MAT 212 Brief Calculus

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MAT 212Brief Calculus

• Section 6.1
• Results of Change and Area Approximation

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Recall Distance Velocity x Time
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You are speeding down a very perilous mountain
road in your brand new fancy yellow Porche when
you spot a fuzzy brown bear cub sitting in the
middle of the road 400 feet directly ahead of
you. You immediately apply the brakes. Your
velocity decreases throughout the 10 seconds it
takes you to stop. Do you hit the bear?
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Do you hit the bear? How far has the Porche
traveled? Recall distance rate x time It goes
at most 1002200 feet in the first 2 seconds And
at most 642128 feet in the next 2 seconds And
so on So during that 10-second period, it goes
at most 1002 642 362 162 42 440
feet
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Do you hit the bear? How far has the Porche
traveled? You could also say It goes at least
642128 feet in the first 2 seconds And at least
36272 feet in the next 2 seconds And so on So
during that 10-second period, it goes at least
642 362 162 42 02 240 feet
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Do you hit the bear? Therefore, 240 feet
Total distance traveled 440 feet There is a
difference of 200 feet between the upper and
lower estimates!!!!!
Upper Estimate
Lower Estimate
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Do you hit the rabbit? How could we get a more
accurate estimate?
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Do you hit the bear? Here is some information
that may help you. v(t) gives the speed of the
car (in ft/sec) as a function of time (in sec)
since the brakes were applied
v(t)
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Consider the graph of v(t)
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Consider the graph of v(t)
The area of the first purple rectangle is
1002200, the upper estimate of the distance
traveled during the first two seconds. The area
for the second purple rectangle is 642128, the
lower estimate during the next two seconds. The
total area of the purple rectangles represents
the upper estimate for the total distance moved
during the ten seconds.
Left-hand sums
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Consider the graph of v(t)
The area of the first green rectangle is
642128, the lower estimate of the distance
traveled during the first two seconds. The area
for the second green rectangle is 36272, the
lower estimate during the next two seconds. The
total area of the green rectangles represents the
lower estimate for the total distance moved
during the ten seconds.
Right-hand sums
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Consider the graph of v(t)
Based on these rectangles, how could you
calculate the difference between the two
estimates?
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Consider the graph of v(t)
Based on these rectangles, how could you
calculate the difference between the two
estimates?
100
2
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The difference in the upper and lower estimates
for the two-second data is 200 feet. What is the
difference in the upper and lower estimates for
the one-second data?
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Consider the graph of v(t)
One-Second Data
100
1
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The difference in the upper and lower estimates
for the two-second data is 200 feet. The
difference in the upper and lower estimates for
the one-second data is 100 feet. What would the
difference be if the velocity were given every
tenth of a second? Every hundredth of a
second? Every thousandth?
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Sothe more measurements we take (the more
rectangles we use), the more accurate our
estimate will be. The total distance traveled is
the area between the velocity curve and the
x-axis Note If the curve is above the x-axis,
the area is considered positive. If the curve is
below the x-axis, the area is considered
negative.
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Units