A bicyclist is pedaling along a straight road with velocity v given in the graph

Question

A bicyclist is pedaling along a straight road with velocity v given in the graph below. Suppose that the cyclist starts 5 miles from a lake, and that positive velocities take her away from the lake and negative velocities toward the lake. Does the cyclist initially start pedaling toward or away from the lake? When does she change direction? At what time(s) is her velocity = 0? At what time(s) is her acceleration a(t) = 0? Estimate the area between v(t) and the t-axis on the interval [0, 1/3]. What measurable quantity does this area represent? Be specific. What are the units for this area? Estimate the area between v(t) and the t-axis on the interval [1/3, 1]. What measurable quantity does this area represent? Be specific. What are the units for this area? Estimate v(t) dt. Estimate |v(t)| dt. What is the total (not net) distance she travels? When is the cyclist farthest from the lake, and how far away is she then? Does she ever return to her starting position? If so, when?

Explanation / Answer

a) towards from the lake as initially velocity is negative b) at 20 minutes (1/3 hours) (at that point velocity becomes 0 and then positive thereafter) c) at 20 mins (1/3 hours) d) acceleration is 0 when the slope of the curve is zero i.e. at t = 40 minutes (2/3 hours) e) The area can be calculated from the exact graph or from the equation of the curve by integration. The unit is miles and it represents distance traveled. f) Same method as above part. g) get the equation of the curve and then integrate it with the limis (0,1) h) integrate it in 2 parts from 0 to 1/3 and then from 1/3 to 1. Make sure to make the sign of the first integral positive. i) The total net distance traveled is the value of the integral in part (h) j) get the value of g and then find its maxima. k) She does return to the starting point. In the function of part (g) put the value equal to 5 and calculate x.

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