Chapter 1 Section 2 Rate of Change 2

Sales of digital video disc (DVD) players have been increasing since they were introduced in early To measure how fast sales were increasing, we calculate a rate of change of the form: Page 103

At the same time, sales of video cassette recorders (VCRs) have been decreasing. See Table 1.11 below:1.11 Year VCR sales (million $) DVD player sales (million $) Page 104

To calculate the rate of change of DVD players: Average rate of change of DVD player sales (1998-> 2003)= Year VCR sales (million $) DVD player sales (million $) Page 105

To calculate the rate of change of DVD players: Average rate of change of DVD player sales (1998-> 2003)= Year VCR sales (million $) DVD player sales (million $) Page 106

To calculate the rate of change of DVD players: Average rate of change of DVD player sales (1998-> 2003)= Year VCR sales (million $) DVD player sales (million $) Page 107

To calculate the rate of change of DVD players: Average rate of change of DVD player sales (1998-> 2003)= Year VCR sales (million $) DVD player sales (million $) Page 108

Graphically, here is what we have: Page 109

To calculate the rate of change of VCR players: Average rate of change of VCR player sales (1998-> 2003)= Year VCR sales (million $) DVD player sales (million $) Page 1010

To calculate the rate of change of VCR players: Average rate of change of VCR player sales (1998-> 2003)= Year VCR sales (million $) DVD player sales (million $) Page 1011

To calculate the rate of change of VCR players: Average rate of change of VCR player sales (1998-> 2003)= Year VCR sales (million $) DVD player sales (million $) Page 1012

To calculate the rate of change of VCR players: Average rate of change of VCR player sales (1998-> 2003)= Year VCR sales (million $) DVD player sales (million $) Page 1013

Graphically, here is what we have: Page 1014

In general, if Q = f(t), we write ΔQ for a change in Q and Δt for a change in t. We define: The average rate of change, or rate of change, of Q with respect to t over an interval is: Average rate of change over an interval Page 1115

The average rate of change of the function Q = f(t) over an interval tells us how much Q changes, on average, for each unit change in t within that interval. On some parts of the interval, Q may be changing rapidly, while on other parts Q may be changing slowly. The average rate of change evens out these variations. Page 1116

DVD Player Sales: Average rate of change is POSITIVE on the interval from 1998 to 2003, since sales increased over this interval. An increasing function. VCR Player Sales: Average rate of change is NEGATIVE on the interval from 1998 to 2003, since sales decreased over this interval. A decreasing function. Page 1117

In general terms: If Q = f(t) for t in the interval a ≤ t ≤ b: f is an increasing function if the values of f increase as t increases in this interval. f is a decreasing function if the values of f decrease as t increases in this interval. Page 1118

And if Q=f(t): If f is an increasing function, then the average rate of change of Q with respect to is positive on every interval. If f is a decreasing function, then the average rate of change of Q with respect to t is negative on every interval. Page 1119

The function A = q(r) = πr 2 gives the area, A, of a circle as a function of its radius, r. Graph q. Explain how the fact that q is an increasing function can be seen on the graph. Page 11 (Example 1)20

The function A = q(r) = πr 2 gives the area, A, of a circle as a function of its radius, r. Graph q. rA Graph climbs as we go from left to right. Page 1221

rAΔrΔrΔAΔAΔA/Δr Page 1222

rAΔrΔrΔAΔAΔA/Δr Page 1223

rAΔrΔrΔAΔAΔA/Δr Page 1224

rAΔrΔrΔAΔAΔA/Δr Page 1225

rAΔrΔrΔAΔAΔA/Δr Note: A increases as r increases, so A=q(r) is an increasing function. Also: Avg rate of change (ΔA/Δr) is positive on every interval. Page 1226

Carbon-14 is a radioactive element that exists naturally in the atmosphere and is absorbed by living organisms. When an organism dies, the carbon-14 present at death begins to decay. Let L = g(t) represent the quantity of carbon-14 (in micrograms, μg) in a tree t years after its death. See Table Explain why we expect g to be a decreasing function of t. How is this represented on a graph?1.12 Page 12 (Example 2)27

