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Cornette - Calculus for the Life Sciences 1/e (Homework)

James Finch

Math - College, section 1, Fall 2019

Instructor: Dr. Friendly

Current Score : 56 / 63

Due : Sunday, January 27, 2030 00:00 EST

Last Saved : n/a Saving...  ()

Ask Your Teacher Print Assignment
Question
Points
1 2 3 4 5 6 7 8 9 10
13/13 6/8 9/9 2/4 1/1 15/17 2/2 2/2 2/3 4/4
Total
56/63 (88.9%)

  • Instructions

    Calculus for the Life Sciences: A Modeling Approach, by James L. Cornette and Ralph A. Ackerman, helps life science students understand the relevance and importance of mathematics to their world and involves modeling living systems with difference and differential equations. Through partnership with the Mathematical Association of America, WebAssign is pleased to offer online question content alongside interactive step-by-step tutorials for this title. All questions include reading links to the eBook for an integrated student experience.

    Question 1 is a multi-part question and shows a range of input options, including equations, exact expressions, and numeric answers.

    Question 2 utilizes special trigonometric grading that forces students to enter simplified trigonometric functions.

    Question 3 displays the answers in a table to match how a student would work this problem on paper.

    Question 4 uses the matrix tool that lets students enter the matrices in a standard format.

    Question 6 lets a student enter coordinate points in a natural (x, y) form. It also has the student plot the points using the graphing tool.

    Question 7 asks the student to compute a mathematical model of given data.

    Question 8 enforces simplification.

    Question 10 includes a stepped out tutorial that walks the student through each step to ultimately find all x values that make the function have a horizontal tangent line.

    View the complete list of WebAssign questions available for this textbook. This demo assignment allows many submissions and allows you to try another version of the same question for practice wherever the problem has randomized values.

    The answer key and solutions will display after the first submission for demonstration purposes. Instructors can configure these to display after the due date or after a specified number of submissions.

Assignment Submission

For this assignment, you submit answers by question parts. The number of submissions remaining for each question part only changes if you submit or change the answer.

Assignment Scoring

Your last submission is used for your score.

1. 13/13 points  |  Previous Answers CornetteApCalc1 1.2.002. My Notes
Question Part
Points
Submissions Used
1 2 3 4 5 6 7 8 9 10 11 12 13
1/1 1/1 1/1 1/1 1/1 1/1 1/1 1/1 1/1 1/1 1/1 1/1 1/1
3/100 2/100 2/100 2/100 2/100 2/100 5/100 2/100 2/100 2/100 1/100 1/100 2/100
Total
13/13
 

problem

The equation
Bt − Bt − 1 = rBt − 1
 carries the same information as
Bt + 1 − Bt = rBt.
(a)
Write the first four instances of
Bt − Bt − 1 = rBt − 1
 using
t = 1,
t = 2,
t = 3,
 and
t = 4.
√564​
34​
√B3​−B2​=B2​r
B4​−B3​=B3​r
(b)
Cascade these four equations to get an expression for B4 in terms of r and B0.
B0​(r+1)4
(c)
Write solutions and compute B40 for the following. (Round
B40
 to the nearest integer for parts (i), (ii), (iii), and round
B40
 to three decimal places for part (iv).)
(i)
B0 = 60    Bt − Bt − 1 = 0.2Bt − 1
60·1.2t
(ii)
B0 = 60    Bt − Bt − 1 = 0.1Bt − 1
60·1.1t
(iii)
B0 = 60    Bt − Bt − 1 = 0.03Bt − 1
60·1.03t
(iv)
B0 = 60    Bt − Bt − 1 = −0.1Bt − 1
60·0.9t

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2. 6/8 points  |  Previous Answers CornetteApCalc1 7.2.005. My Notes
Question Part
Points
Submissions Used
1 2 3 4 5 6 7 8
1/1 1/1 1/1 1/1 0/1 1/1 0/1 1/1
2/100 3/100 2/100 2/100 2/100 1/100 2/100 1/100
Total
6/8
 
(a)
Use
cot(x) = 
cos(x)
sin(x)
 and the quotient rule to show that
[cot(x)]' = −csc2(x).
[cot(x)]' = 
cos(x)
sin(x)
'
 
 = 
(−sin(x)·sin(x))
Correct: Your answer is correct. webMathematica generated answer key
 − cos(x)cos(x)
sin2(x)
Correct: Your answer is correct. webMathematica generated answer key
 = 
−1
sin2(x)
Correct: Your answer is correct. webMathematica generated answer key
 = 
−csc2(x)
Correct: Your answer is correct. webMathematica generated answer key
(b)
Use
sec(x) = 
1
cos(x)
 = (cos(x))−1
 and the power chain rule to show that
[sec(x)]' = sec(x) tan(x).
[sec(x)]' = 
(cos(x))−1
'
 
