#$&* course MTH 173 2027 11MAR2013
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Given Solution: ** The denominator would never be 0, since x^2 must always be positive. So you could never have division by zero, and the function is therefore defined for every value of x. The function also has a smooth graph on this interval and is therefore continuous. The same is true of the correct Problem 4, which is 1 / `sqrt(2x-5) on [3,4]. On this interval 2x-5 ranges continuously from 2*3-5=1 to 2*4-5=3, so the denominator ranges continuously from 1 to `sqrt(3) and the function itself ranges continuously from 1 / 1 to 1 / `sqrt(3). ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q query problem 1.7.24 5th; 1.7.20 4th (was 1.11.9) continuity of sin(x) / x, x<>0; 1/2 for x = 0. Where is the function continuous and where is it not continuous? Your solution: It is continuous on the interval: (-inf, 0] and [0, inf) Where the graph breaks at zero it has a point of y = ½. But because it does break at zero, it is not continuous along all points. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: ** Division by zero is not defined, so sin(x) / x cannot exist at x = 0. The function is, however, not defined at x = 0 by sin(x) / x; the definition says that at x = 0, the function is equal to 1/2. It remains to see what happens to sin(x) / x as x approaches zero. Does the function approach its defined value 1/2, in which case the value of the function at x = 0 would equal its limiting value x = 0 and the function would be continuous; does it approach some other number, in which case the limiting value and the function value at x = 0 would not the equal and the function would not be continuous; or does the limit at x = 0 perhaps not exist, in which case we could not have continuity. Substituting small nonzero values of x into sin(x) / x will yield results close to 1, and the closer x gets to 0 the closer the result gets to 1. So we expect that the limiting value of the function at x = 0 is 1, not 1/2. It follows that the function is not continuous. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q Query problem Find lim (cos h - 1 ) / h, h -> 0. What is the limit and how did you get it? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: The limit as h approaches 0 is y = 0. We can get this by making a table with h values of .1, .01, .001, .0001, -.0001, -.001, -.01, -.1 and solving for y. We see that the sign of the y values changes as it crosses the origin, and that if we use zero for h, we get an undefined answer since we can’t divide by zero. Thus the limit as h approaches 0 is 0. confidence rating #$&*: 3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: ** For h = .1, .01, .001 the values of (cos(h)-1 ) / h are -0.04995834722, -0.004999958472, -0.0005. The limit is zero. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique Rating: ********************************************* Question: `q Query Add comments on any surprises or insights you experienced as a result of this assignment. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): #### I wasn’t too shocked by this because I have studied limits before. #### ------------------------------------------------ Self-critique Rating: STUDENT QUESTION: Is the limit also where the function becomes discontinuous? INSTRUCTOR RESPONSE: A function is continuous at a certain x value if, as you approach that x value, the limiting value of the function is equal to its value at the point. This is equivalent to the following two conditions: If the limiting value of a function y = f(x), as you approach a certain x value, doesn't equal the value of the function, then the function is not continuous. If the function doesn't have a limit at a certain x value, then the function is not continuous at that x value. "" " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: STUDENT QUESTION: Is the limit also where the function becomes discontinuous? INSTRUCTOR RESPONSE: A function is continuous at a certain x value if, as you approach that x value, the limiting value of the function is equal to its value at the point. This is equivalent to the following two conditions: If the limiting value of a function y = f(x), as you approach a certain x value, doesn't equal the value of the function, then the function is not continuous. If the function doesn't have a limit at a certain x value, then the function is not continuous at that x value. "" " Self-critique (if necessary): ------------------------------------------------ Self-critique rating: #*&!