course Mth163 Submitted Friday, June 30, 2009 at 7:00 pm. If your solution to stated problem does not match the given solution, you should self-critique per instructions at
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Given Solution: In terms of the symbol x1 the coordinates of the x = x1 point are ( x1, 1.1 x1 + .8) and the coordinates of the x = x2 point are ( x2, 1.1 x2 + .8). The rise between the two points is therefore rise = (1.1 x2 + .8) - (1.1 x1 + .8) = 1.1 x2 + .8 - 1.1 x1 - .8 = 1.1 x2 - 1.1 x1. The run is run = x2 - x1. The slope is therefore (1.1 x2 - 1.1 x1) / (x2 - x1) = 1.1 (x2 - x1) / (x2 - x1) = 1.1. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok Self-critique Rating: ********************************************* Question: `q002. For the function y = 3.4 x + 7, what are the coordinates of the x = x1 point, in terms of the symbol x1? What are the coordinates of the x = x2 point, in terms of the symbol x2? What therefore is the rise between these two points, and what is the run? What is the average slope of the graph between these two points? Be sure to simplify your result. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: In the function y = 3.4 x + 7, if x = x1 & x2, your coordinates would be: (x1, 3.4 x1+ 7) and (x2, 3.4 x2 + 7). The rise is: (3.4 x2 + 7) (3.4 x1 + 7) = 3.4 x2 3.4 x3 The run is: x2 x1. The average slope is: (3.4 x2 3.4 x1) / (x2 x1) = 3.4(x2 x1) / (x2 x1) = 3.4 Confidence Assessment: positive ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: In terms of the symbol x1 the coordinates of the x = x1 point are ( x1, 3.4 x1 + 7) and the coordinates of the x = x2 point are ( x2, 3.4 x2 + 7). The rise between the two points is therefore rise = (3.4 x2 + 7) - (3.4 x1 + 7) = 3.4 x2 + 7 - 3.4 x1 - 7 = 3.4 x2 - 3.4 x1. The run is run = x2 - x1. The slope is therefore (3.4 x2 - 3.4 x1) / (x2 - x1) = 3.4 (x2 - x1) / (x2 - x1) = 3.4. 009. `query 9 &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok Self-critique Rating: ********************************************* Question: `qSymbolic calculation of slope, preliminary exercise What was the function, between which two points were you to calculate the average slope and how did you get this slope? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The function is: y = .1x^2 3, between x = -2 and x = 7. The slope is (y2 y1) / (x2 x1), we have both the xs coordinates, we need to plug x in the equation to find the ys coordinates: y = .1(-2)^2 3 = -2.6 and y = .1(7)^2 3 = 1.9: giving the coordinates of (-2, -2.6) (7, 1.9) Slope = (y2 y1) / (x2 x1) = (1.9 - -2.6) / (7 - -2) = 4.5 / 5 = .9 Confidence Assessment: positive ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: ** For the function y = .1 x^2 - 3 between x = -2 and x = 7 we get: slope = (y2 - y1) / (x2 - x1). For x1 = 2 and x2 = 7 we have y2 = .1 * 7^2 - 3 = 1.9 and y1 = .1 * 2^2 - 3 = -2.6, so slope = (1.9 - (-2.6) ) / ( 7 - 2) = 4.5 / 5 = .9. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok Self-critique Rating: ********************************************* Question: `qproblem 2 symbolic expression for slope, fn depth(t). What is the expression for the slope between the two specified t values? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: For the function depth(t), given that t = 10 and t = 30 the expression for slope would be[ depth(30) depth(10)] / (30 10) = [depth(30) depth(10)] / 20 Confidence Assessment: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: positive ** The function is given a name: depth(t). t values are 10 and 30. So rise = depth(30) - depth(10) and run = 30 - 10 = 20. Thus slope = [ depth(30) - depth(10) ] / 20 . ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok Self-critique Rating: ********************************************* Question: `qWhat is the rise between the two specified t values? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The two specified t values is 10 & 30 and the rise is the difference of the two depth(t) so the rise would look like depth(30) depth(10) Confidence Assessment: positive ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: ** The rise is the change in depth. The two depths are depth(10) and depth(30). The change in depth is final depth - initial depth, which gives us the expression depth(30)-depth(10) ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I did mean the change in depth instead of the difference in depth. Self-critique Rating: ********************************************* Question: `qWhat is the run between the two specified t values? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The two specified t values is t = 10 and t = 30: the run is the change in time, which is 30 10 = 20 Confidence Assessment: positive ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: ** run = 30 - 10 = 20 ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok Self-critique Rating: ********************************************* Question: `qWhat therefore is the slope and what does it mean? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: Using the information from the last two questions to establish the slope: Slope is the change in depth divided by the change in time = [depth(30) depth(10)] / 30 10 = [depth(30) depth(10)] / 20 Confidence Assessment: positive ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: ** rise = depth(30)-(depth(10) indicates change in depth. run = 30 - 10 = 20 = change in clock time. Slope = rise / run = (depth(30) - depth(10) ) / 20, which is the average rate at which depth changes with respect to clock time between t = 10 and t = 30. