Two automobiles are traveling up a long hill with an steepness that doesn

Governor's School Final 2003 Part 1

1. Two automobiles are traveling up a long hill with an steepness that
doesn't change until the top, which is very far away, is reached. One
automobile is moving twice as fast as the other. At the instant the faster
automobile overtakes the slower their drivers both take them out of gear and
they coast until they stop, both changing velocity at the same average rate.

Which automobile will take longer to come to a stop? Will that automobile
require about twice as long to stop, more than twice as long or less than
twice as long?
Which automobile will have the greater average coasting velocity? Will its
average coasting velocity by twice as great as the other, more than twice as
great or less than twice as great?
Will the distance traveled by the faster automobile be equal to that of the
slower, twice that of the slower or more than twice that of the slower?

2. An automobile traveling down a hill passes a certain milepost
traveling at a speed of 10 mph, and proceeds to coast to a certain lamppost
further down the hill, with its speed increasing by 2 mph every second. The
time required to reach the lamppost is 10 seconds. It then repeats the
process, this time passing the milepost at a speed of 20 mph.

Will the vehicle require more or less than 10 seconds to reach the lamppost?
Since its initial speed was 10 mph greater than before, does it follow that
its speed at the lamppost will be 10 mph greater than before?

3. Two climbers eat Cheerios for breakfast and then climb up a steep
mountain as far as they can until they use up all their energy from the
meal. All other things being equal, who should be able to climb further up
the mountain, the 200-lb climber who has eaten 12 ounces of Cheerios or the
150-lb climber who has eaten 10 ounces of Cheerios?

When a 100 lb person hangs from a certain bungee cord, the cord
stretches by 5 feet beyond its initial unstretched length. When a person
weighing 150 lbs hangs from the same cord, the cord is stretched by 9 feet
beyond its initial unstretched length. When a person weighing 200 lbs
hangs from the same cord, the cord is stretched by 12 feet beyond its
initial unstretched length.

Based on these figures, would you expect that a person of weight 125 lbs
would stretch the cord more or less than 7 feet beyond its initial
unstretched length?

4. When given a push of 10 pounds, with the push maintained through a
distance of 4 feet, a certain ice skater can coast without further effort
across level ice for a distance of 30 feet. When given a push of 20 pounds
(double the previous push) through the same distance, the skater will be
able to coast twice as far, a distance of 60 feet. When given a push of 10
pounds for a distance of 8 feet (twice the previous distance) the skater
will again coast a distance of 60 feet.

The same skater is now accelerated by a sort of a slingshot consisting of a
bungee-type cord slung between two posts in the ice. The cord, as one might
expect, exerts greater and greater force as it is pulled back further and
further. Assume that the force increases in direct proportion to pullback
(ie.g., twice the pullback implies twice the force).
When the skater is pulled back 4 feet and released, she travels 20 feet.
When she is pulled back 8 feet and released, will she be expected to travel
twice as far, more than twice as far or less than twice as far as when she
was pulled back 4 feet?

5. Two identical light bulbs are placed at the centers of large and
identically frosted glass spheres, one of diameter 1 foot and the other of
diameter 2 feet.

To a moth seeking light from half a mile away, unable to distinguish the
difference in size between the spheres, will the larger sphere appear
brighter, dimmer or of the same brightness as the first?
To a small moth walking on the surface of the spheres, able to detect from
there only the light coming from 1 square inch of the sphere, will the
second sphere appear to have the same brightness as the first, twice the
brightness of the first, half the brightness of the first, more than twice
the brightness of the first, or less than half the brightness of the first?

Part 1B

1. A sequence is increasing at an increasing rate. What can be said about the first- and second-difference sequences?

2. A sequence is increasing at a decreasing rate. What can be said about the first- and second-difference sequences?

3. A sequence is decreasing at an increasing rate. What can be said about the first- and second-difference sequences?

4. A sequence is decreasing at a decreasing rate. What can be said about the first- and second-difference sequences?

5. The first-difference sequence of a sequence is 2, 5, 9, 14, 20, ... . Can you tell whether the sequence is increasing or decreasing? Can you tell whether this is at an increasing or decreasing rate? Tell all you can about the sequence.

6. The second-difference sequence of a sequence is 2, 5, 9, 14, 20, ... . Can you tell whether the sequence is increasing or decreasing? Can you tell whether this is at an increasing or decreasing rate? Tell all you can about the sequence.

7. The first-difference sequence of a sequence is -2, -5, -9, -14, -20, ... . Can you tell whether the sequence is increasing or decreasing? Can you tell whether this is at an increasing or decreasing rate? Tell all you can about the sequence.

8. The second-difference sequence of a sequence is -2, 5, -9, 14, -20, ... . Can you tell whether the sequence is increasing or decreasing? Can you tell whether this is at an increasing or decreasing rate? Tell all you can about the sequence.

