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

James Finch

Math - College, section 1, Fall 2019

Instructor: Dr. Friendly

Current Score : 11 / 49

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

Last Saved : n/a Saving...  ()

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Question
Points
1 2 3 4 5 6 7 8 9 10 11 12 13 14
6/7 4/4 0/9 –/6 1/1 –/4 –/2 –/7 –/2 –/1 –/1 0/1 –/2 –/2
Total
11/49 (22.4%)

  • Instructions

    Biocalculus: Calculus, Probability, and Statistics for the Life Sciences, 1st edition by James Stewart and Troy Day, published by Cengage Learning, shows students how calculus relates to biology, with a style that maintains rigor without being overly formal. Students will come away with a sound knowledge of mathematics, an understanding of the importance of mathematical arguments, and a clear understanding of how these mathematical concepts and techniques are central in the life sciences. The WebAssign enhancement to this textbook engages students with immediate feedback, rich tutorial content, and an interactive, fully customizable eBook.

    Questions 2, 4, 5 and 6 have Watch Its.

    Questions 4, 5 and 7 have Master Its.

    Question 1 demonstrates interval grading, which can grade any canonically equivalent interval and enforce proper notation.

    Question 3 is a Tools for Enriching Calculus (TEC) question, which is an interactive resource that allows Calculus students to work with animations that deepen their understanding of key concepts by helping them visualize the concepts they are learning.

    Questions 5 and 8 use differential equation grading to test the validity of the answer. Accepts any correct form of the answer and runs student's responses through a series of tests to ensure the assumptions and requirements of the question are met.

    Question 6 uses the matrix tool, which allows student to define the size of the resulting matrix.

    Question 9 illustrates vector grading.

    Question 10 grades the integral and enforces correct form and notation while also allowing all mathematically correct answers.

    Question 11 uses series grading, which allows any correct version of the series based on the summation index.

    Question 12 uses the WebAssign graphing tool to graph the given region.

    Question 13 uses the WebAssign NumberLine tool to graph the solution set on the real number line.

    Question 14 is a QuickPrep (QP) question that reviews the concept of linear functions to help improve student readiness for calculus. There is a link to additional reading on Linear Functions that gives an overview of learning objectives covered in this topic. Assign any of these QuickPrep modules (or any of the questions from the modules) early in the course or whenever the review is most needed in the course.

    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

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Assignment Scoring

Your last submission is used for your score.

1. 6/7 points  |  Previous Answers SCalcLS1 4.2.012. My Notes
Question Part
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1 2 3 4 5 6 7
1/1 1/1 1/1 0/1 1/1 1/1 1/1
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6/7
 
Consider the equation below.
f(x) = 4x3 + 3x2 − 6x + 2
(a) Find the intervals on which f is increasing. (Enter your answer using interval notation.)
(−∞,−1)∪(0.5,∞)
Correct: Your answer is correct. webMathematica generated answer key

Find the interval on which f is decreasing. (Enter your answer using interval notation.)
(−1,12​)
Correct: Your answer is correct. webMathematica generated answer key

(b) Find the local minimum and maximum values of f.
local minimum value    
0.25
Correct: Your answer is correct. webMathematica generated answer key
local maximum value
5
Incorrect: Your answer is incorrect. webMathematica generated answer key


(c) Find the inflection point.
(x, y) = 
−0.25,3.625
Correct: Your answer is correct. webMathematica generated answer key


Find the interval on which f is concave up. (Enter your answer using interval notation.)
(−0.25,∞)
Correct: Your answer is correct. webMathematica generated answer key

Find the interval on which f is concave down. (Enter your answer using interval notation.)
(−∞,−0.25)
Correct: Your answer is correct. webMathematica generated answer key


