Worksheet 2-4
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Arizona State University *
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Course
5A
Subject
Mathematics
Date
Jun 5, 2024
Type
Pages
6
Uploaded by BarristerWorldYak40
Names and ID Numbers
Math 5A
Unit 2 - Worksheet 4
1.
(a) Go to this Desmos link. Use the sliders for
a
,
h
, and
k
to change the values of
α
,
β
, and
γ
(displayed below the sliders) until you get a nonlinear selection function you like. Note that
γ
(gamma) is written as ’g’ in Desmos. Write down your function below in the form
w
(
z
) =
α
+
βz
+
1
2
(
γ
)
z
2
(b) What type of animal’s relative fitness could be described by your nonlinear selection function?
Why?
(c) Based on your function, if an animal has a weight of 14 kg, would losing weight or gaining
weight be beneficial to its survival? Why?
1
(d) Using your function, find the average rate of change in the relative fitness when an animal’s
weight increases from 15 to 17 kg. Write a sentence interpreting the result.
(e) Using your graph in Desmos, determine the weight that maximises the animal’s relative
fitness, and the maximum relative fitness. How do you think you can prove this using your
equation alone and not the graph in Desmos?
2
(f) Find the instantaneous rate of change in relative fitness of an animal weighing 10.5 kg. Write
a sentence to interpret what this means in the context of this problem.
(g) Find the vertex of your function from Exercise 1, which will have the form (
z, w
(
z
) =
(
weight
,
relative fitness
), and then find the instantaneous rate of change in relative fitness
of an animal that weighs as much as the
z
-coordinate of the vertex.
(h) What do you notice about the result in part (g)? Why do you think this is the case?
3
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2. Suppose that oxygen consumption at rest, in cubic centimeters per hour, for a certain small marine
animal is a function of the pH level of the surrounding water. Researchers gathered data, which
can be accessed in this Google Sheet.
Note: Every time you open the sheet, a different
set of data is produced.
(a) Copy the two columns of data and paste them into Desmos. Perform a linear regression to
find the line of best fit using
y
1
∼
mx
1
+
b
. What is the value of the correlation coefficient
r
? What does this indicate about the linear model?
(b) Perform a
quadratic regression
to find the
parabola of best fit
using
y
1
∼
ax
2
1
+
bx
1
+
c
.
Write down the equation for your function. What is the value of
R
2
? What does this indicate
about the quadratic model?
(c) The domain of a quadratic function is all real numbers, but explain why that cannot be the
domain for your model. What seems to be a meaningful domain for your model?
4
(d) Use your quadratic model from part (b) to estimate the oxygen consumption at rest for a
marine animal in water with a pH level of 8.1. Include proper units.
(e) Use your quadratic model from part (b) to estimate the pH level of the water if the marine
animal is consuming 0.25 cubic centimeters of oxygen per hour at rest. Include proper units.
Hint: The quadratic formula is required.
5
(f) Use your quadratic model from part (b) and the limit definition to find the derivative at the
point where
x
= 7.
(g) Write a sentence interpreting the result in part (f).
6
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