NAME______
Analyzing the point of inflection-Answer Key
1)Hugo Rossi wrote: "In the fall of 1972 President Nixon announced that the rate of increase of inflation was decreasing. This was the first time a sitting president used the third derivative to advance his case for reelection."
Mathematics Is an Edifice, Not a Toolbox, Notices of the AMS, v. 43, no. 10, October 1996.
How does concavity and points of inflection fit into this conversation if at all? Explain.
Students should recognize that the inflation is a rate of change. They should comment that just because the rate of increase was decreasing, that no maximum is guaranteed.
2)Is it possible for the first derivative to be increasing over an interval while the function is concave down over the same interval? Explain. Students should say no. They should comment that if the first derivative is increasing then the second derivative must be positive—therefore it should be concave up.
3)The Chart below shows the average US home price as the composite of various large cities throughout the nation. The last date on graph is October of 2008. Suppose that based upon the other market conditions and pressures, you believe that you are currently at the point of inflection in October of 2008. Explain what this means for home prices after this date. Will they increase or decrease? How will they increase or decrease? Predict any future relative extrema? How do you know?
It is possible for a minimum to be in the future. Students should explain that no minimum is guaranteed. The function could change to concave up and continue to decrease.
4)The graph below is the most current information for home prices. Notice the current increasing trend. From the current graph estimate the previous two points of inflection that have occurred. Explain why you chose those points.
Students should explain that there is a change in concavity at the points marked to the left. The answers could be somewhat off based upon their judgment. They should be close though.
5)Consider the following statement:
Oil prices rose sharply during the first half of the year but have since begun to level off.
If we would graph the price of oil as a function of the time, would we have an inflection point? Do we know anything about concavity? Explain. Students should interpret this scenario as producing a point of inflection. If we would graph price versus time as possible graph is below.
6)Consider the following flask: Suppose water is added to flask so that its volume increases at a constant rate with respect to time. Sketch a graph that depicts the depth of water as a function of time labeling features of the flask along the x-axis. Examine the rate at which the water level rises. Identify concavity and any points of inflections. Provide a justification. Students should look at the narrowest portion of the flask as the corresponding point of inflection. The depth would continue to increase beyond this point but since the flask begins to widen, it would be increasing at a slower rate.
7)Given the graphs below, determine the intervals where f would be concave up. Estimate any points of inflection.
Students should comment that the x values of -2 and 2 appear to be points of inflection because the second derivative is equal to zero.
8) Given the graph of f ' below, state anything that you know concerning the concavity and points of inflection for f.
Since the first derivative is decreasing from 1 to 3, the second derivative would be negative and thus concave down. The other intervals are concave up since the derivative is increasing.
9)For the following function, determine the intervals of concave up and concave down as well as any points of inflection using analytic methods.