Student Session Topic: Average and Instantaneous Rates of Change
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1 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 )). f ( b) f ( a) o The average rate of change of is 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 Instantaneous rate of change b a f ( x) dx b a 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 )). f ( x) f ( a) o The instantaneous rate of change of f( x ) at x = a is lim f '( a) xa 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
2 Multiple Choice Non-Calculator 2 x 2 1. What is the instantaneous rate of change at x = 2 of the function f given by f( x)? x 1 (A) -2 (B) 1 (C) 1 (D) 2 (E) Calculator 2 2. Let f be a function given by f ( x) 3e x and let g be the function given by what value of x do the graphs of f and g have parallel tangent lines? 3 g( x) 6x. At (A) (B) (C) (D) (E) Let f be the function defined by f ( x) x ln 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) (B) (C) (D) (E) Let f be a function given by f ( x) 2e x. 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) (B) (C) (D) (E) 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) Page 2
3 Page The College Board. All rights reserved.
4 2011 The College Board. All rights reserved. Page 4
5 2004 AB/BC The College Board. All rights reserved. Page 5
6 2004 Form B AB The College Board. All rights reserved. Page 6
7 using 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 The College Board. All rights reserved. Page 7
8 Student Session Topic: Average and Instantaneous Rates of Change Answers The multiple choice answers are 1D, 2C,3C, 4A, 5A
9 2007 The College Board. All rights reserved. Page 8
10 2011 The College Board. All rights reserved. Page AB/BC The College Board. All rights reserved. Page 10
11 2004 Form B AB The College Board. All rights reserved. Page 11
12 1999 The College Board. All rights reserved.
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