Mathematical Modeling 2001

Assignment 0702

1. The velocity of an automobile is given in meters / sec by the function v(t) = 3 t + 1, where t is clock time in seconds.

• Find the velocity at t = 4 and at t = 7, and use these velocities and the 3-second time interval to find the average rate of velocity change between t = 4 and t = 7, as well as the change in the position of the automobile.
• On a graph of velocity vs. clock time, locate the t = 4 and t = 7 points. From the t = 4 point sketch a projection line to the t = 4 point on the t axis and label this line with its length, writing the length just to the right of the line and about halfway up. Do the same for the t = 7 point of the graph. Then connect the t = 4 and t = 7 points.

• Find the slope of the line connecting the t = 4 and t = 7 points, and label that line segment with the slope, placing the slope in a rectangular box just above the middle of the line segment.
• The figure you have constructed is a trapezoid, bounded below by the t axis, having as its vertical sides the projection lines you constructed, and the sloping line segment for its top. Its area is equal to the product of its average altitude and its width. What is its area? Label the area by writing it in the center of the trapezoid and placing a circle around it.

• What is the meaning, in terms of velocities, rate of change of velocity, change in position, of the slope and of the area of this graph?

• What is the position function for the given velocity function (hint: the velocity function is the rate-of-change function for the position function)? By how much does this function change between t = 4 and t = 7?

2. The velocity of an automobile is given in meters/sec by the function v(t) = .4 t^2 + 3 t + 1. Sketch a v vs. t graph and locate the t = 4, t = 7 and t = 10 points of the graph.

• For the t = 4 and t = 7 points, construct a trapezoid like the one you constructed in the preceding problem. Find slope and area, and label everything as you did in that problem.
• Do the same for the t = 7 and t = 10 points.

• At what average rate did velocity change between the t = 4 and t = 7 points? At what average rate did velocity change between the t = 7 and t = 10 points? At what specific clock time do you estimate that the rate is actually equal to the average rate?
• By approximately how much did the position change during each of the two time intervals? Do you think your estimates are overestimates or underestimates of the actual changes in position?

• You could more closely estimate changes in position using t intervals from 4 to 4.1, then to 4.2, then to 4.3, etc.. It wouldn't be practical to use an increment of .1 to find the changes from t = 4 to t = 7 by hand, but it wouldn't be any problem to use Excel to do this. How could set up Excel to find these results, and the total change from t = 4 to t = 7?

• The velocity function is the rate-of-change function for the position function. Can you speculate on what the position function might be for the given velocity function?
• If you found a position function, how close are its changes to the changes predicted by the trapezoidal graph?

3. The normal curve is given by the function N(z) = 2 / sqrt(2 `pi) * e^-(z^2 / 2). For a graph of N(z) vs. z, find and plot the graph points corresponding to z = -2, -1, 0, 1, 2. Sketch the trapezoids corresponding to these points, and label heights, slopes and areas according to the conventions of the preceding problems.

• What proportion of the total area lies between z = 0 and z = 1?
• What proportion of the total area lies between z = 1 and z = 2?
• How do these proportions compare with the proportions originally given for the normal curve?

• Refine your graph between z = 0 and z = 1 by using the graph points corresponding to z = 0, .5 and 1. Construct the three trapezoids defined by these points, and find the total area lying between z = 0 and z = 1.

4. There are 10 Christmas trees is a circle. Every two seconds, each tree looks at each of its nearest neighbors to see whether their lights are on or off. If exactly one of its neighbors has its lights on, it turns its light switch to the 'on' position. Otherwise it turns its light switch to the 'off' position.

• If all the trees have their lights on at the beginning, what will happen during the next 20 seconds?
• If all the trees except the one in the middle have their lights on at the beginning, what will happen during the next 20 seconds?
• If only the middle three trees have their lights on at the beginning, what will happen during the next 20 seconds?

• Answer the same questions if the rule is that a tree will place its switch in the 'on' position if at least one of the neighbors has its lights on.
• Answer the same questions if the rule is that a tree will place its switch in the 'on' position if at most one of the neighbors has its lights on.