The median home price in the U.S. over the period 2004–2009 can be approximated by P(t)=-5t^2+75t-30 thousand dollars (4≤t≤9) where t is time in years since the start of 2000.

Question

A) What was the average rate of change of the median home price from the start of 2007 to the start of 2009?

B) What, if anything, does your answer in part (a) say about the median home price in 2008? Explain?

Answer 1

A) The average rate of change of a general function y = f(x) between two points x = a and x = b is:

Avg rate of change = (change in y) / (change in x) = [f(b) - f(a)] / (b - a)

Using our function P(t) provided in the problem, and over the range t = 7 to t = 9, we can find the average rate of change as follows:

P(7) = -5*(7)² + 75*(7) - 30 = 250 (thousand dollars, it is always good to keep units in the back of our mind)

P(9) = -5*(9)² + 75*(9) - 30 = 240

Avg rate of change = [P(9) - P(7)] / (9 - 7) = [(240) - (250)] / (9 - 7) = (-10) / (2) = -5 thousand dollars per year

B) To the best of my knowledge, part B is trying to relate to the Mean Value Theorem (MVT). This theorem states for a function f that is continuous on [a,b] and differentiable on (a,b), then there is at least one value x=c such that a<c<b and

f '(c) = [f(b) - f(a)] / (b-a)

In relation to this problem, we know that from 2007 to 2009 the average rate of change was -5 thousand dollars per year. Was it always changing by a constant -5 thousand dollars per year over the entire interval? Likely not, unless the curve is a straight line (you can look at the graph of the curve and see that the slope of the tangent line certainly changes from t = 7 to t = 9). But the MVT basically says that at some point, c, during the interval, the instantaneous rate of change must be equal to the average rate of change (-5 thousand). At this particular time, they actually do match.

Lets apply the MVT and see where this special point c occurs.

P '(t) = -10t + 75

The right side of the MVT equation looks a lot like the average rate of change we already calculated in part A (hint: it is). So let's set the two sides equal and solve for t:

-10t + 75 = -5

-10t = -80

t = 8

So at t = 8 (or 2008), the instantaneous rate of change of the median home price equals the average rate of change over the interval (7,9) (or 2007 to 2009). If we graph the curve and draw a secant line between the points (7, 250) and (9, 240), the slope of that secant line will equal the slope of the tangent line at t = 8.