Day 4 Questions:
1. If you pass milepost 50 at 3:00 p.m. and milepost 400 at 9:00 p.m. then at what average
rate is your position changing?
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2. Draw a possible graph indicating your position vs. clock time for this trip.
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3. At a certain college a study showed that students who studied 200 hours per
semester had a grade point average of 2.13 while those who studied 400 hours per semester
had a grade point average of 3.43 and those who studied 600 hours had a grade point
average of 3.82. At what average rate does grade point average change with respect to
hours studied between 200 and 400 hours of study? At what average rate does grade point
average change with respect to hours studied between 400 and 600 hours of study?
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4. Draw a corresponding graph indicating your velocity vs. clock time for this trip.
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5. Coasting down a hill you attain a speed of 20 feet / second when you are 80 feet
down the hill, and 5 seconds later your speed is 40 feet / second. What is your best
estimate of how far you have gone down the hill at the instant your speed is 40 feet /
second?
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6. A marble requires 5 seconds to roll from rest a distance of 100 cm down a uniform
incline. What is its average velocity?
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7. Sketch a graph of marble velocity vs. clock time for the preceding situation,
assuming that velocity changes at a constant rate. According to your graph what is the
marble's maximum velocity on the incline, and what is its rate of change of velocity?
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8. Using the formula T = .2 sqrt(L) what are the periods of pendulums 20 cm, 30 cm
and 40 cm long?
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9. For each of the length intervals in the preceding problem, at what average rate
is the period changing with respect to length?
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10. Evaluate the function y = .01 x^2 - 3 x + 500 for x = 30 and for x = 90. At what
average rate does y change with respect to x between x = 30 and x = 90?
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11. If y stands for the yield (in tons) of an acre of corn and x for the number of
inches of rain that falls on the field during the season, then what does the rise between
two points on a graph of y vs. x stand for? What does the run stand for? What therefore
does the slope stand for?
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Day 5 Questions
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1. Plug into y = x^3 - 3 x^2 + 7 x + 10 the values x = 0, 1, 2, 3, 4. Be sure to
save these results for a later problem (and always make some notes so you have your main
results available for subsequent problems). Write your results as a sequence. Show that
the third difference of the sequence is constant.
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2. Consider the sequence 4, 10, 28, 64, 124, 214. Find the first and second
differences of this sequence. Are the first differences positive or negative, and what
does this tell you about the graph? Are the first derivatives increasing or decreaseing
and what does this tell you about the graph? Are the second differences positive or
negative and what does this tell you about the graph?
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3. Use the values you got when you substituted into x^3 - 3 x^2 + 7 x + 10 to
determine the average rate of change of y with respect to x on the intervals from x = 0 to
x = 2 and from x = 2 to x = 4.
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4. Sketch and label the trapezoidal approximation graph corresponding to your
results from the preceding problem.
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5. If the period of a pendulum is T = .2 sqrt(L) then at what average rate does the
period change with respect to length between lengths of 16 cm and 36 cm, and between 16 cm
and 25 cm? Do you think that the rate at which period is changing with respect to clock
time at the precise instant t = 16 is greater than or less than your two results? Why do
you think so?
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6. What is the average rate at which the velocity of an automobile changes if its
velocity changes at a constant rate as it rolls 30 meters down a hill, starting from rest,
in 15 seconds? Sketch a v vs. t graph to support your answer.
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7. What is the average rate of change of the frequency of a pendulum, with frequency
measured in cycles / minute, with respect to length between lengths 25 cm and 64 cm?
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8. If the number of sane people on an island starts at 200 and during every year
gets 60% closer to 500 than at the beginning of the year, then what are the sane
populations at the end of each of the first three years? Sketch a trapezoidal graph of
sane population vs. time in years and label the slopes.
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9. The area of any solid having uniform cross-sectional area is V = A * h, where A
is the uniform cross-sectional area and h the height of the solid. Does this definition
apply to a right circular cylinder and why? What is the area of a right circular cylinder
whose base has area 50 cm^2 and whose height is 10 cm? What is the area of a right
circular cylinder whose base is a circle of radius 3 cm and whose height is 8 cm?
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Day 6
1. Suppose the depth vs. clock time results for a uniform cylinder
of radius 4 cm are depths 25, 18, 13, 10 cm at clock times 0, 5, 10 and 15 seconds. Find
the average rate of depth change with respect to clock time for each of these intervals
and sketch and label a trapezoidal approximation graph representing depth vs. clock time.
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2. For the depth vs. clock time situation of the preceding problem (uniform cylinder
of radius 4 cm are depths 25, 18, 13, 10 cm at clock times 0, 5, 10 and 15 seconds, and if
you didn't have all that written down get in the habit because you won't get a reminder
next time) how many cm^3 of water flowed out between the 25 cm depth and the 18 cm depth?
How much water flowed out between t = 5 seconds and t = 10 seconds? At what average rate,
in cm^3 / second, was water flowing out between t = 10 s and t = 15 s? At what average
rate, in cm^3/second, was water flowing out during the first time interval?
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3. Sketch a trapezoidal approximation graph depicting the volume of water in the
container vs. clock time and label the slopes. What is the meaning of the slopes?
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4. If the water flowing out of the container between t = 5 sec and t = 10 sec flowed
into a tube whose cross-sectional area is .1 cm^2 how long would the water column in the
tube be? How many cm of the tube would therefore fill, per second?
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5. What did you get when you divided 1 by 43 in your homework? Describe the graph
you got when you paired the digits and plotted them. Is your graph evenly distributed?
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6. At what average rate does the period of a pendulum change with respect to length
between lengths of 9 cm and 16 cm? At what average rate does it change between lengths of
16 cm and 25 cm? What is your best estimate, based only on these results, of how much the
period changes between lengths of 15 cm and 17 cm? How accurate do you think this estimate
will be (e.g., within .03 seconds, or within .0007 seconds, or whatever you think; explain
your thinking)?
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7. If the average rate of depth change in a cylinder is -.3 cm / second at the
instant t = 8 seconds, and -.1 cm/second at t = 14 seconds, then what is your best
estimate of how much depth changes between these two clock
times?.....!!!!!!!!...................................
8. Suppose that the rate at which the depth of water in a container changes is -.2
cm/s * sqrt(y), where y is the depth in cm. If the depth is initially 100 cm then at what
rate is the depth changing at that instant? If this rate holds for the next 10 seconds
then how much does the depth change during this 10 seconds? What will then be the depth at
that instant and at what rate will the depth be changing? Is there any inconsistency in
these results?
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9. If water is flowing into a cylinder of radius 4 cm at the rate of 100 cm^3 /
second, then at what rate is depth changing with respect to clock time?