#$&* course MTH 173 12/18 23:40
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Given Solution: `a** There is a set S of numbers for which f(x) < 0, and since for the increasing function f(x) we have f(1.7) < 0 < f(1.8), as you correctly point out, it should be obvious that that set S contains 1.7 and does not contain 1.8 or any number > 1.8. Therefore 1.8 is an upper bound for that set. The set S is nonempty (it contains 1.7) and it has an upper bound. Therefore by the Completeness Axiom it has a least upper bound--of all the numbers, 1.8 included, that bound the set from above, there is a smallest number among all the upper bounds. Since S is the set of all values for which f(1.7) < 0, then if r is the least upper bound of the set, we must have f(r) = 0. If f(r) > 0 then by the continuity and increasing nature of f(x) there would be some smaller x, i.e., x < r, for which f(x) > 0, in which case r couldn't be the least upper bound. So f(r) <=0. But f(r) can't be less than 0 because then there would exist and x > r for which f(r) < 0 and r wouldn't be an upper bound at all. So were stuck: if r is the least upper bound then f(r) = 0. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `qWhat does the Nested Interval Theorem tell us? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Given an infinite sequence of closed intervals, [an; bn], each one contained within the previous one, then there is at least one number in all the intervals. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** It tells us that an infinite sequence of nested closed intervals contains at least one common point. These intervals, which contain their endpoints (i.e., are closed), can't just to put away to nothing. We cannot construct, even if we try, and infinite sequence of closed nested intervals for which there is not a common point. The sequence can't just dwindle away to nothing. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: What does the Intermediate Value Theorem tell us, and how does this help assure us that the polynomial of the first question has a zero between x = 1.7 and x = 1.8? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: According to the intermediate value theorem Suppose f is continuous on a closed interval [a; b]. If k is any number between f (a) and f (b), then there is at least one number c in [a; b] such that f (c) = k. We know that f(1.7) is < 0 and f(1.8) > 0 and thus function is continuously increasing with in the interval. According to the intermediate value theorem since 0 is between f(1.7) and f(1.8), then there is atleast one number c such in [a,b] such that f(c) = 0. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: If f(x) is continuous between two points a and b, then f(x) takes every value between f(a) and f(b). Since the polynomial is negative and one of the given x values and positive at the others, and since a polynomial is continuous at every value of x, it follows from the Intermediate Value Theorem that the polynomial must take the value 0 somewhere between these x values. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `qQuery Theory 1, Problem 2 (was Problem 2 p. 85) r is contained in a sequence of nested intervals whose width approaches 0 as n -> infinity; prove r is unique. How do you prove that r is unique? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: We assume that there are 2 r’s, r1 and r2 and since they are not equal |r2 - r1| > 0 Now as the width of the nested intervals approach 0, that is n tends to infinity, the width value decreases to a value much smaller than any value we can imagine. Thus it is obvious that the width will be less than |r2 - r1| for some large large value of n. When the width of that interval is smaller than |r2 - r1| it becomes impossible for the interval to contain both r1 and r2 in it. This proves that there can eventually be only one value r that can exist in all the ever decreasing intervals. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aSTUDENT SOLUTION: This is all that I have: lim as n app infinity of b of n - a of n r is greater than or equal to a of n for all n r is less than or equal to b of n for all n when r app zero, there is a unique number r INSTRUCTOR CRITIQUE: r doesn't approach zero, the width of the intervals approaches zero. Suppose there were two values, r1 and r2, and that |r2 - r1| wasn't zero. Since the sequence of nested intervals has width approaching 0, eventually the width will be less than any number we might choose. In particular the width will eventually be < | r2 - r1 |, meaning that for some N, if n > N the width of (an, bn) must be < |r2 - r1|. The two values r2 and r1 therefore cannot both exist in any of these intervals, which contradicts the assumption that there could be two different numbers both contained in all the intervals. