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Question 3

The rate of fuel consumption, in gallons per minute, recorded during an airplane flight is given by a

twice-differentiable and strictly increasing function R

of time t. The graph of R and a table of selected

values of R ( t ) , for the time interval O ~ t ~ 90

minutes, are shown above.

(a) Use data from the table to find an approximation

for R'( 45) . Show the computations that lead to

your answer. Indicate units of measure.

(b) The rate of fuel consumption is increasing fastest at

R"( 45) ? Explain your reasoning.

x

(minutes)

R(t)

(gallons l)er minute)

O

30

0

50

70

20

30

40

55

65

T'jme

time t - 45 minutes. What is the value of

(c) ~ thevalue of J0gq R(t) dt using a left Riemann sum with the five sub~ rvals indicated

by the data in the table. Is this numerical approximation less than the value of l~ R(t) dt ?

Explain your reasoning.

(d) For O b ~ 90 minutes, explain the meaning of fob n ( t ) dt in terms of fuel consumption for the

plane. Explain the meaning of ~ ~ R ( t) dt in terms of fuel consumption for the plane. Indicate

units of measure in both answers.

( \ DI{ A ~ \ ~ , R(50) - R(40) 55 - 40 I : a difference quotient using

taJ ~ t4~J ~ ~ -- -- s ~

` ' - ` - ' 50 ` 40 10 | numbers from table and

, 1.5 gal/min2 2 !

interval that contains 45

[ I : 1.5 gal/min2

(b) R"(45) -` 0 since R'(t) has a maximum at [ I : R"(45) = 0

2:{

t 45 . [ I : reason

(c) fo9o R(t) dt ~- (30)(20) + (10)(30) + (10)(40) 2 : _ 11 ~ ~al~ o~ ehftreRieaso~ nu sum

+(20)(55) + (20)(65) - 3700 `

Yes, this approximation is less because the

graph of R is increasing on the interval.

(d) ~ R(t) dt is the total amount of fuel in 2 : meanings

gallons consumed for the first b minutes. I : meaning of ~ R(t)dt

~fbo R(t) dt is the average value of the rate of 3 : 1 : meaning of ~ fbo R(t)dt

fuel consumption in gallons/min during the if no reference to time b

first b minutes. I : units in both answers

Copyright ~ 2003 by College Entrance Examination Board. All rights reserved.

Available at apcentral.collegeboard.com

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