Questions: b) Assuming the pattern repeats, find the average rate of change over the one-year intervals from 2005 to 2006 and from 2006 to 2007. What is the average rate of change from 2005 to 2006? What is the average rate of change from 2006 to 2007? c) Study the table of values. What can be inferred about the function R(t) from t=0 to t=5?

Solution Steps

To find the average rate of change over the one-year intervals from 2005 to 2006 and from 2006 to 2007, we need to calculate the difference in the function values at the endpoints of each interval and then divide by the length of the interval (which is 1 year in this case).

We need to find the average rate of change of a function over specified intervals. The average rate of change of a function \( R(t) \) over an interval from \( t = a \) to \( t = b \) is given by: \[ \text{Average Rate of Change} = \frac{R(b) - R(a)}{b - a} \]

Assume we have the following values for \( R(t) \):

• \( R(2005) = R_1 \)
• \( R(2006) = R_2 \)
• \( R(2007) = R_3 \)

Using the formula for the average rate of change: \[ \text{Average Rate of Change from 2005 to 2006} = \frac{R(2006) - R(2005)}{2006 - 2005} = \frac{R_2 - R_1}{1} \] \[ \text{Average Rate of Change from 2005 to 2006} = R_2 - R_1 \]

Similarly, for the interval from 2006 to 2007: \[ \text{Average Rate of Change from 2006 to 2007} = \frac{R(2007) - R(2006)}{2007 - 2006} = \frac{R_3 - R_2}{1} \] \[ \text{Average Rate of Change from 2006 to 2007} = R_3 - R_2 \]

Final Answer