Chapter 9.9, Problem 32HP

To determine

What type of function will have constant third differences but not constant second difference

Answer to Problem 32HP

The given data represents an exponential model

Explanation of Solution

Given:

A function will have constant third differences but not constant second differences

Concept Used:

If first difference of y-values is equal, then model be linear

If second difference be equal, then the model be quadratic

If first and second differences are not equal then the model will be exponential

Calculation:

Let’s assume a data table which is below

  

In order to check whether the assume data is linear, quadratic or an exponential; $\begin{array}{c}\text{First differences:}\\ d=y_2-y_1\\ =6-3\\ =3\\ d=y_3-y_2\\ =12-6\\ =6\\ d=y_4-y_3\\ =24-12\\ d=12\\ \text{Second differences:}\\ d=6-3\\ d=3\\ d=12-6\\ d=6\\ \text{Ratio of y-values:}\\ r_1=\frac{a_2}{a_1}\\ =\frac{6}{3}=2\\ r_2=\frac{a_3}{a_2}\\ =\frac{12}{6}=2\end{array}$

Thus, the first difference of y-values is not equal but the second difference is also no equal , so the given data represents a exponential model