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
Related questions
Question
Do all these parts of this problem
![in this particular scenario as follows. Be sure to include all units in your
explanations, but keep in mind that the explanations require more than just
units.
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 is
the 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 scales
on the axes. Explain your reasoning regarding why you think your curve is
realistic.
(d) Shade the appropriate region in your graph from part (c) whose signed
area represents the integral in part (a). Explain.
(e) Given that the integrand g(u)f'(u) is positive for u € [0,2], why is it
necessary 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, write
an expression for a function G = h(t), that expresses the number of gallons
of 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 that
takes the trip odometer reading as its input and returns the number of gallons
of gas in the tank as its output. Use t, the number of hours past noon, as the
parameter.
(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,](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faba01501-73d9-4793-a41a-2a1148a0bdf5%2Fa9a21d0b-f6c5-4f82-927c-7a9f0fa24cea%2Flzujod_processed.jpeg&w=3840&q=75)
Transcribed Image Text:in this particular scenario as follows. Be sure to include all units in your
explanations, but keep in mind that the explanations require more than just
units.
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 is
the 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 scales
on the axes. Explain your reasoning regarding why you think your curve is
realistic.
(d) Shade the appropriate region in your graph from part (c) whose signed
area represents the integral in part (a). Explain.
(e) Given that the integrand g(u)f'(u) is positive for u € [0,2], why is it
necessary 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, write
an expression for a function G = h(t), that expresses the number of gallons
of 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 that
takes the trip odometer reading as its input and returns the number of gallons
of gas in the tank as its output. Use t, the number of hours past noon, as the
parameter.
(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,
Transcribed Image Text: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)
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