Mathematics 133 – Intensive Calculus for Science 1 Discussion 6 – Using f′(x) and f′′(x) November 1, 2000

Background

Now that we have learned how to differentiate all of the elementary functions (the power, exponential, and logarithm functions, trigonometric functions and their inverses, and functions built up from them by sums, products, quotients, and compositions), we will look next at several applications of differential calculus. The idea will be to use all the information that the first and second derivatives of a function can tell us to get information about the original function. We already know most of the key ideas here, but we need to “synthesize” everything and give statements of some of the general patterns that can be deduced from what we have learned.

A function f(x) is said to have a local maximum at x = a if f(a) ≥ f(x) for all x near a (for instance all x on some interval containing a). Similarly, f(x) is said to have a local minimum at x = a if f(a) ≤ f(x) for all x near a. (Note: since these words are taken from the Latin language, the plurals are formed according to Latin rules: the plural of “maximum” is “maxima” and the plural of “minimum” is “minima”.)

Discussion Questions

A) Estimate the x-coordinates of the local maxima and local minima of the following function:

B) What is true about the derivative of f at each of the points you found in part A? (Note: there are two different possibilities. Be careful!)

C) Suppose you had a formula or a graph of the derivative function f′(x). How could you use only that to determine where the local maxima and local minima of the function f(x) are located? Give a collection of “rules” to follow, in your own words, to determine where the maxima and where minima are, using only information from

f′(x). Explain with the following example, showing y = f′(x):

D) Now, suppose that in addition to the formula or graph for f′(x) we also had a formula or graph of the second derivative f′′(x). How could you use this information to locate the maxima and minima of f(x), assuming that f′(a) exists whenever x = a is a local maximum or minimum of the function f in question? Again, give a collection of “rules” to follow, in your own words, to determine where the maxima and where minima are, but now using both f′(x) and f′′(x). Explain using the example from question C. (You’ll need first to use that graph to sketch a qualitative graph of y = f′′(x).)

E) Find all local maxima and minima of f(x) = x^{2}e^{−2x} and f(x) = x^{4}−6x^{2}+1. following your “rules” from both questions C and D. Is either of your methods clearly simpler than the other?

Assignment

Group write-ups due in class on Monday, November 6.