A train is moving west along a long straight track toward a station. An observer is located at a point 0.75 miles directly south of the train station with a camera (see figure below). 0.75 me (a) Let x represent the distance of the train to the station. Express x in terms of e (b) (c) The observer notices that when the train is 2.5 miles from the station, he is having to rotate his camera at a rate of 0.06 radians minute to keep the train in focus. Write an appropriate mathematical statement for each of these two pieces of information using the variables defined in this problem. Use Leibniz notation for any derivatives. Use implicit differentiation to find d when the train is 2.5 miles from the station means what about x? the observer rotating his camera 0.06 radians per minute means what about the angle ? (d) How fast is the train traveling at the instant described in part (c)? Give your answer to the nearest miles per hour.

4. As a tornado moves, its speed increases. The function S(d) = 93 log d + 65 a) Graph this function. b) Calculate the average rate of change for the speed of the wind at the centre of a tornado from mile 10 to mile 100. c) Estimate the rate at which the speed of the wind at the centre of a tornado is changing at the moment it has travelled its 10th mile and its 100th mile. d) Use your graph to discuss how the rate at which the speed of the wind at the centre of a tornado changes as the distance that the tornado travels increases.

Table Q3 shows the data from astronomical observations of a type of variable star called a Cepheid variable and represent variations in its apparent magnitude with time. Table Q3 Time, t 0.0 0.2 0.5 0.8 1.1 Apparent magnitude, f(t) 0.302 0.185 0.106 0.093 0.081 From the table above, estimate the apparent magnitude of the star when t = 0.10 and t = 0.95, using Newton divided difference interpolation method.

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