A team of researchers finds that there is a relationship between the overall health of a forest they're studying and the population of a certain beetle which feeds on the bark of old-growth trees. Due to the warming climate, the beetles are now able to reproduce one more time per year than they previously did, exacerbating the stress caused by the drought 13x³ +67 of the past few seasons. They use the function f (x) = to model the relationship between the two populations, .6x5+39 where x is in tens of thousands of beetles and f is in thousands of trees. They note that there is a beneficial contribution to the ecosystem as a whole for a smaller population of the beetles, but that the number of healthy trees sharply declines as the beetle population rises too much. 1. Graph f(x) using technology. a. What appears to be the maximum number of healthy trees, to the nearest thousand? What is the estimated beetle population corresponding to this, to the nearest ten thousand? b. What is the slope of the tangent line to the graph of f (x) at the point corresponding to the maximum number of healthy trees (and the optimal population of beetles)? How do you know? c. How can f'(x) help us find the actual coordinates of the point we just estimated? (Don't actually find it yet, just describe how f'(x) is useful for answering this question, what steps you would take to do this and why.)
A team of researchers finds that there is a relationship between the overall health of a forest they're studying and the population of a certain beetle which feeds on the bark of old-growth trees. Due to the warming climate, the beetles are now able to reproduce one more time per year than they previously did, exacerbating the stress caused by the drought 13x³ +67 of the past few seasons. They use the function f (x) = to model the relationship between the two populations, .6x5+39 where x is in tens of thousands of beetles and f is in thousands of trees. They note that there is a beneficial contribution to the ecosystem as a whole for a smaller population of the beetles, but that the number of healthy trees sharply declines as the beetle population rises too much. 1. Graph f(x) using technology. a. What appears to be the maximum number of healthy trees, to the nearest thousand? What is the estimated beetle population corresponding to this, to the nearest ten thousand? b. What is the slope of the tangent line to the graph of f (x) at the point corresponding to the maximum number of healthy trees (and the optimal population of beetles)? How do you know? c. How can f'(x) help us find the actual coordinates of the point we just estimated? (Don't actually find it yet, just describe how f'(x) is useful for answering this question, what steps you would take to do this and why.)
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter4: Calculating The Derivative
Section4.EA: Extended Application Managing Renewable Resources
Problem 2EA
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Transcribed Image Text:Application - Forestry
A team of researchers finds that there is a relationship between the overall health of a forest they're studying and the
population of a certain beetle which feeds on the bark of old-growth trees. Due to the warming climate, the beetles are
now able to reproduce one more time per year than they previously did, exacerbating the stress caused by the drought
13x³ +67
of the past few seasons. They use the function f(x)
to model the relationship between the two populations,
.6x5 +39
where x is in tens of thousands of beetles and f is in thousands of trees. They note that there is a beneficial
contribution to the ecosystem as a whole for a smaller population of the beetles, but that the number of healthy trees
sharply declines as the beetle population rises too much.
-
1. Graph f(x) using technology.
a. What appears to be the maximum number of healthy trees, to the nearest thousand? What is the estimated
beetle population corresponding to this, to the nearest ten thousand?
b. What is the slope of the tangent line to the graph of f (x) at the point corresponding to the maximum number of
healthy trees (and the optimal population of beetles)? How do you know?
c. How can f¹ (x) help us find the actual coordinates of the point we just estimated? (Don't actually find it yet, just
describe how f¹ (x) is useful for answering this question, what steps you would take to do this and why.)
2. Write down the limit definition of ƒ¹ (x). Do not actually solve this limit.
a. Would you want to use the limit definition of the derivative to find ƒ' (x)? Why or why not?
b. Use the differentiation rules we learned in this unit to find f¹ (x), describing which rules you're using at each
step and telling me the story of your process.
3. Use ƒ' (x) to find the point you described (you can use technology to find it, just please tell me what you're having
the technology do for you).
4. Find the equation of the tangent line to the graph of ƒ (x) at this point, telling me the story of your process.
5. Use the equation of the tangent line to estimate the tree population when the beetle population is 30,000, telling
me the story of your process.
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