{Use of Tech} Spring oscillations A spring hangs from the ceiling at equilibrium with a mass attached to its end. Suppose you pull downward on the mass and release it 10 inches below its equilibrium position with an upward push. The distance x (in inches) of the mass from its equilibrium position after t seconds is given by the function x(t) = 10sin t−10cos t, where x is positive when the mass is above the equilibrium position. <IMAGE>
c. At what times is the velocity of the mass zero?

Interpreting the derivative Find the derivative of each function at the given point and interpret the physical meaning of this quantity. Include units in your answer.

An object dropped from rest falls d(t)=16t² feet in t seconds. Find d′(4).

A feather dropped on the moon On the moon, a feather will fall to the ground at the same rate as a heavy stone. Suppose a feather is dropped from a height of 40 m above the surface of the moon. Its height (in meters) above the ground after t seconds is s = 40−0.8t². Determine the velocity and acceleration of the feather the moment it strikes the surface of the moon.

Position, velocity, and acceleration Suppose the position of an object moving horizontally along a line after t seconds is given by the following functions s = f(t), where s is measured in feet, with s > 0 corresponding to positions right of the origin.

b. Find and graph the velocity function. When is the object stationary, moving to the right, and moving to the left?

f(t) = 18t-3t²; 0 ≤ t ≤ 8

The position (in meters) of a marble, given an initial velocity and rolling up a long incline, is given by s = 100t / t+1, where t is measured in seconds and s=0 is the starting point.

b. Find the velocity function for the marble.

{Use of Tech} Spring oscillations A spring hangs from the ceiling at equilibrium with a mass attached to its end. Suppose you pull downward on the mass and release it 10 inches below its equilibrium position with an upward push. The distance x (in inches) of the mass from its equilibrium position after t seconds is given by the function x(t) = 10sin t - 10cos t, where x is positive when the mass is above the equilibrium position. <IMAGE>

b. Find dx/dt and interpret the meaning of this derivative.  

A woman attached to a bungee cord jumps from a bridge that is 30 m above a river. Her height in meters above the river t seconds after the jump is y(t) = 15(1+e^-t cos t), for t ≥ 0.

Determine her velocity at t = 1 and t = 3. 

Determine whether the following statements are true and give an explanation or counterexample.

c. It is impossible for the instantaneous velocity at all times a≤t≤b to equal the average velocity over the interval a≤t≤b.

Throwing a stone Suppose a stone is thrown vertically upward from the edge of a cliff on Earth with an initial velocity of 32 ft/s from a height of 48 ft above the ground. The height (in feet) of the stone above the ground t seconds after it is thrown is s(t) = -16t²+32t+48.

b. When does the stone reach its highest point?