Questions: Rate of Change The blue curve shows national health expenditures in billions of dollars. The red curve shows its derivative. National health expenditures (in billions of dollars) from 1980 to 1998 are given by the function f(t) in the graph to the left. (a) How much money was being spent at the beginning of 1983 ? 500 billion (b) Approximately how fast was the rate of spending increasing in 1994 ? billion per year

Solution Steps

• The problem states that the national health expenditures are given from 1980 to 1998.
• The x-axis represents the years after 1980.
• For part (a), we need to find the expenditure at the beginning of 1983, which is 3 years after 1980.

• Locate the point on the blue curve (y = f(t)) at x = 3 (3 years after 1980).
• From the graph, the value of y at x = 3 is approximately 500 billion dollars.

• For part (b), we need to find the rate of spending increase in 1994, which is 14 years after 1980.

• Locate the point on the red curve (y' = f'(t)) at x = 14 (14 years after 1980).
• From the graph, the value of y' at x = 14 is approximately 200 billion dollars per year.

Final Answer