Find the Slope-Intercept form of the line which passes

#15 p (-1,5), Q(7,5)

Example 2.1.7. Hecall from page $2, the revenue from selling & units at a price p per unit is given by the formula It = xp. Suppose we are in the scenario of Examples 2.1.5 and 2.1.6. 1. Find and simplify an expression for the weekly revenue M(r) as a function of weekly sales r. 2. Find and interpret the average rate of change of R(x) over the interval [0, 50]. 3. Find and interpret the average rate of change of R(x) as a changes from 50 to 100 and compare that to your result in part 2. 4. Find and interpret the average rate of change of weekly revenue as weekly sales increase from 100 PortaBoys to 150 PortaBoys. Balution. 1. Since H = rp, we substitute p(r) = -1.5x + 250 from Example 2.1.6 to get A(x) = r(-1.5x+ 250) = -1.5x3 + 250x. Since we determined the price demand function p(x) is restricted to 0 5 x 5 166, R(x) is restricted to these values of z as well. 2. Using Definition 2.3, we get that the average rate of change is AR R(50) - R(0) 8750 -0 - = 175. 50 60 0 Interpreting this slope as we have in similar situations, we conclude that for every additional PortaBloy sold during a given week, the weekly revenue increases $175 3. The wording of this part is slightly different than that in Definition 2.3, but its meaning is to find the average rate of change of R over the interval [50, 100] To find this rate of change. we compute AR #(100) - 7(50) 10000 - 8750 = 25 100 -50 50 162 LINEAR AND QUADRATIC FUNCTIONS In other words, for each additional PortaBoy sold, the revenue increases by $25. Note that while the revenue is still increasing by selling more game systems, we aren't getting as much of an increase as we did in part 2 of this example. (Can you think of why this would happen?) 4. Translating the English to the mathematics, we are being asked to find the average rate of change of I over the interval [100, 150). We find AR R(150) - R(100) 3750 - 10000 Ar -125. 150 - 100 50 This means that we are losing $125 dollars of weekly revenue for each additional PortaBoy sold. (Can you think why this is possible?) We close this section with a new look at difference quotients which were first introduced in Section 1.4. If we wish to compute the average rate of change of a function f over the interval [3, r + 4), then we would have [_ /(xth) - f(x) _f(xth) - f(1) AT As we have indicated, the rate of change of a function (average or otherwise) is of great importance in Calculus. Also, we have the geometric interpretation of difference quotients which was promised to you back on page 81 - a difference quotient yields the slope of a secant line