Calculus: The Language of Change

Exercise Set 9.2
Graphs Without Formulas

by John Robeson

Exercise 1) Rough Sketch

Make a qualitative rough sketch of a graph of the distance traveled as a function of time on the following hypothetical trip:

(a)  You travel a total of 100 miles in 2 hours.

(b)  Most of the trip is on rural interstate highway at the 65 mph speed limit.  What qualitative feature or shape does the graph of distance vs. time have when speed is 65 mph?

(c)  You start from your house at rest, gradually increase your speed to 25 mph, slow down, and stop at a stop sign.

(d)  What shape is the graph of distance vs. time while you are stopped?

(e)  You speed up again to 25 mph, travel a while, and enter the interstate.  At the end of the trip, you exit, slow to 25 mph, stop at a stop sign, and proceed to your final destination.

(a)

(a)  You travel a total of 100 miles in 2 hours.

Solution to (a)

Our graph of distance vs. time should start at (0, 0) and end at (2, 100), where (0, 0) is the start of the trip at your house and (2, 100) is our destination 100 miles away after 2 hours.

(b)

(b)  Most of the trip is on rural interstate highway at the 65 mph speed limit.  What qualitative feature or shape does the graph of distance vs. time have when speed is 65 mph?

Solution to (b)

Recall that speed is (distance/time).  So our speed at any instant is 
reflected in the slope of our distance vs. time graph.  Therefore, for this portion of the trip our graph should be a straight line with a positive slope of 65 (miles/hour).

(c)

(c)  You start from your house at rest, gradually increase your speed to 25 mph, slow down, and stop at a stop sign.

Solution to (c)

As stated in part (a) our graph starts at (0, 0).  The slope at this point is 0 since initially we are stopped and our speed is zero.  The graph then curves upward until our slope becomes 25 mph.  The graph continues upward linearly at this rate until we begin to slow down, at which time the graph curves to a flat section for the time we are stopped.

(d)

(d)  What shape is the graph of distance vs. time while you are stopped?

Solution to (d)

As indicated in part (c), the graph is flat when we are stopped, since our speed, and hence the slope of our graph, is zero.

(e)

(e)  You speed up again to 25 mph, travel a while and enter the interstate.  At the end of the trip, you exit, slow to 25 mph, stop at a stop sign, and proceed to your final destination.

Solution to (e)

Again, our graph curves upward from a slope of zero to a slope of 25 mph and continues at this rate until we enter the interstate at which time the graph curves upward more until our slope becomes 65 mph.  It is linear at this slope of 65 mph until we exit and the graph curves again to a slope of 25 mph, then curves even more to a flat section when we are again stopped.  The final section of our graph ends at the coordinate (2, 100) where our slope is again zero (stopped).

Note that our graph is always going up or is flat since we never go backwards, but of course, the slope changes many times and is much steeper at 65 mph than at 25 mph.