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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.
Page 1
VASS, 2013 AP* is a trademark of the College Board. The College Board was not involved in the production of this packet.
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
Page 2
VASS, 2013 AP* is a trademark of the College Board. The College Board was not involved in the production of this packet.
Page 3
©2007 The College Board. All rights reserved.
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 4
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.
Page 5
VASS, 2013 AP* is a trademark of the College Board. The College Board was not involved in the production of this packet.
2004 Form B AB 2
©2004 The College Board. All rights reserved.
Page 6
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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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 Answers
The multiple choice answers are 1D, 2C,3C, 4A, 5A
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
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
VASS, 2013 AP* is a trademark of the College Board. The College Board was not involved in the production of this packet.
2004 Form B AB 2
©2004 The College Board. All rights reserved.
Page 11
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.