A particle moves according to a law of motion

s = f(t), t ≥ 0, where *t* is measured in seconds and *s* in feet. (If an answer does not exist, enter DNE.)

f(t) = t^{3} − 9t^{2} + 24t

1) Draw a diagram to illustrate the motion of the particle and use it to find the total distance (in ft) traveled during the first 6 seconds.

2) Find the acceleration (in ft/s^{2}) at time *t *and after 1 second.

3) When is the particle speeding up and slowing down? (Enter your answer using interval notation.)

A particle moves according to a law of motion *s* = *f*(*t*) = *t*^{3} – 12*t*^{2} + 36*t*, *t* 0, where *t* is measured in seconds and *s* in feet.

(a) Find the velocity at time *t*.

*v*(*t*) =

ft/s

(b) What is the velocity after 5 s?*v*(5) = ft/s

(c) When is the particle at rest?*t* = s (smaller value)*t* = s (larger value)

(d) When is the particle moving in the positive direction? (Enter your answers in ascending order. If you need to use -∞ or ∞, enter -INFINITY or INFINITY.)

( , ) ∪ ( , )

(e) Find the total distance traveled during the first 7 s.

feet

(f) Draw a diagram to illustrate the motion of the particle. (Do this on paper. Your instructor may ask you to turn in this graph.)

(g) Find the acceleration at time *t* and after 5 s.*a*(*t*) =

*a*(5) = ft/s^{2}

(h) Graph the position, velocity, and acceleration functions for 0 *t* 7. (Do this on paper. Your instructor may ask you to turn in this graph.)

(i) When is the particle speeding up? (Enter your answers in ascending order. If you need to use -∞ or ∞, enter -INFINITY or INFINITY.)

A particle moves according to a law of motion *s* = *f*(*t*) = 0.01*t*^{4} – 0.08*t*^{3}, *t* > 0, where *t* is measured in seconds and *s* in feet.

(a) Find the velocity at time *t*.

*v*(*t*) =

ft/s

(b) What is the velocity after 3 s?*v*(3) = ft/s

(c) When is the particle at rest?*t* = s (first time)*t* = s (second time)

(d) When is the particle moving in the positive direction? (If you need to use -∞ or ∞, enter -INFINITY or INFINITY.)

( , )

(e) Find the total distance traveled during the first 8 s.

feet

(f) Draw a diagram to illustrate the motion of the particle. (Do this on paper. Your instructor may ask you to turn in this graph.)

(g) Find the acceleration at time *t* and after 3 s.*a*(*t*) =

*a*(3) = ft/s^{2}

(h) Graph the position, velocity, and acceleration functions for 0 < *t* 8. (Do this on paper. Your instructor may ask you to turn in this graph.)

(i) When is the particle speeding up? (If you need to use -∞ or ∞, enter -INFINITY or INFINITY.)

A particle moves according to a law of motion *s* = *f*(*t*), *t* ≥ 0, where *t* is measured in seconds and *s* in feet.

f(t) = 0.01t^{4} − 0.03t^{3}

(a) Find the velocity at time *t* (in ft/s).

*v*(*t*) =

(b) What is the velocity after 1 second(s)?*v*(1) = ft/s

(c) When is the particle at rest?

t = s (smaller value) |

t = s (larger value) |

(d) When is the particle moving in the positive direction? (Enter your answer using interval notation.)

(e) Find the total distance traveled during the first 11 seconds. (Round your answer to two decimal places.)

ft

(f) Find the acceleration at time *t* (in ft/s^{2}).

a(t) =

Find the acceleration after 1 second(s).

a(1) = ft/s^{2}

(g) Graph the position, velocity, and acceleration functions for the first 11 seconds.

(h) When, for

0 ≤ t < ∞,

is the particle speeding up? (Enter your answer using interval notation.)

When, for

0 ≤ t < ∞,

is it slowing down? (Enter your answer using interval notation.)

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