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VASS, 2013 AP* is a trademark of the College Board. The College
Board was not involved in the production of this packet.
Student Session Topic: Average and Instantaneous Rates of
Change
The concepts of average rates of change and instantaneous rates
of change are the building blocks of
differential calculus. The AP exams tend to incorporate these
concepts in application problems both
with and without a calculator. There can be a straightforward
question in the multiple‐choice section. More typically, on the
free‐response section a rate of change question will be part of a
longer question.
Basic concepts
Average rate of change
o The average rate of change of ( )f x on [a, b] is the slope of
the line through the points
( , ( ))a f a and ( , ( ))b f b .
o The average rate of change of is ( ) ( )f b f a
b a
o The average rate of change of ( )f x on [a, b] is NOT the same
as the average value of ( )f x
on a, b]. Average value of ( )f x on [a, b] is
( )
b
a
f x dx
b a
.
.
Instantaneous rate of change
o The instantaneous rate of change of ( )f x at x = a is the
slope of the line tangent to the graph of
( )y f x at the point ( , ( ))a f a .
o The instantaneous rate of change of ( )f x at x = a is ( ) (
)
lim '( )x a
f x f af a
x a
What Students Should Be Able To Do
Distinguish between average and instantaneous rates of
change.
Find the average rate of change of a dependent variable with
respect to an independent variable on an interval
Find the instantaneous rate of change of a dependent variable
with respect to an independent variable at a point.
Distinguish between an average rate of change and an average
value of a function.
In this session, you will practice with 5 multiple choice
problems and all or part of the following free
response problems: 2007 Form B AB/BC3, 2011 Form B AB/BC 1, 2004
AB/BC 1, 2004 Form B AB 2,
1999 AB 6.
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Multiple Choice
Non-Calculator
_____1. What is the instantaneous rate of change at x = 2 of the
function f given by
2 2( )
1
xf x
x
?
(A) -2 (B) 1
6 (C)
1
2 (D) 2 (E) 6
Calculator
_____2. Let f be a function given by 2( ) 3 xf x e and let g be
the function given by 3( ) 6g x x . At
what value of x do the graphs of f and g have parallel tangent
lines?
(A) -0.701 (B) -0.567 (C) -0.391 (D) -0.302 (E) -0.258
_____3. Let f be the function defined by ( ) lnf x x x . What is
the value of c for which the
instantaneous rate of change of f at x = c is the same as the
average rate of change of f
over [1, 4]?
(A) 0.456 (B) 1.244 (C) 2.164 (D) 2.342 (E) 2.452
_____4. Let f be a function given by 24( ) 2 xf x e . For what
value of x is the slope of the line tangent
to the graph of f at , ( )x f x equal to 3?
(A) 0.168 (B) 0.276 (C) 0.318 (D) 0.342 (E) 0.551
_____5. Let ( )f x x . If the rate of change of f at x = c is
twice its rate of change at x = 1, then c =
(A) 1
4 (B) 1 (C) 4 (D)
1
2 (E)
1
2 2
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VASS, 2013 AP* is a trademark of the College Board. The College
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Page 3
©2007 The College Board. All rights reserved.
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VASS, 2013 AP* is a trademark of the College Board. The College
Board was not involved in the production of this packet.
©2011 The College Board. All rights reserved.
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VASS, 2013 AP* is a trademark of the College Board. The College
Board was not involved in the production of this packet.
2004 AB/BC 1
©2004 The College Board. All rights reserved.
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2004 Form B AB 2
©2004 The College Board. All rights reserved.
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VASS, 2013 AP* is a trademark of the College Board. The College
Board was not involved in the production of this packet.
1999 AB 6
(d) Suppose that w is increasing at the constant rate of 7 units
per second. When w = 5, what is the rate of
change of the area of ΔPQR with respect to time? Determine
whether the area is increasing or decreasing at
this instant.
using
©1999 The College Board. All rights reserved.
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Student Session Topic: Average and Instantaneous Rates of Change
Answers
The multiple choice answers are 1D, 2C,3C, 4A, 5A
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VASS, 2013 AP* is a trademark of the College Board. The College
Board was not involved in the production of this packet.
©2007 The College Board. All rights reserved. Page 8
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VASS, 2013 AP* is a trademark of the College Board. The College
Board was not involved in the production of this packet.
©2011 The College Board. All rights reserved. Page 9
2004 AB/BC 1
©2004 The College Board. All rights reserved.
Page 10
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2004 Form B AB 2
©2004 The College Board. All rights reserved.
Page 11
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VASS, 2013 AP* is a trademark of the College Board. The College
Board was not involved in the production of this packet.
©1999 The College Board. All rights reserved.
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