Section 1.3 Rates of Change and Behavior of Graphs

Question

Let . Find a number c such that the average rate of change of the function f on the interval (1,c) is  

Answer 1

[ 1/c - 1] / (c-1) = -1/8

(1-c)/c divided by (c-1) = -1/8

(1-c)/c * 1/(c-1) = -1/8

(1-c)/ [c(c-1)] = -1/8

c(c-1) (-1) = 8(1-c)

(c^2-c)(-1) = 8(1-c)

c - c^2 = 8 - 8c

0 = 8 - 8c -c + c^2

0 = c^2 - 9c + 8

0 = (c - 8 )(c - 1 )

c-8 = 0 or c-1=0

c=8 or c=1

so the average rate of change on (1,8) of the function y=1/x is

(1/8-1) / (8-1) =

-7/8 divided by 7 =

-1/8

c=8 is the official answer

Answer 2

Let a , b be real numbers, and let f: {real numbers} -> {real numbers}. Remember that the average rate of change for a function on an interval ( a , b ) is,

[ f( b ) - f( a ) ] / [ b - a ] .

We are given the function f is defined as

f( x ) = 1 / x.

Using the above definition, since we are given the interval ( 1 , c ), we want to find

[ f( c ) - f( 1 ) ] / [ c - 1 ] = -1 / 8

we perform the following,

[ ( 1 / c ) -( 1 / 1 ) ] / [ c - 1 ] = -1 / 8 by definition of the function;

implies

[ ( 1 / c ) -( c / c ) ] / [ c - 1 ] = -1 / 8 ;

implies

[ ( 1 - c ) / c ] / [ c - 1 ] = -1 / 8 ;

implies

[ 1 - c ] / [ c( c - 1 ) ] = - 1 / 8 ;

implies

[ c - 1 ] / [ c( c - 1) ] = 1 / 8 ;

implies

1 / c = 1 / 8

implies

c = 8.

Therefore, c = 8

Q.E.D.