Urgent Help in Calc

[dagreat45]

[h=1]Find an overestimate and an underestimate.?[/h]For time, t, in hours, 0≤t≤1, a bug is crawling at a velocity, v, in meters/hour given by the function v(t) shown below. Use Δt=0.2 to estimate the distance that the bug crawls during this hour. Find an overestimate and an underestimate. Then average the two to get a new estimate. Include units with each answer.

v(t)=[4]/[9+t]

a) The underestimate is:____________ . help (units)

b) The overestimate is:_____________ . help (units)

c) The average estimate is:____________________________ . help (units)

I think I know the answer is the underestimate .2 and then Im kinda of stuck with the rest as to solving the whole question

 

[MarkFL]

For the underestimate, you want to let the bug's velocity for a particular sub-interval be the value found from the velocity function at the end of the sub-interval since the velocity is decreasing.

Hence, the underestimate would be:

\(\displaystyle s(t)=\dfrac{1}{5}\sum_{k=1}^5\left(\dfrac{4}{9+ \dfrac{k}{5}} \right)=4\sum_{k=1}^5\left(\dfrac{1}{45+k} \right)\)

Can you now use a similar expression to find the overestimate?