course Mth 173 Calculus I11-25-2007
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00:24:07 query class notes #33 Give the definition of a limit and explain what it means.
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RESPONSE --> A limit is what happens to a function when a variable nears a certain number.
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00:25:14 STUDENT RESPONSE: A limit is the resulting answer as x approaches some number or infinity. The limit is not that quantity exactly, but it is said to be as close as we can get without being exact. It is taken from an infinitely small interval from either side of what x approaches. INSTRUCTOR COMMENTS: That's a good expression of what it means. The formal definition, which is very necessary if we are to be sure we're on a solid foundation, is that lim{x -> a} f(x) = L if for any `epsilon, no matter how small, we can find a `delta such that whenever | x - a | < `delta, | f(x) - L | < `delta. **
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RESPONSE --> okay
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00:27:16 Give the definition of continuity and explain what it means.
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RESPONSE --> continuity is continous nature of a function, are there gaps in the function? A way of thinking about continuity is can the graph be drawn without the pen being lifted off the paper?
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00:29:53 STUDENT RESPONSE: For a function to be continuous, it has to have a limit. Continuity can be applied to sums, differences, products, constant multiples, and quotients where the denominator doesn`t equal zero. INSTRUCTOR CRITIQUE: ** The key is that at a given value x = a the limit of the function f(x) as we approach that a is equal to the value of the function at a--i.e., lim{x -> a} f(x) = f(a). If this is true for every x value on some interval, then the function is said to be continuous. **
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RESPONSE --> I described it more in the terms of a graph but the algebriac oriented definition makes sense.
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00:30:41 Given definition of differentiability and explain what it means.
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RESPONSE --> is the comparision of the function to it's dirivatives at the functions limits.
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00:41:23 STUDENT RESPONSE: Differentiability can be found by taking the limit as x approaches some value by both sides. A function has to have a limit and therefore have a derivative to be differentiable. INSTRUCTOR CRITIQUE: When we considered differentiability of f(x) at x = a we look at the limit of the expression [ f(a + `dx) - f(a) ] / `dx, specifically lim { `dx -> 0} [ f(a + `dx) - f(a) ] / `dx. If this limit exists, then the function is differentiable at x = a. In order to exist, the limit as `dx -> 0+ (i.e., 'from above' or through positive values of `dx) must exist and must equal the limit as `dx -> 0- (i.e., 'from below' or through negative values of `dx), which of course must also exist. **
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RESPONSE --> okay
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00:45:28 Query Theory 2, Differentiability and Continuity, Problem 6 (was problem 6 page 142 ) Q = C for t<0 and Ce^(-t/(RC)) for t>=0. Is the charge Q a continuous function of time?
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RESPONSE --> Both of the functions are continous and the question of the singular function continuity arises at the point at which they meet, so the limits for the fuctions at 0 where they meet needs to match for continuity. the limit is zero for both of them so the function is continous.
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00:50:13 You know that the exponential function Ce^(-t/(RC)) is continuous, and the constant function Q = C is continuous. Therefore for t < 0 and for t > 0 the function is continuous. The question arises at the point where the two functions meet--i.e., at t = 0. Do the right-and left-hand limits exist, and are the equal? The left-hand limit is that of the constant function Q = C. No matter how close you get to t = 0, this function will equal C and its limit will therefore equal C. You should be able to state and prove this in terms of `epsilons and `deltas. The right-hand limit is that of the exponential function, which continuously and smoothly approaches its t = 0 value C. You should think this through in terms of `epsilons and `deltas also, though a rigorous algebraic proof might be difficult at this stage. Therefore both limits exist and are equal, and the function is therefore continuous at t = 0.
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RESPONSE --> okay
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00:51:56 Is the current I = dQ / dt defined for all times?
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RESPONSE --> No because when t=0 the dirivative is undefined and therefore not continous.
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00:52:32 ** For t < 0, dQ / dt is the derivative of a constant function and is therefore zero. The derivative of the exponential function at t = 0 is not zero. Therefore the left-hand limit and the right-hand limit of the derivative differ at t = 0, and the derivative does not exist. **
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RESPONSE --> okay
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00:56:49 What is the derivative of Q = C for t < 0 and what is the derivative of C e^(-t/(RC)) for t>0? Does the derivative approach the same limit at t = 0 from the left as from the right? What does this have to do with the definition of the derivative?
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RESPONSE --> for T<0 the dirivative is 0 because it's a constant function and there for has no change so no slope for T>0 the dirivative is 1 / R e^(-t / (RC)) which isn't continous at t=0
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00:59:02 For t < 0 we have Q = C. The derivative of the Q = C function is at all points 0, since this is a constant function. So as we approach t = 0 from the left the limiting value of the derivative is zero. For t > 0 we have the function C e^(-t / (RC) ), which has derivative 1 / R e^(-t / (RC) ). At t -> 0 from the right this derivative, which is continuous, approaches -1 / R e^(-0 / R C) = -1 / R * 1 = -1 / R. This value is not 0. Since the derivative approaches 0 from the left and -1 / R from the right it is not continuous at t = 0. It follows that the derivative is not defined at t = 0.
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RESPONSE --> okay
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01:10:42 Query Theory 2, Differentiability and Continuity, Problem 0 (was problem 9 page 142) g(r) = 1 + cos(`pi r / 2) -2 <= r <= 2, 0 elsewhere. Is g continuous at r = 2? Explain.
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RESPONSE --> yes because the limit as r->2 is the same from both sides
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01:11:23 ** g(2) = 0, as is easily seen by substituting. Since the cosine function is continuous for r < 2, the limiting value of the function as r -> 2 form the left is 0. Since g(r) = 0 for r > 2 the limit of g(r) as r -> 2 from the right is 0. Since the limiting values are identical as we approach r = 2 from the right and left, the function is continuous at r = 2. **
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RESPONSE --> okay
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01:11:52 Is g differentiable at r = -2? Explain.
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RESPONSE --> g'= `pi / 2sin(`pi r / 2) and as r->-2 is 0 coming from both sides
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01:12:32 ** The derivative of g(r) is g ' (r) = 1 + cos(`pi r / 2) is `pi / 2 * sin(`pi r / 2). g(-2) = 0 and this derivative function is continuous, so as r -> -2 from the left this has limit zero. The derivative of y = 0, which is the function for r < -2, is 0. Since the limiting value of the derivative as we approach r = -2 is 0 from both sides, and the same is true at at r = 2, the derivative is continuous. **
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RESPONSE --> okay
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01:12:59 Query Add comments on any surprises or insights you experienced as a result of this assignment.
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RESPONSE --> This is sort of reinforcing the algebriac aspects of the derivative, limits, and tangent lines.
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01:13:25 I dealt better with this focus on theory than the other in the chapter. This shows a lot about how to prove what you have already known, and it shows you why things work in a particular way. Again there is more relevance and meaning added to the tangent approximation, and the factor of error.
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RESPONSE --> okay
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