The Product and Quotient Rules

Example 1. (page 131 #56.) The temperature T of food placed in a refrigerator is modeled by where t is the time (in hours). What is the initial temperature of the food? Find the rate of change of T with respect to t when (a) t=1, (b) t=3, (c) t=5, and (d) t=10.

The initial temperature is T(0)=75. Next, we shall find or Note that if we set f(t)=4t²+16t+75, and g(t)=t²+4t+10. Then and we have the followings:

We noticed that the rate of change getting smaller (in absolute vaule) and smaller (close to 0) when time goes by. This means the rate of change is going toward 0 or the temperature is getting more steady. This can be observed by the graph of T(t), shown below: click here

Example 2: (page 132, #63) The ordering and transportation cost C (in thousands of dollars) of the components used in manufacturing a product is C(x)=100(200/x ² + x/(x+30)), for x greater than or equal to 1, where x is the order size (in hundreds). Find the rate of change of C with respect to x for the following order sizes: (a) x=10, (b) x=15, (c) x=20

First, we find We obtain and

Therefore the rate of change of C decreases ( in terms of absolute value) when the order sizes increase. Can you use Maple to graph this function?