3.4 Derivatives as rates of change  (Page 4/15)

$s\left(t\right)=2t^3-3t^2-12t+8$

$s\left(t\right)=2t^3-15t^2+36t-10$

$s\left(t\right)=\frac{t}{1+t^2}$

A rocket is fired vertically upward from the ground. The distance $s$ in feet that the rocket travels from the ground after $t$ seconds is given by $s\left(t\right)=\mathrm{-16}{t}^2+560t.$

1. Find the velocity of the rocket 3 seconds after being fired.
2. Find the acceleration of the rocket 3 seconds after being fired.

A ball is thrown downward with a speed of 8 ft/s from the top of a 64-foot-tall building. After t seconds, its height above the ground is given by $s\left(t\right)=\mathrm{-16}{t}^2-8t+64.$

1. Determine how long it takes for the ball to hit the ground.
2. Determine the velocity of the ball when it hits the ground.

The position function $s\left(t\right)=t^2-3t-4$ represents the position of the back of a car backing out of a driveway and then driving in a straight line, where $s$ is in feet and $t$ is in seconds. In this case, $s\left(t\right)=0$ represents the time at which the back of the car is at the garage door, so $s\left(0\right)=\mathrm{-4}$ is the starting position of the car, 4 feet inside the garage.

1. Determine the velocity of the car when $s\left(t\right)=0.$
2. Determine the velocity of the car when $s\left(t\right)=14.$

The position of a hummingbird flying along a straight line in $t$ seconds is given by $s\left(t\right)=3t^3-7t$ meters.

1. Determine the velocity of the bird at $t=1$ sec.
2. Determine the acceleration of the bird at $t=1$ sec.
3. Determine the acceleration of the bird when the velocity equals 0.

A potato is launched vertically upward with an initial velocity of 100 ft/s from a potato gun at the top of an 85-foot-tall building. The distance in feet that the potato travels from the ground after $t$ seconds is given by $s\left(t\right)=\mathrm{-16}{t}^2+100t+85.$

1. Find the velocity of the potato after $0.5\text{s}$ and $5.75\text{s}.$
2. Find the speed of the potato at 0.5 s and 5.75 s.
3. Determine when the potato reaches its maximum height.
4. Find the acceleration of the potato at 0.5 s and 1.5 s.
5. Determine how long the potato is in the air.
6. Determine the velocity of the potato upon hitting the ground.