 = 
(−cos(X)−2)
Incorrect: Your answer is incorrect. webMathematica generated answer key
(−sin(x))
 = 
1p(X)​​​·p(X)​​p(X)​​​
(c)
Show that
[csc(x)]' = −csc(x) cot(x).
[csc(x)]' = 
(sin(x))−1
'
 
 = 
−syn(X)−2
Incorrect: Your answer is incorrect. webMathematica generated answer key
(cos(x))
 = 
−1p(X)​​​·p(X)​​p(X)​​​

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3. 9/9 points  |  Previous Answers CornetteApCalc1 2.8.001. My Notes
Question Part
Points
Submissions Used
1 2 3 4 5 6 7 8 9
1/1 1/1 1/1 1/1 1/1 1/1 1/1 1/1 1/1
3/100 1/100 1/100 3/100 3/100 3/100 1/100 1/100 2/100
Total
9/9
 
Which of the three graphs below are periodic? For any that are periodic, find the period and the amplitude. (If a graph is not periodic, enter NP for its period and amplitude.)
graph 1
graph 2
graph 3
Graph Periodicity Period Amplitude
1
2
Correct: Your answer is correct. webMathematica generated answer key
1
Correct: Your answer is correct. webMathematica generated answer key
2 Correct: Your answer is correct. seenKey

not periodic
NP
Correct: Your answer is correct. webMathematica generated answer key
NP
Correct: Your answer is correct. webMathematica generated answer key
3 Correct: Your answer is correct. seenKey

periodic
1
Correct: Your answer is correct. webMathematica generated answer key
12​
Correct: Your answer is correct. webMathematica generated answer key

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4. 2/4 points  |  Previous Answers CornetteApCalc1 15.6.003a. My Notes
Question Part
Points
Submissions Used
1 2 3 4
1/1 0/1 0/1 1/1
2/100 2/100 2/100 1/100
Total
2/4
 
For the given matrix, A, compute the characteristic roots of A using
r2 − (a1, 1 + a2, 2)r + a1, 1a2, 2 − a1, 2a2, 1 = 0.
 (Enter your answers as a comma-separated list.)
A =
21
32
r =
2+√3,2−√3
Correct: Your answer is correct. webMathematica generated answer key
Compute A−1 using
det A = det
a1, 1a1, 2
a2, 1a2, 2
= a1, 1a2, 2 − a1, 2a2, 1
or
A−1 = 
1
det A
a2, 2−a1, 2
−a2, 1a1, 1
A−1
 =

Incorrect: Your answer is incorrect. seenKey

[2, -1; -3, 2]
Compute
A ✕ A−1.
A ✕ A−1
=

Incorrect: Your answer is incorrect. seenKey

[1,0;0,1]
Compute
A−1 ✕ A.
A−1 ✕ A
=

Correct: Your answer is correct. seenKey

[1,0;0,1]

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5. 1/1 points  |  Previous Answers CornetteApCalc1 1.10.001. My Notes
Question Part
Points
Submissions Used
1
1/1
1/100
Total
1/1
 

problem

Jack Wolfe applied his data (see figure below) to resolve a dispute about estimates of ambient temperatures during the last 65 million years. Fossil leaves from strata ranging in age back to 65 million years were examined for the percent of smooth-leafed species in each stratum. Under the hypothesis that the relation between the percent of smooth-leafed species and the temperature in modern species persisted over the last 65 million years, he was able to estimate the past temperatures. Wolfe examined leaves from four stages of Gulf of Alaska strata, the early, middle and late Ravenian, and the Kummerian, and found percent smooth as shown below.
early Ravenian middle Ravenian late Ravenian Kummerian
65 54 20 55
Ravenian and Kummerian are names from paleobotanists, maybe specific to Wolfe. The standard names are Priabonian and Rupelian, respectively.
The Ravenian stages are in the Eocene epoch (56 to 33.9 million years ago), the Kummerian stage is in the Oligocene epoch (33.9 to 23 million years ago). Describe the temperature change during the late Ravenian. (Select all that apply.)
Correct: Your answer is correct.

