assignment 5

course mth 272

end program|楖|ޭ{k

Your work has been received. Please scroll through the document to see any inserted notes (inserted at the appropriate place in the document, in boldface) and a note at the end. The note at the end of the file will confirm that the file has been reviewed; be sure to read that note. If there is no note at the end, notify the instructor through the Submit Work form, and include the date of the posting to your access page.

assignment #005

005. Calculus

qa initial problems

06-07-2008

......!!!!!!!!...................................

17:26:29

`q001. The graph of a certain function is a smooth curve passing through the points (3, 5), (7, 17) and (10, 29).

Between which two points do you think the graph is steeper, on the average?

Why do we say 'on the average'?

......!!!!!!!!...................................

RESPONSE -->

the graph is steeper between the points (7,17) and (10,29) because the difference in slope is greater than that of (3,5) to (7,17).

confidence assessment: 3

.................................................

......!!!!!!!!...................................

17:28:22

2. Answer without using a calculator: As x takes the values 2.1, 2.01, 2.001 and 2.0001, what values are taken by the expression 1 / (x - 2)?

1. As the process continues, with x getting closer and closer to 2, what happens to the values of 1 / (x-2)?

2. Will the value ever exceed a billion? Will it ever exceed one trillion billions?

3. Will it ever exceed the number of particles in the known universe?

4. Is there any number it will never exceed?

5. What does the graph of y = 1 / (x-2) look like in the vicinity of x = 2?

......!!!!!!!!...................................

RESPONSE -->

lesser values.

self critique assessment:

.................................................

......!!!!!!!!...................................

17:28:36

2. Answer without using a calculator: As x takes the values 2.1, 2.01, 2.001 and 2.0001, what values are taken by the expression 1 / (x - 2)?

1. As the process continues, with x getting closer and closer to 2, what happens to the values of 1 / (x-2)?

2. Will the value ever exceed a billion? Will it ever exceed one trillion billions?

3. Will it ever exceed the number of particles in the known universe?

4. Is there any number it will never exceed?

5. What does the graph of y = 1 / (x-2) look like in the vicinity of x = 2?

......!!!!!!!!...................................

RESPONSE -->

self critique assessment: 1

.................................................

......!!!!!!!!...................................

17:30:31

For x = 2.1, 2.01, 2.001, 2.0001 we see that x -2 = .1, .01, .001, .0001. Thus 1/(x -2) takes respective values 10, 100, 1000, 10,000.

It is important to note that x is changing by smaller and smaller increments as it approaches 2, while the value of the function is changing by greater and greater amounts.

As x gets closer in closer to 2, it will reach the values 2.00001, 2.0000001, etc.. Since we can put as many zeros as we want in .000...001 the reciprocal 100...000 can be as large as we desire. Given any number, we can exceed it.

Note that the function is simply not defined for x = 2. We cannot divide 1 by 0 (try counting to 1 by 0's..You never get anywhere. It can't be done. You can count to 1 by .1's--.1, .2, .3, ..., .9, 1. You get 10. You can do similar thing for .01, .001, etc., but you just can't do it for 0).

As x approaches 2 the graph approaches the vertical line x = 2; the graph itself is never vertical. That is, the graph will have a vertical asymptote at the line x = 2. As x approaches 2, therefore, 1 / (x-2) will exceed all bounds.

Note that if x approaches 2 through the values 1.9, 1.99, ..., the function gives us -10, -100, etc.. So we can see that on one side of x = 2 the graph will approach +infinity, on the other it will be negative and approach -infinity.

......!!!!!!!!...................................

RESPONSE -->

ok

self critique assessment: 2

.................................................

......!!!!!!!!...................................

17:34:37

`q003. One straight line segment connects the points (3,5) and (7,9) while another connects the points (10,2) and (50,4). From each of the four points a line segment is drawn directly down to the x axis, forming two trapezoids. Which trapezoid has the greater area? Try to justify your answer with something more precise than, for example, 'from a sketch I can see that this one is much bigger so it must have the greater area'.

......!!!!!!!!...................................

RESPONSE -->

I dont know other than by sketching it out and seeing that (10,2) and (50,4) is the greater trapezoid.

confidence assessment: 1

.................................................

......!!!!!!!!...................................

17:37:14

Your sketch should show that while the first trapezoid averages a little more than double the altitude of the second, the second is clearly much more than twice as wide and hence has the greater area.

