#$&*
course Mth 163
6/6 around 12:00 am.
003. PC1 questions
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Question: `q001 A straight line connects the points (3, 5) and (7, 17), while another straight line continues on from (7, 17) to the point (10, 29). Which line is steeper and on what basis to you claim your result?
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Your solution:
The steeper line is the one that connects the points (7, 17) and (10, 29) because it has a bigger slope compared to the line that connects the points (3, 5) and (7, 17). For example if we calculate the slope of the points (7, 17) and (10, 29) by using the slope formula (y2- y1)/(x2-x1) it becomes (29-17)/(10-7) which is 4. On the other hand, if we do the same thing for the points (3, 5) and (7, 17) it becomes (17-5)/(7-3) which is 3. Therefore a slope of 4 is steeper than a slope of 3 which means that the steeper line is the one that connects the points (7, 17) and (10, 29)
confidence rating #$&*: 3
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Self-critique (if necessary): OK
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Self-critique Rating: OK
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Question: `q002. The expression (x-2) * (2x+5) is zero when x = 2 and when x = -2.5. Without using a calculator verify this, and explain why these two values of x, and only these two values of x, can make the expression zero.
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Your solution:
The reason why only when x = 2 and when x = -2.5 the expression (x-2) * (2x+5) is zero is because when I replaced the 2 into the expression it became (2-2) * (2(2) + 5) = 0 * 9 = 0. Also, when I replaced the -2.5 into the expression it became (-2.5 - 2) * (2(-2.5) +5) = -4.5 * (-5 + 5) = -4.5 * 0 = 0. The reason why the result is 0 on both times is because one of the terms is 0 so when multiplying by the other term even though it isn“t 0 any number multiplied by 0 is 0.
confidence rating #$&*: 3
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Self-critique (if necessary): OK
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Self-critique Rating: OK
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Question: `q003. For what x values will the expression (3x - 6) * (x + 4) * (x^2 - 4) be zero?
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Your solution:
The expression (3x - 6) * (x + 4) * (x^2 - 4) will be zero for the x values of x = 2, x = -4, and x = 2 or x = -2. The reason for this result is because in order to find the x values for when the expression is 0 I need to write each of the terms equal to zero. For example, when 3x - 6 = 0 it becomes x = 2; when x + 4 = 0 it becomes x = -4; and when x^2 - 4 = 0 it becomes x = 2 or x = -2. So in the original expression (3x - 6) * (x + 4) * (x^2 - 4) if I substitute any of these x values into the expression the result is 0.
confidence rating #$&*: 3
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Self-critique (if necessary): OK
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Self-critique Rating: OK
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Question: `q004. 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:
The trapezoid with the bigger area is the one that has the points (10,2) and (50,4) because when I measured the first trapezoid“s width it was just 4 units wide, and the second was 50 units wide which means the second one has the greater area.
confidence rating #$&*:: 3
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Self-critique (if necessary): I did not include in my answer the fact that the average altitude of the first trapezoid was more than the average of the second trapezoid although that doesn“t change the fact that the second trapezoid has a greater area than the first.
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Self-critique Rating: 3
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Question: `q005. Sketch graphs of y = x^2, y = 1/x and y = `sqrt(x) [note: `sqrt(x) means 'the square root of x'] for x > 0. We say that a graph increases if it gets higher as we move toward the right, and if a graph is increasing it has a positive slope. Explain which of the following descriptions is correct for each graph:
As we move from left to right the graph increases as its slope increases.
As we move from left to right the graph decreases as its slope increases.
As we move from left to right the graph increases as its slope decreases.
As we move from left to right the graph decreases as its slope decreases.
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Your solution:
The first description is correct for the graph of y = x^2 because for each unit we move in the x axis, the values of the y axis almost double; therefore the graph is increasing at an increasing rate. For example, by observing the points (1, 1), (2, 4), (3, 9), and (4, 16) we can see that the y values are increasing very fast. So as we move from left to right the graph increases as its slope increases.
The last description is correct for the graph of y = 1/x because when I graphed the points (1, 1), (2, ½), (3, 1/3), and (4, ¼) I noticed that the y values were becoming less and less compared to the x values that for each unit we move right in the x axis the y values decrease by more than half each time. So as we move from left to right the graph decreases as its slope decreases.
The third description is correct for the graph y = `sqrt(x) because when I graphed the points (1, 1), (2, 1.414), (3, 1.732), and (4, 2) I noticed that while the line was still increasing the y values were increasing very slowly, for each unit we moved right in the x axis, the y axis values increased just a little bit. So as we move from left to right the graph increases as its slope decreases.
confidence rating #$&*: 3
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Self-critique (if necessary): I did not mention the fact that as the numbers decrease they get closer to zero which leads to a negative slope which is why the second graph is decreasing at a decreasing rate.
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Self-critique Rating: 3
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Question: `q006. If the population of the frogs in your frog pond increased by 10% each month, starting with an initial population of 20 frogs, then how many frogs would you have at the end of each of the first three months (you can count fractional frogs, even if it doesn't appear to you to make sense)? Can you think of a strategy that would allow you to calculate the number of frogs after 300 months (according to this model, which probably wouldn't be valid for that long) without having to do at least 300 calculations?
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Your solution:
I started to calculate 20 frogs by 10 % like this 20 * 0.1 which is 2, and then I added the initial 20 frogs which is 22. This means that by the end of the first month I have 22 frogs. By the end of the second month I have 24.2 by multiplying the 22 by 0.1 and adding the 22. By the end of the third month I have 26.62 by doing the same process I did before.
confidence rating #$&*: 2
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Self-critique: I did not know how to do the second task of the exercise which was to think of a strategy that would allow me to calculate the number of frogs after 300 months. When I read the given solution it made sense to me and I was able to understand that multiplying 20 by 1.1 is the same as multiplying 20 by 0.1 and then adding 20. So by multiplying 20 times 1.1 to the 300th power we have the answer to how many frogs there are after 300 months which by my calculations is 5.234*10^13.
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Self-Critique rating: 3
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Question: `q007. Calculate 1/x for x = 1, .1, .01 and .001. Describe the pattern you obtain. Why do we say that the values of x are approaching zero? What numbers might we use for x to continue approaching zero?What happens to the values of 1/x as we continue to approach zero? What do you think the graph of y = 1/x vs. x looks for x values between 0 and 1?
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Your solution:
When I calculated 1/x for x=1 I got 1, for x=.1 I got 10, for x=.01 I got 100, and for x=.001 I got 1000. The reason why we say that the values of x are approaching zero is because they are .01, .001, etc. Other numbers may be .0001, .00001, etc. The values of y=1/x are multiples of 10 like 100, 1000, 10000, etc. the graph looks like a very steep line increasing from left to right.
confidence rating #$&*: 3
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Self-critique (if necessary): OK
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Self-critique Rating: OK
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Question: `q008. At clock time t the velocity of a certain automobile is v = 3 t + 9. At velocity v its energy of motion is E = 800 v^2. What is the energy of the automobile at clock time t = 5?
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Your solution:
v=3(5)+9=24
E=800*24^2= 800 * 576= 460,800
The energy of the automobile is 460,800.
confidence rating #$&*: 3
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Self-critique (if necessary): OK
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Self-critique Rating: OK
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Question: `q009. Continuing the preceding problem, can you give an expression for E in terms of t?
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Your solution:
E=800(3 t + 9)^2
confidence rating #$&*: 3
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Self-critique (if necessary): OK
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Self-critique Rating: OK
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