course Mth 173 assignment #023£l±Ý‘ñئ‰’íÀUxÍ´x‚óq ê‹T{©àŹ§
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11:27:28 Query 4.3.34 (3d edition extra problem): Sketch a possible graph for a function which is positive, continuous, with a global maximum at (3,3); the 1st and 2d derivatives have the same sign for x<3, opposite signs for x > 3. Describe your graph, telling where it is increasing in decreasing, where it is concave up where it is concave down, and where (if anywhere) it has local maxima and minima.
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RESPONSE --> for t(x): we have a curving line concave down that has a maximum at point (3,3) for t(x)': we have a positive value for x<3 and a negative value for x>3 indicating that the function will slowly gain and then at x=3 reach a point of inflection which is the global maximum then goes to a negative slope. for t''(x): we have a continusly decreasing slope.
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11:31:08 ** The function would have to be increasing for x < 3, which would make the first derivative positive. The second derivative could also be positive, with the function starting out with an asymptote to the negative x axis and gradually curving upward to reach (3,3). It would then have to start decreasing, which would make the first derivative negative, so the second derivative would have to be positive. The function would have be sort of 'pointed' at (3,3). The graph, which would have to remain positive, could then approach the positive x axis as an asymptote, always decreasing and always concave up. The horizontal asymptotes would not have to be at the x axis and could in fact by at any y < 3. The asymptote to the right also need not equal the asymptote to the left. **
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RESPONSE --> How can you tell when the graph will become asymptotic without the having an actual function?
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11:36:27 Query problem 4.3.31 (3d edition 4.3.29) f(v) power of flying bird vs. v; concave up, slightly decreasing for small v; a(v) energy per meter. Why do you think the graph as the shape it does?
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RESPONSE --> because the as the bird flies it uses more energy to add to it's speed and further fight gravity.
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11:38:25 ** the graph actually doesn't give energy vs. velocity -- the authors messed up when they said that -- it gives the rate of energy usage vs. velocity. They say this in the problem, but the graph is mislabeled. The graph says that for high velocities the rate of energy usage, in Joules / second, increases with increasing velocity. That makes sense because the bird will be fighting air resistance for a greater distance per second, which will require more energy usage. To make matters worse for the bird, as velocity increases the resistance is not only fought a greater distance every second but the resistance itself increases. So the increase in energy usage for high velocities isn't too hard to understand. However the graph also shows that for very low velocities energy is used at a greater rate than for slightly higher velocities. This is because low velocities imply hovering, or near-hovering, which requires more energy than the gliding action the bird achieves at somewhat higher velocities. **
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RESPONSE --> okay that makes sense, the way the book sad it was an energy vs. distance graph was a little confusing.
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11:43:16 Query Add comments on any surprises or insights you experienced as a result of this assignment.
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RESPONSE --> Optimization is essintially the analysis of everything to find it's maximum effciency. The most effcient point is either the minumum or maxium of the function or the combined ""average"" minimum of of two or more functions.
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11:43:32 I had some difficulty with the graphical interpretations, but I think going over more notes can give me a better understanding
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RESPONSE --> okay
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