To justify this a little more precisely, the first trapezoid, which runs from x = 3 to x = 7, is 4 units wide while the second runs from x = 10 and to x = 50 and hence has a width of 40 units. The altitudes of the first trapezoid are 5 and 9,so the average altitude of the first is 7. The average altitude of the second is the average of the altitudes 2 and 4, or 3. So the first trapezoid is over twice as high, on the average, as the first. However the second is 10 times as wide, so the second trapezoid must have the greater area.

This is all the reasoning we need to answer the question. We could of course multiply average altitude by width for each trapezoid, obtaining area 7 * 4 = 28 for the first and 3 * 40 = 120 for the second. However if all we need to know is which trapezoid has a greater area, we need not bother with this step.

......!!!!!!!!...................................

RESPONSE -->

okay, you find the altitude and width of each trapezoid. Even though trap. 1 has a greater altitude, the width of trap 2 outweighs that when computing area.

self critique assessment: 2

.................................................

......!!!!!!!!...................................

17:40:23

`q004. If f(x) = x^2 (meaning 'x raised to the power 2') then which is steeper, the line segment connecting the x = 2 and x = 5 points on the graph of f(x), or the line segment connecting the x = -1 and x = 7 points on the same graph? Explain the basisof your reasoning.

......!!!!!!!!...................................

RESPONSE -->

the line segment connecting the x=-1 and x=7 points is steeper because when all 4 numbers are raised to power 2. (1 and 49) has a steeper slope than (4 and 25).

confidence assessment: 2

.................................................

......!!!!!!!!...................................

17:41:54

The line segment connecting x = 2 and the x = 5 points is steeper: Since f(x) = x^2, x = 2 gives y = 4 and x = 5 gives y = 25. The slope between the points is rise / run = (25 - 4) / (5 - 2) = 21 / 3 = 7.

The line segment connecting the x = -1 point (-1,1) and the x = 7 point (7,49) has a slope of (49 - 1) / (7 - -1) = 48 / 8 = 6.

The slope of the first segment is greater.

......!!!!!!!!...................................

RESPONSE -->

oh, oops. I looked at that totally wrong. I understand what it was asking now.

I know rise/run.

self critique assessment: 3

.................................................

......!!!!!!!!...................................

17:45:13

`q005. Suppose that every week of the current millenium you go to the jewler and obtain a certain number of grams of pure gold, which you then place in an old sock and bury in your backyard. Assume that buried gold lasts a long, long time ( this is so), that the the gold remains undisturbed (maybe, maybe not so), that no other source adds gold to your backyard (probably so), and that there was no gold in your yard before..

1. If you construct a graph of y = the number of grams of gold in your backyard vs. t = the number of weeks since Jan. 1, 2000, with the y axis pointing up and the t axis pointing to the right, will the points on your graph lie on a level straight line, a rising straight line, a falling straight line, a line which rises faster and faster, a line which rises but more and more slowly, a line which falls faster and faster, or a line which falls but more and more slowly?

2. Answer the same question assuming that every week you bury 1 more gram than you did the previous week.

{}3. Answer the same question assuming that every week you bury half the amount you did the previous week.

......!!!!!!!!...................................

RESPONSE -->

The points on the graph will lie on a leve straight line because y=the # of grams of gold in the backyard remains the same, whereas t=number of weeks is changing.

So with the y not moving and the x changing, you get a straight line to the right.

confidence assessment: 2

&#

Your response did not agree with the given solution in all details, and you should therefore have addressed the discrepancy with a full self-critique, detailing the discrepancy and demonstrating exactly what you do and do not understand about the given solution, and if necessary asking specific questions (to which I will respond).

&#

.................................................

......!!!!!!!!...................................

17:45:31

1. If it's the same amount each week it would be a straight line.

2. Buying gold every week, the amount of gold will always increase. Since you buy more each week the rate of increase will keep increasing. So the graph will increase, and at an increasing rate.

3. Buying gold every week, the amount of gold won't ever decrease. Since you buy less each week the rate of increase will just keep falling. So the graph will increase, but at a decreasing rate. This graph will in fact approach a horizontal asymptote, since we have a geometric progression which implies an exponential function.

......!!!!!!!!...................................

RESPONSE -->

self critique assessment: 3

.................................................

......!!!!!!!!...................................

17:47:15

`q006. Suppose that every week you go to the jewler and obtain a certain number of grams of pure gold, which you then place in an old sock and bury in your backyard. Assume that buried gold lasts a long, long time, that the the gold remains undisturbed, and that no other source adds gold to your backyard.

