The distance s (in feet) covered by a motorcycle traveling...

The distance $s$ (in feet) covered by a motorcycle traveling in a straight line and starting from rest in $t$ sec is given by the function $$s(t)=-0.1 t^{3}+2 t^{2}+24 t \quad(0 \leq t \leq 3)$$ Calculate the motorcycle's average velocity over the time interval $[2,2+h]$ for $h=1,0.1,0.01,0.001,0.0001,$ and 0.00001 and use your results to guess at the motorcycle's instantaneous velocity at $t=2$

The distance $s$ (in feet) covered by a motorcycle traveling in a straight line and starting from rest in $t$ sec is given by the function $$s(t)=-0.1 t^{3}+2 t^{2}+24 t \quad(0 \leq t \leq 3)$$ Calculate the motorcycle's average velocity over the time interval $[2,2+h]$ for
$h=1,0.1,0.01,0.001,0.0001,$ and 0.00001 and use your results to guess at the motorcycle's instantaneous velocity at $t=2$

Key Concepts

Derivative

The derivative of a function at a given point quantifies the rate at which the function is changing at that point. In the context of motion, the derivative of the displacement function with respect to time provides the instantaneous velocity. It is conceptually defined as the limit of the difference quotient as the interval length approaches zero.

Instantaneous Velocity

Instantaneous velocity is the velocity of an object at a specific moment in time. It is obtained by taking the limit of the average velocity as the time interval approaches zero. In calculus, this limit process is equivalent to finding the derivative of the position (or displacement) function with respect to time.

Average Velocity

Average velocity is the total displacement divided by the total time taken over a specified interval. It gives a measure of the overall speed and direction of an object’s motion between two points in time. In problems involving continuous motion, it is computed by taking the difference in the object's position at the end points of the time interval and dividing by the length of the interval.

Difference Quotient

The difference quotient is a mathematical expression that represents the average rate of change of a function over an interval. It is given by [f(t + h) - f(t)] / h and serves as the foundation for defining the derivative. This quotient is used to approximate how a function changes with a small change in its input.

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