Calculate the instantaneous velocity for the indicated value...

Calculate the instantaneous velocity for the indicated value of the time (in s) of an object for which the displacement (in $\mathrm{ft}$ ) is given by the indicated function. Use the method of Example 3 and calculate values of the average velocity for the given values of $t$ and note the apparent limit as the time interval approaches zero. $s=4 t+10 ;$ when $t=3 ;$ use values of $t$ of 2.0,2.5,2.9,2.99,2.999

Calculate the instantaneous velocity for the indicated value of the time (in s) of an object for which the displacement (in $\mathrm{ft}$ ) is given by the indicated function. Use the method of Example 3 and calculate values of the average velocity for the given values of $t$ and note the apparent limit as the time interval approaches zero.
$s=4 t+10 ;$ when $t=3 ;$ use values of $t$ of 2.0,2.5,2.9,2.99,2.999

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Key Concepts

Limit Process

The limit process is a fundamental concept in calculus that involves analyzing the behavior of a function as the input value approaches a particular point. In the context of velocity, taking the limit of the average velocity as the time interval shrinks to zero yields the instantaneous velocity.

Difference Quotient

The difference quotient is a formula used to compute the average rate of change of a function over an interval. It is expressed as [f(t + ?t) – f(t)]/?t, and taking the limit of this expression as ?t approaches zero gives the derivative, which is the instantaneous rate of change.

Average Velocity

Average velocity is calculated as the change in displacement divided by the change in time over a given time interval. It approximates instantaneous velocity when the selected interval becomes sufficiently small.

Instantaneous Velocity

This is the velocity of an object at a specific moment in time. It is defined as the limit of the average velocity over a time interval as the interval approaches zero, and mathematically it is the derivative of the displacement function with respect to time.

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