Question

Given the function f(x)=4(2)2, Section A is from x=1 to x=2 and section B is from x=3 to x=4. Part A: find the average rate of change of each section. Part B: How many times greater is the average rate of change of section B than section A? Explain why one rate of change is greater than the other.

Answer 1

Answer:

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Step-by-step explanation:

I'm sure you want your functions to appear as perfectly formed as possible so that others can help you.  f(x) = 4(2)x should be written with the " ^ " sign to denote exponentation:  f(x) = 4(2)^x

                                                                                     f(b) - f(a)

The formula for "average rate of change" is a.r.c. = --------------

                                                                                          b - a

                                   change in function value

This is equivalent to  ---------------------------------------

                                           change in x value

For Section A:  x changes from 1 to 2 and the function changes from 4(2)^1 to  4(2)^2:  8 to 16.  Thus, "change in function value" is 8 for a 1-unit change in x from 1 to 2.  Thus, in this Section, the a.r.c. is:

                8

              ------ = 8 units    (Section A)

                 1

Section B:  x changes from 3 to 4, a net change of 1 unit:  f(x) changes from

4(2)^3 to 4(2)^4, or 32 to 256, a net change of 224 units.  Thus, the a.r.c. is

       224 units

     ----------------- = 224 units (Section B)

           1 unit

The a.r.c for Section B is 28 times greater than the a.r.c. for Section A.

This change in outcome is so great because the function f(x) is an exponential function; as x increases in unit steps, the function increases much faster (we say "exponentially").