Let L = g(t) represent the quantity of carbon-14 (in micrograms, μg) in a tree t years after its death. See Table Explain why we expect g to be a decreasing function of t. How is this represented on a graph?1.12 t, time L, carbon Page 1228

tLΔtΔtΔLΔLΔL/Δt Like in the last example, let’s fill in the table on the right, one column at a time: Page 1229

tLΔtΔtΔLΔLΔL/Δt Like in the last example, let’s fill in the table on the right, one column at a time: Page 1230

tLΔtΔtΔLΔLΔL/Δt Like in the last example, let’s fill in the table on the right, one column at a time: Page 1231

tLΔtΔtΔLΔLΔL/Δt Like in the last example, let’s fill in the table on the right, one column at a time: Page 1232

Since the amount of carbon-14 is decaying over time, g is a decreasing function. In Figure 1.10, the graph falls as we move from left to right and the average rate of change in the level of carbon-14 with respect to time, ΔL/Δt, is negative on every interval.1.10 Page 1233

tLΔtΔtΔLΔLΔL/Δt Here you can again see what was said on the last slide. (Lower values of t result in higher values of L, and vice versa. And ΔL/Δt is negative on every interval.) Page 1234

In general, we can identify an increasing or decreasing function from its graph as follows: The graph of an increasing function rises when read from left to right. The graph of a decreasing function falls when read from left to right. Page 1235

On what intervals is the function graphed in Figure 1.11 increasing? Decreasing? 1.11 Page 1336

On what intervals is the function graphed in Figure 1.11 increasing? Decreasing? 1.11 Inc Page 1337

On what intervals is the function graphed in Figure 1.11 increasing? Decreasing? 1.11 Inc Dec Page 1338

On what intervals is the function graphed in Figure 1.11 increasing? Decreasing? 1.11 Inc Dec Page 1339

On what intervals is the function graphed in Figure 1.11 increasing? Decreasing? 1.11 Inc Dec Inc Page 1340

On what intervals is the function graphed in Figure 1.11 increasing? Decreasing? 1.11 Inc Dec Inc Dec Page 1341

Using inequalities, we say that f is increasing for −3<x<−2, for 0<x<1, and for 2<x<3. Similarly, f is decreasing for −2<x<0 and 1<x<2. Inc Dec Inc Dec Inc Page 1342

Function Notation for the Average Rate of Change Suppose we want to find the average rate of change of a function Q = f(t) over the interval a ≤ t ≤ b. On this interval, the change in t is given by: Page 1343

At t = a, the value of Q is f(a), and at t = b, the value of Q is f(b). Therefore, the change in Q is given by: Function Notation for the Average Rate of Change Page 1344

Using function notation, we express the average rate of change as follows: The average rate of change of Q = f(t) over the interval a ≤ t ≤ b is given by: Function Notation for the Average Rate of Change Page 1345

Let’s review: Page 1346

Page 1347

Page 1448

Calculate the avg rate of change of the function f(x) = x 2 between x = 1 and x = 3 and between x = -2 and x = 1. Show results in a graph: Page 14 (Example 4)49

Calculate the avg rate of change of the function f(x) = x 2 between x = 1 and x = 3 and between x = -2 and x = 1. Show results in a graph: Between x=1 and x=3: Page 1450

Calculate the avg rate of change of the function f(x) = x 2 between x = 1 and x = 3 and between x = -2 and x = 1. Show results in a graph: Between x=1 and x=3: Page 1451

Calculate the avg rate of change of the function f(x) = x 2 between x = 1 and x = 3 and between x = -2 and x = 1. Show results in a graph: Between x=-2 and x=1: Page 1452

Calculate the avg rate of change of the function f(x) = x 2 between x = 1 and x = 3 and between x = -2 and x = 1. Show results in a graph: Between x=-2 and x=1: Page 1453

Calculate the avg rate of change of the function f(x) = x 2 between x = 1 and x = 3 and between x = -2 and x = 1. Show results in a graph: Page 1454

End of Section