Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

The displacement (in meters) of an object moving in a straight line is given by:

$$f(t)= 1 + 2t + \frac{1}{4}t^2$$

where $t$ is measured in seconds.

Find the average velocity over the following time periods:

(i) $[1, 1.2]$

(ii) $[1, 1.1]$

(iii) $[1, 1.01]$

(iv) $[1, 1.001]$

Also use the information above to estimate the instantaneous velocity when $t = 1$ second.

share|improve this question
add comment

2 Answers

Average velocity of $f(t)$ between $t=a$ and $t=b$ is simply the slope of the line that travels through these points. Thus

$$\frac{\Delta d}{\Delta t}=\frac{f(1.2)-f(1)}{1.2-1}$$

Instantaneous velocity at a point $t = p$ is the same as average velocity where the two points are both very close to $p$, i.e.

$$\lim_{\Delta t \to 0}\frac{\Delta d}{\Delta t}= \lim_{\Delta t \to 0}\frac{f(1 + \Delta t)-f(1)}{\Delta t}$$

So you may thus choose $\Delta t \approx 0$ to get an approximation, or, if you know how, differentiate $f$ at $1$.

share|improve this answer
add comment

I will assume that this is homework.

Average velocity is determined by dividing the distance travelled by the time it takes to travel that far. Therefore, for the interval $[ 1, 1.2 ]$ we have $$\dfrac{\textrm{distance}}{\textrm{time}} = \dfrac{f(1.2) - f(1)}{1.2 - 1} = \dfrac{3.76 - 3.25}{0.2} = 2.55 ~m/sec.$$

The other parts are quite similar. To estimate the instantaneous velocity at 1, we estimate what the values you receive from parts i-iv seem to be getting closer to.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.