SECTION 2.5   Concavity

The graph of a linear function is a straight line because the average rate of change is a constant. Page 84

The graph of a linear function is a straight line because the average rate of change is a constant. Of course, not all graphs are straight lines; they may bend up (concave up) or down (concave down). Page 84

Consider the following: t (years) S ($1,000's) Rate of change (ΔS/Δt) 40 10 72 20 128 30 230 411 Page 84

What can we say about the Rate of Change? t (years) S ($1,000's) Rate of change (ΔS/Δt) 40 3.2 10 72 5.6 20 128 10.2 30 230 18.1 411 Page 84

Concave Up: Page 84

Example #1 The following table shows Q, the quantity of carbon-14 (in μg) in a 200μg sample remaining after t thousand years. t (thousand years) Q (μg) 200   5 109 10 60 15 33 Page 84 Example 1

What is Q? An Increasing or Decreasing Function? Example #1 What is Q? An Increasing or Decreasing Function? t (thousand years) Q (μg) 200   5 109 10 60 15 33 Page 84

Example #1 What is Q? An Increasing or Decreasing Function? Decreasing... what does this say about the rate of change? t (thousand years) Q (μg) 200   5 109 10 60 15 33 Page 84

Example #1 What is Q? An Increasing or Decreasing Function? Decreasing... what does this say about the rate of change? Always negative. t (thousand years) Q (μg) Rate of change ΔQ/Δt 200   −18.2 5 109 −9.8 10 60 −5.4 15 33 Page 85

Example #1 What can we say about the concavity of the graph, and what does this mean about the rate of change of the function? Page 85

Example #1 The graph bends upward, so it is concave up. We can see that the rate of change of the function is increasing, because the rate is becoming less negative. Page 85

Example #1 The graph bends upward, so it is concave up. We can see that the rate of change of the function is increasing, because the rate is becoming less negative. The slope is negative Page 85

Example #1 The graph bends upward, so it is concave up. We can see that the rate of change of the function is increasing, because the rate is becoming less negative. The slope is negative & increasing. Page 85

Example #1 The rate of change of the function is increasing, because the rate is becoming less negative. t (thousand years) Q (μg) Rate of change ΔQ/Δt 200   −18.2 5 109 −9.8 10 60 −5.4 15 33 Page 85

Example #2 Table 2.18 gives the distance traveled by a cyclist, Karim, as a function of time: Page 85 Example 2

What can we say about the Rate of Change? time (hours) d, distance (miles) Avg speed, Δd/Δt (mph)   20 mph 1 20 15 mph 2 35 10 mph 3 45 7 mph 4 52 5 mph 5 57 Page 85

Decreasing. t, time (hours) d, distance (miles) Avg speed, Δd/Δt (mph)   20 mph 1 20 15 mph 2 35 10 mph 3 45 7 mph 4 52 5 mph 5 57 Page 85

What can we say about the graph below? Example #2 What can we say about the graph below? Page 85

Example #2 Decreasing speed leads to a decreasing slope and a graph which bends downward. Page 85

Example #2 Decreasing speed leads to a decreasing slope and a graph which bends downward; thus the graph is concave down. Page 85

Summary: Increasing and Decreasing Functions; Concavity Page 86

Summary: Increasing and Decreasing Functions; Concavity If f is a function whose rate of change increases (gets more positive as we move from left to right), then the graph of f is concave up. That is, the graph bends upward. Page 86 Blue Box

Concave Up: Page 84

Summary: Increasing and Decreasing Functions; Concavity If f is a function whose rate of change increases (gets less negative as we move from left to right), then the graph of f is concave up. That is, the graph bends upward. Page 86 Blue Box

Page 85

Summary: Increasing and Decreasing Functions; Concavity If f is a function whose rate of change decreases (gets less positive as we move from left to right), then the graph of f is concave down. That is, the graph bends downward. Page 86 Blue Box

Page 85

Summary: Increasing and Decreasing Functions; Concavity If f is a function whose rate of change decreases (gets more negative as we move from left to right), then the graph of f is concave down. That is, the graph bends downward. Page 86 Blue Box

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Summary: Increasing and Decreasing Functions; Concavity Finally, if a function has a constant rate of change, its graph is a line and it is neither concave up nor concave down. Page 86

End of Section 2.5