course Mth 173 4/9 3 Question: `qQuery 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.
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Given Solution: `a** 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. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary):*wouldn’t the graph be concave down???
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Given Solution: `a** 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. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): Yes, a bird flying more slowly will use more energy in real life. ------------------------------------------------ Self-critique rating #$&* 3 ********************************************* Question: `qQuery Add comments on any surprises or insights you experienced as a result of this assignment. I had some difficulty with the graphical interpretations, but I think going over more notes can give me a better understanding &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): I like the realism. ------------------------------------------------ Self-critique rating #$&* 3