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course PHY 232
7/3
Question: `q001. There are 12 questions in this document.
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'?
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Your solution:
Slope = rise / run.
Between points (7, 17) and (10, 29) we get rise / run = (29 - 17) / (10 - 7) =12 / 3 = 4.
The slope between points (3, 5) and (7, 17) is 3 / 1. (17 - 5) / (7 -3) = 12 / 4 = 3.
The segment with slope 4 is the steeper. The graph being a smooth curve, slopes may vary from point to point. The slope obtained over the interval is a specific type of average of the slopes of all points between the endpoints.
We say on the average because the slope of the curve is continuously changing and we have to take and average measure.
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?
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Your solution:
1. As X gets closer to 2, the value of the expression tends to reach infinity.
2. If the above mentioned process is allowed to continue, yes it can reach those numbers.
3. Yes, it can.
4. Nope.
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Question: `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'.
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Your solution:
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.
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Question: `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 basis of your reasoning.
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Your solution:
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.
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Question: `q005. Suppose that every week of the current millennium you go to the jeweler 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.
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Your solution:
This situation would be represented by a line which will be a rising straight line. Since the gold is being added periodically, hence the straight line will increase at an increasing rate.
If 1 gram more is buried each week, the rise of the line will increase.
If half the amount is buried, then the line will still increase at an increasing rate but at a lesser rate than before. Or almost half the rate.
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Question: `q006. Suppose that every week you go to the jeweler 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. the 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.
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Your solution:
The graph will be a horizontal straight line since same amount of jewellery is bought each week.
Straight line because the rate of increase is same each week.
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.
``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?
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Your solution:
Depth = 100-2t+0.t^2
To find the depth at a particular time, we can substitute that value of t in the equation to find out the time.
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.
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Question: `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?
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Your solution:
To find the rate at which the water descends, we can plug in the value of t in the equation.
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 8.5 cm/s.
The average of the rates is 8.5 cm/sec. In 10 sec that would imply a change of 85 cm.
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Question: `q009. Sketch the line segment connecting the points (2, -4) and (6, 4), and the line segment connecting the points (2, 4) and (6, 1). The first of these lines if the graph of the function f(x), the second is the graph of the function g(x). Both functions are defined on the interval 2 <= x <= 6.
Let h(x) be the function whose value at x is the product of the values of these two functions. For example, when x = 2 the value of the first function is -4 and the value of the second is 4, so when x = 2 the value of h(x) is -4 * 4 = -16.
Answer the following based just on the characteristics of the graphs you have sketched. (e.g., you could answer the following questions by first finding the formulas for f(x) and g(x), then combining them to get a formula for h(x); that's a good skill but that is not the intent of the present set of questions).
What is the value of h(x) when x = 6?
Is the value of h(x) ever greater than its value at x = 6?
What is your best description of the graph of h(x)?
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Your solution:
The value of h(x) is1 when x=6.
No the value of h(x) cannot be greater than at x=6 because the range of the function is 2 <= x <= 6.
H(x) is the inverse function of f(x)
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Question: `q010. A straight line segment connects the points (3,5) and (7,9), while the points (3, 9) and (7, 5) are connected by a curve which decreases at an increasing rate. From each of the four points a line segment is drawn directly down to the x axis, so that the first line segment is the top of a trapezoid and the second a similar to a trapezoid but with a curved 'top'. Which trapezoid has the greater area?
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Your solution:
Since both trapezoids have the same base and height, they will have the same area. (Based on the judgment from figure).
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Question: `q011. Describe the graph of the position of a car vs. clock time, given each of the following conditions:
• The car coasts down a straight incline, gaining the same amount of speed every second
• The car coasts down a hill which gets steeper and steeper, gaining more speed every second
• The car coasts down a straight incline, but due to increasing air resistance gaining less speed with every passing second
Describe the graph of the rate of change of the position of a car vs. clock time, given each of the above conditions.
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Your solution:
1) Since the car travels equal distance in equal intervals of time, the graph will be a straight horizontal line.
2) The graph will increase at an increasing rate.
3) The graph will decrease at an decreasing rate.
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