Do all these parts of this problem in this particular scenario as follows. Be sure to include all units in yourexplanations, but keep in mind that the explanations require more than justunits.i) Explain the meaning of the differential du.ii) Explain the meaning of the product f'(u) du.iii) Explain the meaning of the product g(u)(f'(u) du).iv) Finally, explain the meaning of the definite integral.(b) The definite integral in part (a) is a function of what variable? What isthe appropriate domain for this function?(c) Sketch by hand a realistic possibility for the parametrically-defined curve(s,r) = (f(t), g(t)), 0≤ t ≤ 2.Label the axes with the appropriate variable names and units. Put scaleson the axes. Explain your reasoning regarding why you think your curve isrealistic.(d) Shade the appropriate region in your graph from part (c) whose signedarea represents the integral in part (a). Explain.(e) Given that the integrand g(u)f'(u) is positive for u € [0,2], why is itnecessary to speak of signed area instead of just area in part (d)?(f) Rembering that there are 10 gallons of gas in the tank at 1:00pm, writean expression for a function G = h(t), that expresses the number of gallonsof gas in the tank at time t hours past noon. Include the appropriate domain.Do this as simply as possible and explain your reasoning.(g) Use your result from part (f) to parameterize the graph of a function thattakes the trip odometer reading as its input and returns the number of gallonsof gas in the tank as its output. Use t, the number of hours past noon, as theparameter.(h) Choose specific functions f ang g so that(i) the speed is always between 10 miles/hour and 70 miles/hour,(ii) the variation in the speed is at least 40 miles/hour during the trip, andThe continuously differentiable functionss = f(t), 0≤ t ≤ 2,r = g(t), 0≤t≤ 2,respectively express your trip odometer reading (in miles) and the rate atwhich your car burns fuel (in gal/mile), in terms of the number of hours pastnoon on a two-hour trip.The tank has 10 gallons of gas at 1:00pm. The trip odometer reads 0miles at noon. Your car never comes to a stop during the trip.(a) Interpret the meaning of the definite integral[*9(u)f'(u) du = [[*9(u) (f' (u) du)

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and The continuously differentiable functions s = f(t), 0≤t≤ 2, r=g(t), 0≤t≤ 2, respectively express your trip odometer reading (in miles) and the rate at which your car burns fuel (in gal/mile), in terms of the number of hours past noon on a two-hour trip. The tank has 10 gallons of gas at 1:00pm. The trip odometer reads 0 miles at noon. Your car never comes to a stop during the trip. (a) Interpret the meaning of the definite integral [*9(u)f'(u) du = [*9(u)(f' (u) du)

and The continuously differentiable functions s = f(t), 0≤t≤ 2, r=g(t), 0≤t≤ 2, respectively express your trip odometer reading (in miles) and the rate at which your car burns fuel (in gal/mile), in terms of the number of hours past noon on a two-hour trip. The tank has 10 gallons of gas at 1:00pm. The trip odometer reads 0 miles at noon. Your car never comes to a stop during the trip. (a) Interpret the meaning of the definite integral [9(u)f'(u) du = [9(u)(f' (u) du)

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Do all these 3 parts of this short problem