How to get the instantaneous velocity and maximum velocity?

Question

The graph below shows the total distance , in miles, traveled by a bicyclist after hours.

(b) Estimate the bicyclist's instantaneous velocity at the following times.

At hour, the instantaneous velocity is approximately ....... mi/hr.

At hours, the instantaneous velocity is approximately ..........mi/hr.

At hours, the instantaneous velocity is approximately ......... mi/hr.

(c) Estimate the bicyclist's maximum velocity and the specific time at which it occurs.

The maximum velocity is approximately........ mi/hr, which occurs when ......... hr.

Answer 1

The instantaneous velocity is the value of the slope of the tangent line at the value of "t" in question. This is not an exact science, just do the best you can as follows:

Draw your best tangent line to the curve at t = 0.5 like the red line in this sketch:

The point of tangency is the blue dot and occurs at t = 1/2 hr.

Then look for some nice grid intersection points (places on the red tangent line that cross grid lines so they are easily read). I picked (1.5, 25) and (2.5, 40) because when I drew the line, it seemed to cross these grid lines.

Then use the slope equation to calculate the slope: (40 - 25)/(2.5 - 1.5) = 15/1 = 15 miles per hour.

Repeat for the other values of t.