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `qQuery Theory 2 Problem 12 from Limits and Continuity (was problem 12 p 134) various limits at infinity How do you algebraically manipulate the expression (x+3) / (2-x) so that its limit as x -> infinity can be easily obtained, and what is the resulting limit? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Given function (x + 3) / (2 - x) We can see that as x tends to infinity, the numerator tends to infinity and the denominator tends to - infinity. Now if we take x common from the numerator and denominator we get (1 + 3/x) / (-1 + 2/x) Now as x tends to infinity 3/x and 2/x both tend to 0 and the value of the limit tends to -1. Thus lim x tending to 0 (x + 3) / (2 - x) = -1 confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aSTUDENT RESPONSE: Simply by plugging a large number into the equation for x. Then repeating with even larger numbers to verify. resulting limit -1. INSTRUCTOR CRITIQUE: That's not an algebraic manipulation. You can't rigorously prove anything by plugging in numbers--the behavior might be different if you used different numbers. ANOTHER STUDENT RESPONSE: I did this by using p. 129 to simplify and take quantities off to themselves to get an answer I found that this equation ultimately comes down to (inf+3)/(2-inf) The top portion will be positive by adding an infinitely large number, while the bottom portion becomes negative by adding an arbitrarily large number, whochgives +/- or -inf INSTRUCTOR CRITIQUE AND SOLUTION: You have the right idea, but inf (meaning infinity) is not a number so you can't formally deal with a quantity like (inf + 3) / (2 - inf). If you divide both numerator and denominator of (x+3) / (2-x) by x you get ( 1 + 3/x) / ( 2/x - 1). Then as x -> infinity, meaning as x gets as large as you might wish, 3 / x and 2 / x both approach 0 as a limit, meaning that each of these quantities can be made to be as close to zero as we might wish. As a result ( 1 + 3/x) / ( 2/x - 1) can be made as closed as we wish to 1 / -1 = -1. The limit is therefore -1. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `qHow do you algebraically manipulate the expression (2 e^x + 3) / (3 e^x + 2) so that its limit as x -> infinity can be easily obtained, and what is the resulting limit? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Given function (2e^x + 3) / (3e^x + 2) As x tends to infinity, the numerator tends to infinity and so does the denominator Thus to calculate the limit we divide the numerator and the denominator by e^x. On doing so we get (2 + 3/e^x) / (3 + 2/e^x) Now as x tends to infinity 3/e^x and 2/e^x, both tend to 0 and thus the ratio tends to 2/3 Thus lim x tending to infinity (2e^2 + 3) / (3e^x + 2) = 2/3 confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aSTUDENT RESPONSE: As the x value grows arbitrarily larger, both the top and bottom portions will go to infinity, which yields positve infinity as the result INSTRUCTOR CRITIQUE AND SOLUTION The same could be said of x / x^2, but the limit isn't infinite. If we divide both numerator and denominator by e^x we get (3 + 2 / e^x) / (2 + 3 / e^x). The terms 2/e^x and 3 / e^x approach zero so the fraction approaches 3/2.** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `qQuery Theory 2 Problem 18 from Limits and Continuity (was problem 18 p. 135) find positive delta such that the graph leaves the 2*epsilon by 2*delta window by the sides; fn -2x+3, a = 0, b = 3. Give your solution. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Given function f(x) = 2x + 3 The function passes through the point (0,3) where a = 0 and b = 3. We need to find a positive delta value such the f(x) function leaves the window by the sides. The 2 delta window will have delta interval in either sides of x = a which is 0. Thus the interval is (-delta, + delta) This represents a vertical strip along the y axis with the y axis centered on the strip and the strip extending on either sides by delta units. For epsilon similar will be the case so but here the strip will be horizontal with y = 3 in the middle and extending by epsilon lengths on either side of y = 3. Thus the 4 rectangle formed is the intersection of the 2 strips with 3 - epsilon < y < 3 + epsilon and - delta < x < delta We assume small values of epsilon = .1 and delta = .01 Thus 2.9 < y < 3.1 and -0.01 < x < 0.01 The value of the function at x = -0.01 is f(-0.01) = 2(-0.01) + 3 = -0.02 + 3 =2.98 which is within the interval of 2.9 < x < 3.1 The value of the function at x = 0.01 is f(0.01) = 2(0.01) + 3 = 3.02 which is within the interval of 2.9 < x < 3.1 Thus for this combination of epsilon and delta we see that the function enters the 2delta * 2 epsilon window from the sides instead of the bottom. By further reducing the value of delta we get function which passes through the sides instead of the top and bottom. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `a** The graph of this function is a straight line passing through (a, b) = (0, 3) with a slope of -2. The window we are talking about lies within a `delta-interval about x = a--i.e., with the interval (a - `delta, a + `delta). This is an interval that extends to distance `delta on either side of x = a, and is therefore centered at a. You should sketch a picture of such an interval. Since a = 0, the x interval is (-`delta, +`delta). You should sketch a picture of this specific interval, thinking of `delta as a small number which can be made as small as we wish. On the plane, this interval defines a vertical 'strip' centered at the y axis and extending distance `delta on either side. You should sketch a picture of this 'strip'. The number `epsilon defines base similar interval on y values--an `epsilon-interval about b, extending from b-`epsilon to b+`epsilon. You should sketch a picture of such an interval. For the specific value b = 3, the `epsilon-interval is (3 - `epsilon, 3 + `epsilon). You should sketch this interval. This interval defines a horizontal 'strip' centered at the line y = 3 and extending to distance `epsilon on both sides. You should sketch a picture of such a strip. The two strips, one vertical and one horizontal, meet to form a rectangle at their intersection. This rectangle is a 'window' whose width is 2 `delta and whose height is 2 * `epsilon. Sketch the graph of the function in your window. If your `delta is too large for your `epsilon, the graph will exit the window at top and bottom, not at the sides. We regard `epsilon as given, and we adjust `delta so that the graph exits the window at the sides. It should be clear that by making `delta small enough this is possible. It should also be clear that this will be possible no matter how small `epsilon is chosen. Thus FOR EVERY `epsilon, no matter how small, THERE EXISTS a `delta such that WHENEVER | x - a | < `delta, IT FOLLOWS THAT | f(x) - b | < `epsilon. This is what defines and proves the statement that limit {x -> a} (f(x)) = b. ** INSTRUCTOR SUGGESTIONS FOR THE PERPLEXED: If you don't understand this problem, try to following. Actually attempt to do the steps, sketch your window and the graph, before you read the explanation: First verify that the graph of the function y = -2x + 3 passes through the point (a, b) = (0, 3) of the xy plane. Now sketch the 'window' for epsilon = .1 and delta = .2: The 'window' is a rectangle which extends epsilon units above and below the point (a, b), and delta units to the right and left of this point. The point (a, b) will be in the middle of the window. The y coordinate therefore extends .1 unit above and below y = 3, from y = 2.9 to y = 3.1. The x coordinate extends .2 units to the left and right of x = 0, from x = -.2 to x = .2. a So the 'window' consists of the rectangle 2.9 < y < 3.1, -.2 < x < .2. Sketch the graph of the function within this window: For x = -.2 we get y = 3.4, and for x = +.2 we get y = 2.6. The point (-.2, 3.4) lies above the window, and the point (.2, 2.6) lies below, so neither of these points lies on or within the boundary of the window. For y = 2.9, we find that x = .05. So the point (.05, 2.9) lies on the lower boundary of the window. For y = 3.1, we find that x = -.05. So the point (-.05, 3.1) lies on the upper boundary of the window. Thus our graph within the window runs from the point (-.05, 3.1) to the point (.05, 2.9). Does the graph leave the window by the sides, or by the top and/or bottom of the window? The graph leaves the window by the top and bottom. The x = -.2 and x = +.2 points are too far above and below y = 3 to exit by the sides of the window. So this window doesn't fulfill the condition of the problem. We conclude that epsilon = .1 and delta = .2 doesn't fulfill the specified condition. Now try it again for epsilon = .1 and delta = .01. Your window will be the rectangle -.01 < x < .01, 2.9 < y < 3.1. When x = -.01, y = 3.03. The corresponding point (-.01, 3.03) lies on the left-hand boundary of the window. When x = .01, y = 2.97. The corresponding point (.01, 2.97) lies on the right-hand boundary of the window. The graph now leaves the window through its sides. You might want to try a few more combinations of epsilon and delta to get a better feel for this situation. Then read the given solution once more, and feel free to submit additional questions. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK ********************************************* Question: `qQuery Add comments on any surprises or insights you experienced as a result of this assignment. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: `aThis has been by far the most difficult assignment ever in this course. I have not had much experience at proving things that I know to be true, just because they are. I need lots of help on this assignment. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): OK ------------------------------------------------ Self-critique Rating: OK "" " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: "" " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!