Part 1C

1. The sequence 2, 5, 9, 14, ... represents depths of water in cm at 4-second intervals. Sketch a corresponding graph of depth vs. clock time and completely label the graph. Tell specifically what the meaning is of the slope of the second trapezoid, and interpret also the rate of slope change corresponding to the third depth as well as the area and accumulated area of the fourth trapezoid.

2. The sequence 12, 15, 19, 24, ... represents the rates of depth change of water with respect to clock time, in cm/s at 5-second intervals. Sketch a corresponding rate graph of depth change vs. clock time and completely label the graph. Tell specifically what the meaning is of the slope of the second trapezoid, and interpret also the rate of slope change corresponding to the third depth as well as the area and accumulated area of the fourth trapezoid.

3. The sequence 21, 52, 93, 144, ... represents the amounts of water in a container, in cm^3, at 3-second intervals. Sketch a corresponding graph of amount vs. clock time and completely label the graph. Tell specifically what the meaning is of the slope of the second trapezoid, and interpret also the rate of slope change corresponding to the third depth as well as the area and accumulated area of the fourth trapezoid.

4. The sequence 21, 52, 93, 144, ... represents the rates of change of the volume of water in a container with respect to clock time, in cm^3/s at 6-second intervals. Sketch a corresponding graph of rate of change of volume vs. clock time and completely label the graph. Tell specifically what the meaning is of the slope of the second trapezoid, and interpret also the rate of slope change corresponding to the third depth as well as the area and accumulated area of the fourth trapezoid.

5. The sequence 41, 62, 92, 44, ... represents the position of a marble rolling down a track, in cm, at 5-second intervals. Sketch a corresponding graph of position vs. clock time and completely label the graph. Tell specifically what the meaning is of the slope of the second trapezoid, and interpret also the rate of slope change corresponding to the third depth as well as the area and accumulated area of the fourth trapezoid.

6. The sequence 41, 62, 92, 44, ... represents the rates of change of the the position of a marble rolling down a track with respect to clock time, in cm/s at 4-second intervals. Sketch a corresponding graph of rate of change of position vs. clock time and completely label the graph. Tell specifically what the meaning is of the slope of the second trapezoid, and interpret also the rate of slope change corresponding to the third depth as well as the area and accumulated area of the fourth trapezoid.

Part 1D

1. Evaluate the function r = 3 t + 12 at t = 0, 3, 6 and 9 and construct a trapezoidal approximation graph of y vs. t. Make a table of accumulated area up to clock time t vs. t for t = 0, 3, 6, and 9. Construct a trapezoidal graph of accumulated area vs. t labeling the heights, slopes and rates of slope change. If r represents the rate at which depth changes in cm/s vs. clock time t then what does your graph of accumulated area vs. t represent?

On your graph of accumulated area vs. t what does the slope of the second trapezoid represent?

List the slopes of your graph of accumulated area vs. the clock time at the midpoint of each interval, listing a slope and a midpoint clock time for each interval. Sketch a graph of slope vs. midpoint clock time, labeling only the heights and slopes of your new graph. What does this graph represent and what do its slopes represent?

Finally evaluate the function y = 2 t^2 - 3 t + 40 at t = 0, 3, 6 and 9 and construct a trapezoidal approximation graph of y vs. t. What is the midpoint clock time on each interval? Make a table of slope vs. midpoint clock time, listing the slope of each trapezoid vs. the midpoint clock time of that trapezoid. Construct a graph of slope vs. midpoint clock time. If y represents the volume of water in a container in cm^3 vs. clock time t in seconds then what does your graph of slope vs. midpoint t represent?

2. Evaluate the function r = 2 t + 9 at t = 0, 4, 8 and 12 and construct a trapezoidal approximation graph of y vs. t. Make a table of accumulated area up to clock time t vs. t for t = 0, 4, 8 and 12. Construct a trapezoidal graph of accumulated area vs. t labeling the heights, slopes and rates of slope change. If r represents the rate at which the volume of water in a container change in cm^3/s vs. clock time t in seconds then what does your graph of accumulated area vs. t represent?

On your graph of accumulated area vs. t what does the slope of the second trapezoid represent?

Finally evaluate the function y = t^2 - 2 t + 40 at t = 0, 4, 8 and 12 and construct a trapezoidal approximation graph of y vs. t. What is the midpoint clock time on each interval? Make a table of slope vs. midpoint clock time, listing the slope of each trapezoid vs. the midpoint clock time of that trapezoid. Construct a graph of slope vs. midpoint clock time. If y represents the speed of a marble rolling down a curved track then what does your graph of slope vs. midpoint t represent?

