2. Intro to Derivatives

Using the Alternative Formula for Derivatives

Use the formula

f'(x) = lim (z → x) (f(z) − f(x)) / (z − x)

to find the derivative of the functions in Exercises 23–26.

g(x) = 1 + √x

Cylinder pressure If gas in a cylinder is maintained at a constant temperature T, the pressure P is related to the volume V by a formula of the form

P = (nRT / (V − nb)) − (an² / V²),

in which a, b, n, and R are constants. Find dP/dV. (See accompanying figure.)

<IMAGE>

The best quantity to order One of the formulas for inventory management says that the average weekly cost of ordering, paying for, and holding merchandise is

A(q) = (km / q) + cm + (hq / 2),

where q is the quantity you order when things run low (shoes, TVs, brooms, or whatever the item might be); k is the cost of placing an order (the same, no matter how often you order); c is the cost of one item (a constant); m is the number of items sold each week (a constant); and h is the weekly holding cost per item (a constant that takes into account things such as space, utilities, insurance, and security).

Find dA/dq and d²A/dq².

Average single-family home prices P (in thousands of dollars) in Sacramento, California, are shown in the accompanying figure from the beginning of 2006 through the end of 2015.

d. During what year did home prices drop most rapidly and what is an estimate of this rate?

Fruit flies (Continuation of Example 4, Section 2.1.) Populations starting out in closed environments grow slowly at first, when there are relatively few members, then more rapidly as the number of reproducing individuals increases and resources are still abundant, then slowly again as the population reaches the carrying capacity of the environment.

b. During what days does the population seem to be increasing fastest? Slowest?

Temperature The given graph shows the outside temperature T in °F, between 6 a.m. and 6 p.m.

b. At what time does the temperature increase most rapidly? Decrease most rapidly? What is the rate for each of those times?