Solution or Explanation
Click to View Solution
Enhanced Feedback
Please try again, keeping in mind that the function f is increasing on an interval if
f ' > 0
 and is decreasing if
f ' < 0.
 Moreover, the function f has a local minimum or maximum at c if
f '(c) = 0
 and
f '
 changes sign at c. If
f '
 changes from positive to negative then f has a local maximum at c, and if
f '
 changes from negative to positive then f has a local minimum at c. The function f is concave up on an interval if
f '' > 0
 and concave down if
f '' < 0.
 Also, f has an inflection point where it changes concavity.
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2. 4/4 points  |  Previous Answers SCalcLS1 4.4.029. My Notes
Question Part
Points
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1 2 3 4
1/1 1/1 1/1 1/1
2/50 2/50 1/50 1/50
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4/4
 
Ornithologists have determined that some species of birds tend to avoid flights over large bodies of water during daylight hours. It is believed that more energy is required to fly over water than land because air generally rises over land and falls over water during the day. A bird with these tendencies is released from an island that is 7 km from the nearest point B on the shoreline, flies to a point C on the shoreline, and then flies along the shoreline to its nesting area D. Assume that the bird instinctively chooses a path that will minimize its energy expenditure. Points B and D are 9 km apart. (Round your answers to two decimal places.)
(a) In general, if it takes 1.4 times as much energy to fly over water as land, to what point C should the bird fly in order to minimize the total energy expended in returning to its nesting area?
Correct: Your answer is correct. seenKey

7.14

 km from B

(b) Let W and L denote the energy (in joules) per kilometer flown over water and land, respectively. Assuming the bird's energy expenditure is minimized, determine a function for the ratio W/L in terms of x, the distance from B to C.
W
L
 =
√49+x2x​
Correct: Your answer is correct. sqrt(49 + x^2)/x


(c) What should the value of W/L be in order for the bird to fly directly to its nesting area D?
Correct: Your answer is correct. seenKey

1.27

(d) If the ornithologists observe that birds of a certain species reach the shore at a point 2 km from B, how many times more energy does it take a bird to fly over water than land?
Correct: Your answer is correct. seenKey

3.64


Solution or Explanation
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3. 0/9 points  |  Previous Answers SCalcLS1 4.4.TEC.001. My Notes
Question Part
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1 2 3 4 5 6 7 8 9
0/1 –/1 –/1 –/1 –/1 –/1 –/1 –/1 –/1
1/50 0/50 0/50 0/50 0/50 0/50 0/50 0/50 0/50
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0/9
 

  • Instructions for Simulation

    Open the simulation tab below and click the link to launch the simulation. The menu lists six problems. Clicking on one of the problems displays the problem statement along with a diagram. You then have the opportunity to explore different scenarios for the problem numerically and graphically, after which you can make an initial estimate of the solution. When you are ready to compute the solution analytically, hints are provided to guide you.

    Area of a Rectangle in an Ellipse

    After reading the problem statement, click the Numerical Exploration button to display an interactive diagram of the upper half of an ellipse with an inscribed rectangle.

    Drag the a and b slider handles, click on a slider bar, or click on a number next to a slider to change the values in the equation of the ellipse. Similarly, use the x slider to control the width of the inscribed rectangle. The area of the rectangle is displayed and updated as you change the values of a, b, or x.

    Click the Graph Area button to display a graph of the rectangle's area A(x) as a function of x as well as the area for the current value of x (represented by a green dot on the graph).

    When you are ready to solve the problem analytically, click on the Hints for Analytical Solution button to display a sequence of hints to help guide you.

  • Simulation

  • Exercise

    Find the dimensions and the area of the largest rectangle that can be inscribed in the upper half of the ellipse:
    x2
    a2
     + 
    y2
    b2
     = 1

    Numerical Exploration
    For the default values of a = 3 and b = 2.5, what approximate value of x gives the largest rectangle area? (Round your answer to two decimal places.)
    Incorrect: Your answer is incorrect. seenKey

    2.12

    What are the dimensions of this rectangle? (Enter the dimensions as a comma-separated list. Round your answers to two decimal places.)