Part 1E

1. Evaluate the function y = 3^x at x = 1.9, 2, and 2.1. Sketch the corresponding trapezoidal approximation graph, with your x axis extending only from x = 1.9 to x = 2.1, and label the heights and slopes. Be sure you use enough significant figures to show the differences between the slopes.

Repeat using x = 1.99, 2 and 2.01, with the x axis extending from 1.99 to 2.01.

Repeat once more, using x = 1.999, 2 and 2.001, with the x axis extending from 1.999 to 2.001.

What do you think is the precise rate of change of y with respect to x at the point x = 2?

2. Evaluate the function y = 2^x at x = 1.9, 2, and 2.1. Sketch the corresponding trapezoidal approximation graph, with your x axis extending only from x = 1.9 to x = 2.1, and label the heights and slopes. Be sure you use enough significant figures to show the differences between the slopes.

Repeat using x = 1.99, 2 and 2.01, with the x axis extending from 1.99 to 2.01.

Repeat once more, using x = 1.999, 2 and 2.001, with the x axis extending from 1.999 to 2.001.

What do you think is the precise rate of change of y with respect to x at the point x = 2?

3. Evaluate the function y = 4^x at x = 1.9, 2, and 2.1. Sketch the corresponding trapezoidal approximation graph, with your x axis extending only from x = 1.9 to x = 2.1, and label the heights and slopes. Be sure you use enough significant figures to show the differences between the slopes.

Repeat using x = 1.99, 2 and 2.01, with the x axis extending from 1.99 to 2.01.

Repeat once more, using x = 1.999, 2 and 2.001, with the x axis extending from 1.999 to 2.001.

What do you think is the precise rate of change of y with respect to x at the point x = 2?

4. Evaluate the function y = 5^x at x = 1.9, 2, and 2.1. Sketch the corresponding trapezoidal approximation graph, with your x axis extending only from x = 1.9 to x = 2.1, and label the heights and slopes. Be sure you use enough significant figures to show the differences between the slopes.

Repeat using x = 1.99, 2 and 2.01, with the x axis extending from 1.99 to 2.01.

Repeat once more, using x = 1.999, 2 and 2.001, with the x axis extending from 1.999 to 2.001.

What do you think is the precise rate of change of y with respect to x at the point x = 2?

Part 1F

1. Sketch a graph of y = sqrt(t) for t = 0, 2, 4. Label heights and areas.

If you were to repeat this exercise using t = 0, 1, 2, 3, 4 (don't actually do this) do you think the accumulated area under the graph would be more or less than under the original graph?

If you were to again repeat this exercise using t = 0, .1, .2, .3, .4, ..., 3.8, 3.9, 4 do you think the accumulated area under the graph would be more or less than under the preceding graph?

Do you think the change in the accumulated area would be more or less between the second and third than between the first and second graphs?

Do you think that as you use smaller and smaller intervals that there is some value the changing accumulated area will never pass?

2. Sketch a graph of y = t^2 + 10 for t = 0, 2, 4. Label heights and areas.

If you were to repeat this exercise using t = 0, 1, 2, 3, 4 (don't actually do this) do you think the accumulated area under the graph would be more or less than under the original graph?

If you were to again repeat this exercise using t = 0, .1, .2, .3, .4, ..., 3.8, 3.9, 4 do you think the accumulated area under the graph would be more or less than under the preceding graph?

Do you think the change in the accumulated area would be more or less between the second and third than between the first and second graphs?

Do you think that as you use smaller and smaller intervals that there is some value the changing accumulated area will never pass?

3. Sketch a graph of y = 2^t + 10 for t = 0, 2, 4. Label heights and areas.

If you were to repeat this exercise using t = 0, 1, 2, 3, 4 (don't actually do this) do you think the accumulated area under the graph would be more or less than under the original graph?

If you were to again repeat this exercise using t = 0, .1, .2, .3, .4, ..., 3.8, 3.9, 4 do you think the accumulated area under the graph would be more or less than under the preceding graph?

Do you think the change in the accumulated area would be more or less between the second and third than between the first and second graphs?

Do you think that as you use smaller and smaller intervals that there is some value the changing accumulated area will never pass?

4. Sketch a graph of y = 10 + 10 / 2^t for t = 0, 2, 4. Label heights and areas.

If you were to repeat this exercise using t = 0, 1, 2, 3, 4 (don't actually do this) do you think the accumulated area under the graph would be more or less than under the original graph?

If you were to again repeat this exercise using t = 0, .1, .2, .3, .4, ..., 3.8, 3.9, 4 do you think the accumulated area under the graph would be more or less than under the preceding graph?

Do you think the change in the accumulated area would be more or less between the second and third than between the first and second graphs?

Do you think that as you use smaller and smaller intervals that there is some value the changing accumulated area will never pass?