Interpret the meaning of the derivative

Graph the function in the domain of .

Plot the derivative of the function given in the graph below.

Plot the derivative of the graphed function .

A particle's position is defined by the equation for constant values of where . If the particle starts at zero velocity, and the acceleration of the particle at is .  Find the values of and . 

A particle's displacement is shown in the graph below. Provide a graph of the velocity and acceleration for the particle.

An astronomer finds that the position of an asteroid is defined by the equation measured in (astronomical units) for a bound of and is measured in years. If the asteroids speed gets below per year, it will start heading towards the Earth! Will this asteroid head towards Earth?

Assuming that a particle's displacement is a third degree polynomial and given that its displacement has critical points at and . Find the particle's acceleration as a function of time.

Suppose a moss grows on a tree at a rate of at  a warm day where is measured in hours. The next morning, half of all the grown moss dies. Find the rate at which the moss's growth rate is changing. 

Considering that the speed of sound is , find the time at which the sound of a ball striking the bottom of a well at meters in depth would reach the surface.

Find the distance traveled by a free fall (under the effect of gravity) of an object in seconds. Let its initial position be as and its initial speed to be .  

Suppose that represents the diameter (measured in meter) of a circle at (measured in second). What does  mean with respect to the radius?

A biologist measures that the total consumption of a shark in a local area can be modelled as for any given time, . They also measured that the amount of fish in that given area is changing at a rate of . If the change in the shark's consumption rate is the same as the change in the fish's spawning rate, there will not be enough for fish for the sharks. Will there ever be enough for fish for the sharks?

Suppose a cheetah can run at a rate of and a gazelle moves a distance of . When will the speed of a cheetah surpass the speed of a gazelle?

Given that the position of a particle is defined as , what is its acceleration?

Sketch the velocity function graph based on the graph of the position function  shown below.