Chapter 5.1, Problem 56PPSE

Solution

a.

To find: The first differences of the function.

The first differences of the function : $1,2,4,8,16,32,\mathrm{...}$

Given information:

The outputs for a function are

  $1,2,4,8,16,32,\mathrm{...}$

Explanation:

y	First difference
1	$2-1=1$
2	$4-2=2$
4	$8-4=4$
8	$16-8=8$
16	$32-16=16$
32

First difference is same the output of the function $1,2,4,8,16,32,\mathrm{...}$

b.

To find: The second differences of this function.

The second differences of this function are $1,2,4,8,16,32,\mathrm{...}$

Given information:

The outputs for a function are

  $1,2,4,8,16,32,\mathrm{...}$

Explanation:

y	First difference	Second difference
1	1	$2-1=1$
2	2	$4-2=2$
4	4	$8-4=4$
8	8	$16-8=8$
16	16
32

Second differences are again same as the original output which is $1,2,4,8,16,32,\mathrm{...}$

c.

To find: The tenth differences of this function.

The tenth differences are $1,2,4,8,16,32,\mathrm{...}$

Given information:

The outputs for a function are

  $1,2,4,8,16,32,\mathrm{...}$

Explanation:

The first and second differences are exactly same , so on continuing like this the nth difference will be the same as the original outputs.

Thus, the tenth differences of this function will be $1,2,4,8,16,32,\mathrm{...}$

d.

To determine: If any polynomial function matches the original outputs, explain with reasoning.

No, there is does not exist such polynomial function to match the original outputs because all the n differences of the function are same which results into an exponential function $2^{n-1},n$ is any natural number.

Given information:

The outputs for a function are

  $1,2,4,8,16,32,\mathrm{...}$

Explanation:

After getting the first and the second differences exactly same, it indicates that the rate of change of function is constant.

It will result into an exponential function.

The required function will be $2^{n-1},n$ is any natural number.

So, no polynomial function can match the original outputs.