2. SECTION
2.5  
Concavity

3. The graph of a linear function is a straight line because the average rate of change is a constant.
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4. 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).
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5. Consider the following:
t (years)
S ($1,000's)
Rate of change (ΔS/Δt) 40 10 72 20
128 30
230
411
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6. 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
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7. Concave Up:
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8. 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
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9. 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
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10. 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
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11. 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
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12. 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?
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13. 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.
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14. 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
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15. 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.
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16. 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
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17. Example #2
Table 2.18 gives the distance traveled by a cyclist, Karim, as a function of time:
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18. 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
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19. 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
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20. What can we say about the graph below?
Example #2
What can we say about the graph below?
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21. Example #2
Decreasing speed leads to a decreasing slope and a graph which bends downward.
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22. Example #2
Decreasing speed leads to a decreasing slope and a graph which bends downward; thus the graph is concave down.
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23. Summary: Increasing and Decreasing Functions; Concavity
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24. 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.
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25. Concave Up:
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26. 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.
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27. Page 85

28. 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.
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29. Page 85

30. 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.
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31. Page 86

32. 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.
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33. End of Section 2.5