# In a time of t seconds, a particle moves a distance of s meters from its starting point, where...

## Question:

In a time of {eq}t{/eq} seconds, a particle moves a distance of {eq}s{/eq} meters from its starting point, where {eq}s=3t^2{/eq}.

a) Find the average velocity between {eq}t=1{/eq} and {eq}t=1+h{/eq} if:

i. {eq}h=0.1{/eq}, ii. {eq}h=0.01{/eq}, iii. {eq}h=0.001{/eq}

b) Use your answers to part a) to estimate the instantaneous velocity of the particle at time {eq}t=1{/eq}.

## Average value :

Average value of function is also known as mean of function.

Consider a function {eq}\displaystyle f(x) {/eq} .Then average value of the function {eq}\displaystyle f(x) {/eq} on the interval {eq}\displaystyle [a,b] {/eq} is given by

## Average value :

Average value of function is also known as mean of function.

Consider a function {eq}\displaystyle f(x) {/eq} .Then average value of the function {eq}\displaystyle f(x) {/eq} on the interval {eq}\displaystyle [a,b] {/eq} is given by

Average value {eq}\displaystyle = \frac{\int_a^bf(x)dx}{b-a} {/eq}

{eq}\displaystyle = \frac{\int_a^bf(x)dx}{b-a} {/eq}

## Answer and Explanation:

Given a particle moves a distance of s meters from its starting point, {eq}\displaystyle f(t)=s=3t^2 {/eq}

Now, the velocity is obtained by differentiating it,{eq}\displaystyle f'(t)= 6t = v {/eq}

Average velocity in interval {eq}\displaystyle [1,1.1] {/eq} is:

{eq}\displaystyle \begin{align} f'_{avg}&= \frac{\int_a^bf(x)dx}{b-a}\\ &=\frac{\int_1^{1.1}6tdx}{1.1-1}\\ &=\frac{(3t^2)_1^{1.1}}{0.1}\\ &= 6.3\\ \end{align} {/eq}

Average velocity in interval {eq}\displaystyle [1,1.001] {/eq} is:

{eq}\displaystyle \begin{align} f'_{avg}&= \frac{\int_a^bf(x)dx}{b-a}\\ &=\frac{\int_1^{1.001}6tdx}{1.001-1}\\ &=\frac{(3t^2)_1^{1.001}}{0.001}\\ &= 6.003\\ \end{align} {/eq}

Average velocity in interval {eq}\displaystyle [1,1.01] {/eq} is:

{eq}\displaystyle \begin{align} f'_{avg}&= \frac{\int_a^bf(x)dx}{b-a}\\ &=\frac{\int_1^{1.01}6tdx}{1.01-1}\\ &=\frac{(3t^2)_1^{1.01}}{0.01}\\ &= 6.03\\ \end{align} {/eq}

Since the interval 1,1.001 is the smallest , we obtain the most accurate value of instantaneous velocity in this interval.

So by using above results we get the instantaneous velocity of the particle at time t = 1 is 6.003 meters per second