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6. 15/17 points  |  Previous Answers CornetteApCalc1 2.6.008. My Notes
Question Part
Points
Submissions Used
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
1/1 1/1 1/1 1/1 1/1 1/1 1/1 1/1 1/1 1/1 0/1 1/1 1/1 1/1 1/1 1/1 0/1
1/100 1/100 1/100 1/100 1/100 1/100 1/100 1/100 1/100 1/100 3/100 1/100 1/100 1/100 1/100 1/100 3/100
Total
15/17
 
Shown in the figure below is a graph of the function F.
WebAssign Plot
Make a table of F and F −1.
F
F −1
x1
(x, y) = 
0,0.5
(x, y) = 
0.5,0
x2
(x, y) = 
1,2
(x, y) = 
2,1
x3
(x, y) = 
2,3
(x, y) = 
3,2
x4
(x, y) = 
5,5
(x, y) = 
5,5
x5
(x, y) = 
6,6
(x, y) = 
6,6
Plot the points of the inverse.
No Solution
Help
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
Clear Graph
Delete Layer
Fill

Graph LayersToggle Open/Closed

Submission Data

Let G be F −1. Make a table of G−1. (Order your answers from smallest to largest x, then from smallest to largest y.)
G −1
x1
(x, y) = 
0,0.5
x2
(x, y) = 
1,2
x3
(x, y) = 
2,3
x4
(x, y) = 
5,5
x5
(x, y) = 
6,6
Plot the points of G−1.
No Solution
Help
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
Clear Graph
Delete Layer
Fill

Graph LayersToggle Open/Closed

Submission Data

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7. 2/2 points  |  Previous Answers CornetteApCalc1 1.12.006. My Notes
Question Part
Points
Submissions Used
1 2
1/1 1/1
2/100 2/100
Total
2/2
 

problem

The length of an burr oak leaf was measured on successive days in May. The data are shown in the table below.
Length of a burr oak leaf.
Day May 7 May 8 May 9 May 10 May 11
Length (mm) 68 76 86 99 114
Use the equation
L(t) = At2 + Bt + C
 to approximate the data and compute the coefficients of the equation. (Round all numerical values to three decimal places.)
L(t) =
1.214t2+4.214t+62.6
Correct: Your answer is correct. webMathematica generated answer key
Do you have a mathematical model of leaf growth?
     Correct: Your answer is correct.

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8. 2/2 points  |  Previous Answers CornetteApCalc1 2.3.003. My Notes
Question Part
Points
Submissions Used
1 2
1/1 1/1
1/100 3/100
Total
2/2
 
Find a function, L, defined for all numbers (domain is all numbers) such that for all numbers a and b,
L(a + b) = L(a) + L(b).
 (Simplify your answer completely.)
L(x) =
x
Correct: Your answer is correct. webMathematica generated answer key
Is there another such function?
     Correct: Your answer is correct.

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9. 2/3 points  |  Previous Answers CornetteApCalc1 5.1.005a. My Notes
Question Part
Points
Submissions Used
1 2 3
1/1 1/1 0/1
2/100 2/100 2/100
Total
2/3
 
Find (approximately) an equation of the line tangent to the graph of
y = 2.5t
 at
−1, 
2
5
,
 (0, 1), and
1, 
5
2
.
 (Use
h = 0.0001
 for your step size in the centered difference formula. Round your coefficients to three decimal places.)
At
−1, 
2
5
 y(t) =
0.366(t+1)+25​
Correct: Your answer is correct. webMathematica generated answer key At
(0, 1)
 y(t) =
0.916t+1
Correct: Your answer is correct. webMathematica generated answer key At
1, 
5
2
 y(t) =
2.3(t−1)+52​
Incorrect: Your answer is incorrect. webMathematica generated answer key

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10. 4/4 points  |  Previous Answers CornetteApCalc1 6.2.003c.WA.Tut. My Notes
Question Part
Points
Submissions Used
1 2 3 4
1/1 1/1 1/1 1/1
2/100 2/100 5/100 2/100
Total
4/4
 
Find the derivative of the given function.
f(x) = e4 sin(x)
f '(x) =
4cos(x)e4sin(x)
Correct: Your answer is correct. 4cos(x)e^(4sin(x))


What is the domain of f(x)?
     Correct: Your answer is correct.

What is the domain of f '(x)?
     Correct: Your answer is correct.

Find all x-values at which the function has horizontal tangent lines. (Enter your answers as a comma-separated list. Use n to denote any arbitrary integer values. If an answer does not exist, enter DNE.)
x =
(2n+1)π2​
Correct: Your answer is correct. webMathematica generated answer key


Solution or Explanation
Derivative:
f(x) = e4 sin(x). Use 
d
du
 eu = eu where u = 4 sin(x)

f '(x) = e4 sin(x) 
d
dx
 (4 sin(x))
 = 4 cos(x)e4 sin(x).


Domain:
f(x) = e4 sin(x) and f '(x) = 4 cos(x)e4 sin(x)
 are composed of functions defined for all x. Domain of each: (−∞, ∞).

Horizontal Tangent:
f '(x) = 0 when cos(x) = 0
 (since the exponential portion is always positive).
right double arrow implies x = ± 
π
2
, ± 
3π
2
, ± 
5π
2
,  

therefore x = ± 
(2n + 1)π
2
 for n = −2, −1, 0, 1, 2,  

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