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): 0k Self-critique Rating: ********************************************* Question: `q problem 5 graph points corresponding to load1 and load2 What are the coordinates of the requested graph points? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: With load = load1 and load2, being your x points and springlenth(load) = springlenth(load1) and springlenth(load2) being your y points: Your coordinates would be: (load1, springlength(load1)) and (load2, springlength(load2)) Confidence Assessment: positive ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: ** The horizontal axis is the 'load' asix, the vertical axis is the springLength axis. The load axis coordinates are load1 and load2. The corresponding spring lengths are springLength(load1) and springLength(load2). The springLength axis coordinates are springLength(load1) and springLength(load2). The graph points are thereofore (load1, springLength(load1) ) and (load2, springLength(load2) ). ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok Self-critique Rating: ********************************************* Question: `qWhat is your expression for the average slope of the graph between load1 and load2? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The slope would be the change in the springlength(load) divided by the change in the loads: [springlength(load2) springlength(load1)] / (load2 load1) Confidence Assessment: positive ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: ** rise = springLength(load2) - springLength(load1) run = load2 - load1 so slope = [ springLength(load2) - springLength(load1)] / (load2 - load1). ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok Self-critique Rating: ********************************************* Question: `q problem 6 symbolic expression for slope of depth function YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: In the function depth(t), t = t1 and t = t2, the slope would be the change in depth(t) and the change in t: [depth(t2) depth(t1)] / (t2 t1) Confidence Assessment: positive ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: ** the name of the function is depth(t). We need the slope between t = t1 and t = t2. The depths are depth(t1) and depth(t2). Thus rise is depth(t2) - depth(t1) and run is t2 - t1. Slope is [ depth(t2) - depth(t1) ] / (t2 - t1). ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ok Self-critique Rating: ********************************************* Question: `q problem 8 average rate from formula f(t) = 40 (2^(-.3 t) ) + 25 intervals of partition (10,20,30,40) What average rate do you get from the formula? Show your steps. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The average rate would be the change in rate divided by the change in time. Finding the rate change for each interval by using the function above: f(10) = 40*2^(-3 * 10) +25 = 25.00000004 f(20) = 40 * 2^(-3 * 20) + 25 = 25 f(30) = 40 * 2^(-3 * 30) + 25 = 25 f(40) = 40 * 2^(-3 * 40) + 25 = 25 The average rate for the three intervals: [f(20) f(10)] / (20 10) = (25 25.00000004) / 10 this will not solve on my calculator gives me -4E -9 [f(30) f(20)] /( 30 20) (25 25) / 10 = 0 [f(40) f(30)] / (40 30) = (25-25) / 10 = 0 Confidence Assessment: not sure ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: ** ave rate = change in depth / change in t. For the three intervals we get (f(20)-f(10))/(20-10) = (25.625 - 30 )/(20 - 10) = -4.375 / 10 = -.4375 (f(30)-f(20))/(30-20) = (25.07813 - 25.625)/(30 - 20) = -.5469 / 10 = -.05469. (f(40)-f(30))/(40-30) = (25.00977 - 25.07813)/(40 - 30) = -.0684 / 10 = -.00684. ** Add comments on any surprises or insights you experienced as a result of this assignment. YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: I understand according to our worksheet that the change in depth / change in time is the average rate as indicated in my solution. I also understand the it would be f(t2) f(t1) / t2 t1 as indicated in my work. I thought that you had to plug the t= = 10, 20, 30, & 40 into the function given to derive what f(t2) and f(t1) is. I have tried several time a=on my calculator to solve these functions and still dont get what you get for f(t2) and f(t1) also resulting in a different answer. Am I going about finding the change in depth wrong, and if so what is the correct way?
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Given Solution: STUDENT RESPONSE: Ummm I know the slope formula is (y2-y1)/(x2-x1), but I always just put the number into the expression in the order I see them, but that is ok because I keep the order and get the correct answere because the y2,y1,x2,x1 or all relative. I am correct in doing this? INSTRUCTOR COMMENT: In other words you use (y1 - y2) / (x1 - x2) instead of (y2 - y1) / (x2 - x1). It's more conventional to regard, say, 10 as x1 and 20 as x2, so f(20) is y2 and f(10) is y1. If you start from the lower x number and change to the higher the difference is higher - lower, and this is the way we usually think about changes. According to this convention we calculate change in y as y2 - y1 and change in x as x2 - x1. You are doing (y1 - y2) / (x1 - x2) and you get a negative change in x, a negative denominator, and if you are thinking about change from the first quantity to the second this is backwards. However as you say both numerator and denominator follow the same order so you still get the right answer, since (y1-y2)/(x1-x2)= (y2-y1) / (x2-x1). ** "