course Mth 173
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17:36:02 Query problem 2.6.4. s(t) = 5 t^2 + 3 What are the functions for velocity and acceleration as functions of t?
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RESPONSE --> v(t) = s'(t) = 10t km/min a(t) = v'(t) = s''(t) = 10 km/min^2
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17:36:05 ** The velocity function is s ' (t) = 10 t and the acceleration function is s '' (t) = 10. *&*&
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17:39:30 Query problem 2.6.12. Function negative, increasing at decreasing rate. What are the signs of the first and second derivatives of the function?
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RESPONSE --> slope is always positive, so f'(x) is positive slope is decreasing, so f''(x) is negative
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17:39:42 *&*& The function is increasing so its derivative is positive. The slopes are decreasing, meaning that the rate of increase is decreasing. This means that the first derivative, which is represented by the slope, is decreasing. The second derivative is the rate of change of the derivative. Since the derivative is decreasing its rate of change is negative. Thus the second derivative is negative. *&*&
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17:40:45 Query problem 2.6.20 continuous fn increasing, concave down. f(5) = 2, f '(5) = 1/2. Describe your graph.How many zeros does your function have and what are their locations?
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RESPONSE --> the function has just one zero which will occur somewhere when x < 5, most likely close to 5
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17:42:15 ** The average slope between (0,0) and (5, 2) is .4. If the graph goes thru (0,0) then its slope at (5, 2), which is 1/2, is greater than its average slope over the interval, which cannot be the case if the graph is concave down. A constant slope of 1/2, with the graph passing through (5,2), would imply an x-intercept at (1, 0). Since the function is concave down the average slope between the x intercept and (5, 2) must be > 1/2 an x intercept must lie between (1,0) and (2,0). We can't really say what happens x -> infinity, since you don't know how the concavity behaves. It's possible that the function approaches infinity (a square root function, for example, is concave down but still exceeds all bounds as x -> infinity). It can approach an asymptote and 3 or 4 is a perfectly reasonable estimate--anything greater than 2 is possible. However the question asks about the limit at -infinity. As x -> -infinity we move to the left. The slope increases as we move to the left, so the function approaches -infinity as x -> -infinity. f'(1) implies slope 1, which implies that the graph makes an angle of 45 deg with the x axis; it's not horizontal. Because of the downward concavity the ave slope between x = 1 and x = 5 is greater than the slope at x = 5--i.e., greater than 1/2. This is the only restriction. A slope of 1 is therefore indeed possible. A slope of 1/4, or any slope less than 1/2, would be impossible. **
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RESPONSE --> This is far more precise - now I understand how to better approximate using test points such as the origin, and the slope of the function at a known point
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17:43:28 What is the limiting value of the function as x -> -infinity and why must this be the limiting value?
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RESPONSE --> the function approaches negative infinity as x approaches negative infinity. Since the function is only increasing and is doing so at a decreasing rate, it will decrease without bound as x -> -inf.
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17:43:55 STUDENT RESPONSE AND INSTRUCTOR COMMENT: The limiting value is 2, the curve never actually reaches 2 but comes infinitessimally close. INSTRUCTOR COMMENT: The value of the function actually reaches 2 when x = 5, and the function is still increasing at that point. If there is a horizontal asymptote, which might indeed be the case, it would have to be to a value greater than 2, since the function is increasing. **
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RESPONSE --> This wasn't really the question, was it?
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17:45:00 Is it possible that f ' (1) = 1? Is it possible that f ' (1) = 1/4?
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RESPONSE --> f'(1) cannot be one, since the slope is decreasing and f'(5) = 1/2 f'(1) could possibly be 1/4, since this is smaller than its value at 5
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17:46:10 ** f'(1) implies slope 1, which implies that if x and y scales are equal the graph makes an angle of 45 deg with the x axis; it's not horizontal. Because of the downward concavity the ave slope between x = 1 and x = 5 is greater than the slope at x = 5--i.e., greater than 1/2. This is the only restriction. A slope of 1 is therefore indeed possible. A slope of 1/4, or any slope less than 1/2, would be impossible. **
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RESPONSE --> Somehow I mixed these up - no problem, just a silly mistake. I understand quite clearly.
