HELP NEEDED PLEASE !!! After your first semester of calculus, you are lucky enou
ID: 2841551 • Letter: H
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HELP NEEDED PLEASE !!!
After your first semester of calculus, you are lucky enough to land a summer job working for NASA because of your knowledge of the mean value theorem. Given a proposed flight path for a manned rocket, your job is to determine about how rough or smooth the flight may seem to someone on board. Your employer has certain criteria that measure these quantities. The first is the angle between the x-axis and the line tangent to the flight path, which should not change too fast. If the rocket travels in a straight line, say y = mx + b, where m and b are constants, then the angle between the line tangent to the graph of y = mx + b and the x-axis is the same, no matter where the rocket may be. Compute the angle between the x-axis and the line tangent to the curve y = mx + b at any point. Do you get the same answer for different points of tangency? Certainly traveling in a straight line would be considered a smooth ride. Use your answer above to justify this statement. If this angle does change, then the flight path is curved, and the ride is a little rougher. Your supervisor realizes this and has already computed some function A(t), where t represents time, that gives the angle between the tangent to the flight path and some fixed axis. She then explains that part of your job is to determine how quickly A(t) is changing. If the angle A(t) is changing quickly, then the direction the rocket is headed is changing quickly as well. At this point you interrupt and question whether knowing the change in A(t) is enough information. After all, you expound, if the angles has changed 90 degree, then this could be a gradual 90 degree turn or a sharp 90 degree turn. We need to distinguish between these two cases. You continue that for a gradual 90 degree turn, a lot of distance is covered when making the turn, while very little distance is covered for a sharp turn. If we knew the distance traveled, then we could compare the change in the angle with the change in the distance. Impressed with these remarks, your supervisor explains that her engineering team has considered the distance function. However, they refer to this as arc length, L(t), since the distance traveled is the length of the arc of the rocket path. They have decided to consider the ''bending ratio", which is the change in the angle A(t) with respect to arc length. In other words, we would like to know the value of B(t1, t0) = A(t1) - A(t0)/L(t1) - L(t0). In this equation t0 and t1 are two time values that are close to each other. For your calculations of the bending ration, you will use the following lemma. Lemma: If A(t) and L(t) are continuous on [t0, t1], differentiable on (t0,t1), and , then there is some constant c with toExplanation / Answer
HELP NEEDED PLEASE !!! After your first semester of calculus, you are lucky enou
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