If a rock is thrown upward on the planet Mars with a...

If a rock is thrown upward on the planet Mars with a velocity of $10 \mathrm{m} / \mathrm{s}$, its height in meters $t$ seconds later is given by $y=10 t-1.86 t^{2}$ (a) Find the average velocity over the given time intervals: (i) $(1,2]$ (ii) $[1,1.5]$ (iii) $(1,1.1]$ (iv) $[1,1.01]$ (v) $[1,1.001]$ (b) Estimate the instantancous velocity when $t=1.$

If a rock is thrown upward on the planet Mars with a velocity of $10 \mathrm{m} / \mathrm{s}$, its height in meters $t$ seconds later is given by $y=10 t-1.86 t^{2}$
(a) Find the average velocity over the given time intervals:
(i) $(1,2]$
(ii) $[1,1.5]$
(iii) $(1,1.1]$
(iv) $[1,1.01]$
(v) $[1,1.001]$
(b) Estimate the instantancous velocity when $t=1.$

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Key Concepts

Secant Line and Difference Quotient

The concept of the secant line is used to approximate the average rate of change between two points on a graph. The difference quotient formula, (f(b) - f(a))/(b - a), calculates the slope of the secant line, serving as an approximation of the instantaneous rate of change over the interval.

Average Velocity

Average velocity is defined as the total displacement divided by the total time elapsed over a given interval. It represents the slope of the secant line connecting two points on the position-time graph, providing a measure of how fast the position changes on average over that interval.

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, which mathematically corresponds to the derivative of the position function with respect to time.

Derivative

The derivative of a function represents the instantaneous rate of change of the function with respect to its variable. In the context of motion, the derivative of the position function with respect to time gives the instantaneous velocity, which is found by applying limit processes to the difference quotient.

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