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17:48:49 Query problem 5.1.3 speed at 15-min intervals is 12, 11, 10, 10, 8, 7, 0 mph
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RESPONSE --> a) in first 30 sec - lower sum is 5.25 mi, upper is 5.75 mi b) over whole time - lower sum is 11.5 mi, upper is 14.5 mi
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17:49:05 ** your upper estimate would assume that the speed on each time interval remained at the maximum value given for that interval. For example on the first interval from 0 to 15 minutes, speeds of 12 mph and 11 mph are given. The upper estimate would assume the higher value, 12 mph, for the 15 minute interval. The estimate would be 12 mi/hr * .25 hr = 3 miles. For the second interval the upper estimate would be 11 mi/hr * .25 hr = 2.75 mi. For the third interval the upper estimate would be 10 mi/hr * .25 hr = 2.5 mi. For the fourth interval the upper estimate would be 10 mi/hr * .25 hr = 2.5 mi. For the fifth interval the upper estimate would be 8 mi/hr * .25 hr = 2.0 mi. For the sixth interval the upper estimate would be 7 mi/hr * .25 hr = 1.75 mi. The upper estimate would therefore be the sum 14.5 mi of these distances. Lower estimates would use speeds 11, 10, 10, 8, 7 and 0 mph in the same calculation, obtaining a lower estimate of the estimate 11.5 mph. **
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17:55:38 What time interval would result in upper and lower estimates within .1 mile of the distance?
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RESPONSE --> We want to find intervals such that upper - lower <= .1 12 mi/hr x <= .1 mi x = .0083 hr, or about 30 sec
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17:55:58 ** The right- and left-hand limits have to differ over an interval (a, b) by at most ( f(b) - f(a) ) * `dx. We want to find `dx that will make this expression at most .1 mile. Thus we have the equation ( f(b) - f(a) ) * `dx <= .1 mile. Since f(b) - f(a) = 0 - 12 = -12, we have | -12 mph | * `dx <= .1 mile so `dx <= .1 mile / 12 mph = 1/120 hour = 30 sec. **
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17:58:32 Query problem 5.1.13. Acceleration table for vel, estimate vel (at 1-s intervals 9.81, 8.03, 6.53, 5.38, 4.41, 3.61) Give your upper and lower estimates of your t = 5 speed and explain how you obtained your estimates.
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RESPONSE --> upper estimate = 34.16 m/s, which consists of the left-hand sum over the intervals 9.81*1 + 8.03*1 + 6.53*1 + 5.38*1 + 4.41*1 lower estimate = 27.96 m/s, which consists of the right-hand sum over the intervals 8.03*1 + 6.53*1 + 5.38*1 + 4.41*1 + 3.61*1
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17:58:39 For each interval we multiply the maximum or minimum value by the time interval. For each interval the maximum value given happens to be the left-hand value of the acceleration and the minimum is the right-hand value. Left-hand values give us the sum 9.81 m/s^2 * 1 sec +8.03 m/s^2 * 1 sec +6.53 m/s^2 * 1 sec +5.38 m/s^2 * 1 sec +4.41 m/s^2 * 1 sec =34.16 m/s. Right-hand values give us the sum 8.03 m/s^2 * 1 sec +6.53 m/s^2 * 1 sec +5.38 m/s^2 * 1 sec +4.41 m/s^2 * 1 sec +3.61 m/s^2 * 1 sec =27.96 m/s. So our lower and upper limits for the change in velocity are 27.96 m/s and 34.16 m/s. *&*&
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17:59:39 What is the average of your estimates, and is the estimate high or low (explain why in terms of concavity)?
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RESPONSE --> 31.06 m/s This is a high average - since this function is only decreasing, the curve will be concave up and thus lie slightly under our estimated trapezoidal (or average between upper and lower) estimates.
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18:00:00 ** The average of the two estimates is (27.96 m/s + 34.16 m/s) / 2 = 31.06 m/s. The graph of a(t) is decreasing at a decreasing rate. It is concave upward. On every interval a linear approximation (i.e., a trapezoidal approximation) will therefore remain above the graph. So the graph 'dips below' the linear approximation. This means that on each interval the average acceleration will be closer to the right-hand estimate, which is less than the left-hand estimate. Thus the actual change in velocity will probably be closer to the lower estimate than to the upper, and will therefore be less than the average of the two estimates. Another way of saying this: The graph is concave up. That means for each interval the graph dips down between the straight-line approximation of a trapezoidal graph. Your estimate is identical to that of the trapezoidal graph; since the actual graph dips below the trapezoidal approximation the actual velocity will be a bit less than that you have estimated. **
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18:00:02 Query Add comments on any surprises or insights you experienced as a result of this assignment.
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