What is the Rate of change of f (x^2) given rate of change of f (x) Find the average rate of change of f(x^2) on the interval [1,4] given that the average rate of change of f(x) equals 9 on interval [1,16]? This question has two different "answers" according to two different teachers The first give answer of 9 by assuming y=x^2 and applying the formula of rate of change as following Lety=x^2 with I= [1,4] thenf(1)=1,f(4)=16

Kade Rosales

Kade Rosales

Answered question

2022-08-12

What is the Rate of change of f ( x 2 ) given rate of change of f (x)
Find the average rate of change of f ( x 2 ) on the interval [1,4] given that the average rate of change of f ( x ) equals 9 on interval [1,16]?
This question has two different "answers" according to two different teachers
The first give answer of 9 by assuming y = x 2 and applying the formula of rate of change as following Let y = x 2 with I= [1,4] then f ( 1 )=1, f ( 4 )=16
f ( 16 ) f ( 1 ) ( 16 1 )
The second give the answer of 45 as following
f ( 16 ) f ( 1 ) 16 1 = 9
f ( 16 ) f ( 1 ) 15 = 9
f ( 16 ) f ( 1 ) = 9 15
Now we find rate of change of f ( x 2 ) following f ( 16 ) f ( 1 ) 4 1 = 9 15 3 = 45 what is the right answer?
Can we ensure the either answers geometrically? Thank you for helping

Answer & Explanation

Brooks Hogan

Brooks Hogan

Beginner2022-08-13Added 18 answers

Let g ( x ) = f ( x 2 )
The average rate of change of a function f on an interval [a,b] is f ( b ) f ( a ) b a
The average rate of change of f on the interval [1,16] is
f ( 16 ) f ( 1 ) 16 1 = f ( 16 ) f ( 1 ) 15 .
You are told this rate of change equals 9.
The average rate of change of g on the interval [1,4] is thus equal to
g ( 4 ) g ( 1 ) 4 1 = f ( 16 ) f ( 1 ) 3 = 5 f ( 16 ) f ( 1 ) 15 = 45.

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