1. If you graph the rate at which gold is accumulating from week to week vs. tne number of weeks since Jan 1, 2000, will the points on your graph lie on a level straight line, a rising straight line, a falling straight line, a line which rises faster and faster, a line which rises but more and more slowly, a line which falls faster and faster, or a line which falls but more and more slowly?

2. Answer the same question assuming that every week you bury 1 more gram than you did the previous week.

3. Answer the same question assuming that every week you bury half the amount you did the previous week.

......!!!!!!!!...................................

RESPONSE -->

the points on the graph will lie on a rising straight line. The line is straight because you obtain the same # every time from the store.

confidence assessment: 3

.................................................

......!!!!!!!!...................................

17:48:11

This set of questions is different from the preceding set. This question now asks about a graph of rate vs. time, whereas the last was about the graph of quantity vs. time.

Question 1: This question concerns the graph of the rate at which gold accumulates, which in this case, since you buy the same amount eact week, is constant. The graph would be a horizontal straight line.

Question 2: Each week you buy one more gram than the week before, so the rate goes up each week by 1 gram per week. You thus get a risingstraight line because the increase in the rate is the same from one week to the next.

Question 3. Since half the previous amount will be half of a declining amount, the rate will decrease while remaining positive, so the graph remains positive as it decreases more and more slowly. The rate approaches but never reaches zero.

......!!!!!!!!...................................

RESPONSE -->

self critique assessment: 3

&#

This also requires a self-critique.

&#

.................................................

......!!!!!!!!...................................

17:52:28

``q007. If the depth of water in a container is given, in centimeters, by 100 - 2 t + .01 t^2, where t is clock time in seconds, then what are the depths at clock times t = 30, t = 40 and t = 60? On the average is depth changing more rapidly during the first time interval or the second?

......!!!!!!!!...................................

RESPONSE -->

at t=30 depth = 49

at t=40 depth =36

at t=60 depth =16

depth changes more rapidly from t=40 to t=60.

49-36= 13 36-16=20

self critique assessment: 3

.................................................

......!!!!!!!!...................................

17:52:38

At t = 30 we get depth = 100 - 2 t + .01 t^2 = 100 - 2 * 30 + .01 * 30^2 = 49.

At t = 40 we get depth = 100 - 2 t + .01 t^2 = 100 - 2 * 40 + .01 * 40^2 = 36.

At t = 60 we get depth = 100 - 2 t + .01 t^2 = 100 - 2 * 60 + .01 * 60^2 = 16.

49 cm - 36 cm = 13 cm change in 10 sec or 1.3 cm/s on the average.

36 cm - 16 cm = 20 cm change in 20 sec or 1.0 cm/s on the average.

......!!!!!!!!...................................

RESPONSE -->

self critique assessment: 3

&#

You need a detailed self-critique here.

&#

.................................................

......!!!!!!!!...................................

17:55:45

`q008. If the rate at which water descends in a container is given, in cm/s, by 10 - .1 t, where t is clock time in seconds, then at what rate is water descending when t = 10, and at what rate is it descending when t = 20? How much would you therefore expect the water level to change during this 10-second interval?

......!!!!!!!!...................................

RESPONSE -->

t=10 =9

t=20= 8

9+8 = 17/2 =8.5 8.5*10 =85 cm

confidence assessment: 1

.................................................

......!!!!!!!!...................................

17:56:06

At t = 10 sec the rate function gives us 10 - .1 * 10 = 10 - 1 = 9, meaning a rate of 9 cm / sec.

At t = 20 sec the rate function gives us 10 - .1 * 20 = 10 - 2 = 8, meaning a rate of 8 cm / sec.

The rate never goes below 8 cm/s, so in 10 sec the change wouldn't be less than 80 cm.

The rate never goes above 9 cm/s, so in 10 sec the change wouldn't be greater than 90 cm.

Any answer that isn't between 80 cm and 90 cm doesn't fit the given conditions..

The rate change is a linear function of t. Therefore the average rate is the average of the two rates, or 9.5 cm/s.

The average of the rates is 8.5 cm/sec. In 10 sec that would imply a change of 85 cm.

......!!!!!!!!...................................

RESPONSE -->

self critique assessment: 3

Good, but be sure to use units throughout your calculations.

.................................................

&#

This looks good. Let me know if you have any questions. &#