest point?

Accepted Answer

$$t = 30 s$$

Question 9

For the object in the previous question, what is the velocity when it strikes the ground?

Accepted Answer

$$-42 m/s$$

Question 10

A car moves along a straight road with position $$s(t). If$$ $$v(t) \u003c 0$$ and $$a(t) \u003c 0$$, what is happening to the car?

Accepted Answer

It is accelerating in the negative direction.

Question 11

Given $$s(t) = 1024 - 16t^2$, what is the velocity and acceleration of the object when it hits the ground?

Accepted Answer

$$v(8) = -256 ft/s$, a(8) = -32 ft/s$^2$$

Question 12

A particle's position is given by $$s(t) = t^3$ - 6t^2$ + 9t. At $$what times is the particle moving forward?

Accepted Answer

$$t  3$$

Question 13

If $$v(t) = s'(t)$$ and $$a(t) = v'(t)$$, which of the following best describes speed?

Accepted Answer

$$|v(t)|$$

Question 14

A ball is thrown upward with position $$s(t) = -0.7t^2$ + 42t. $$What is the height of the ball at its highest point?

Accepted Answer

$$630 m$$

Question 15

Given $$s(t) = 4t^2$ - 9t$$, what is the general formula for velocity $$v(t)$$?

Accepted Answer

$$v(t) = 8t - 9$$

Question 16

A particle's position is $$s(t) = 12t - 3t^2$. What is the acceleration at any time t$?

Accepted Answer

$$a(t) = -6$$

Question 17

An object is thrown upward with $$s(t) = -0.7t^2$ + 42t. At $$what time does it strike the ground?

Accepted Answer

$$t = 60 s$$

Question 18

If $$v(t) \u003e 0$$, what is the direction of motion of the object?

Accepted Answer

Right or up

Question 19

A particle moves along a line with $$s(t) = t^3$ - 6t^2$ + 9t. $$What is the velocity function $$v(t)$$?

Accepted Answer

$$v(t) = 3t^2$ - 12t + 9$$

Derivatives as Rates of Change 3.6

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