# Interactive video lesson plan for: 2003 ap calculus frq free response question ab bc #3 the rate of fuel consumption if gallons per min

#### Activity overview:

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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
(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 ?
(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

Available at apcentral.collegeboard.com

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