    Now try it with a = 3, b = 4. What approximate value of x gives the largest rectangle area? (Round your answer to two decimal places.)


    What are the dimensions of this rectangle? (Enter the dimensions as a comma-separated list. Round your answers to two decimal places.)


    If you change the value of b, does the same value of x give the rectangle with the largest area?
        

    If you change the value of a, does the same value of x give the rectangle with the largest area?
        

    Does the optimal rectangle depend on a, b, or both?
        

    Analytical Solution
    Find the dimensions and area of the largest rectangle that can be inscribed in the upper half of the ellipse. (Give your answers in terms of a and b. Enter the dimensions as a comma-separated list.)
    x2
    a2
     + 
    y2
    b2
     = 1
    dimensions    
    area    
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4. –/6 points SCalcLS1 5.1.003.MI. My Notes
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–/6
 
(a) Estimate the area under the graph of f(x) = 2 cos(x) from x = 0 to x = π/2 using four approximating rectangles and right endpoints. (Round your answers to four decimal places.)
R4 =

Sketch the graph and the rectangles.

Is your estimate an underestimate or an overestimate?
    

(b) Repeat part (a) using left endpoints.
L4 =

Sketch the graph and the rectangles.

Is your estimate an underestimate or an overestimate?
    

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5. 1/1 points  |  Previous Answers SCalcLS1 7.4.003.MI. My Notes
Question Part
Points
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1
1/1
1/50
Total
1/1
 
Solve the differential equation.
(x2 + 3)y' = xy
y2=C(x2+3)
Correct: Your answer is correct. webMathematica generated answer key

Solution or Explanation
(x2 + 3)y' = xy
dy
dx
 = 
xy
x2 + 3
 
dy
y
 = 
x dx
x2 + 3
    [y ≠ 0]
dy
y
 = 
x dx
x2 + 3
 
ln|y| = 
1
2
 ln(x2 + 3) + C    [u = x2 + 3, du = 2x dx]
 = ln(x2 + 3)1/2 + ln eC
 = ln
eC
x2 + 3
|y| = eC
x2 + 3
y = K
x2 + 3
, where K = ±eC is a constant.
(In our derivation, K was nonzero, but we can restore the excluded case y = 0 by allowing K to be zero.)

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6. –/4 points SCalcLS1 8.5.004. My Notes
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1 2 3 4
–/1 –/1 –/1 –/1
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–/4
 
Construct the corresponding matrix for each of the following matrix diagrams.
(a)    


(b)    


(c)    


(d)    

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7. –/2 points SCalcLS1 9.2.009.MI. My Notes
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–/1 –/1
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–/2
 
Find the first partial derivatives of the function.
f(x, y) = y4 − 5xy
fx(x, y) = 
fy(x, y) = 

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8. –/7 points SCalcLS1 10.3.005. My Notes
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1 2 3 4 5 6 7
–/1 –/1 –/1 –/1 –/1 –/1 –/1
0/50 0/50 0/50 0/50 0/50 0/50 0/50
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–/7
 
Metastasis is the process by which cancer cells spread throughout the body and initiate tumors in various organs. This sometimes happens via the bloodstream, where cancer cells become lodged in capillaries of organs and then move across the capillary wall into the organ. Using C to denote the number of cells lodged in a capillary and I for the number that have invaded the organ, we can model this as
C' = −αC − βC      I' = αC − δI + ρI
where all parameters are positive, α is the rate of movement across the capillary wall, β is the rate of dislodgement from the capillary, δ is the rate at which cancer cells in the organ die, and ρ is their growth rate†.
(a) Find the general solution.
(C(t), I(t)) = 


(b) Classify the origin when
ρ > δ.
    

Classify the origin when
ρ < δ.
    

(c) What is the solution to the initial value problem if
C(0) = C0
 and
I(0) = 0?
(C(t), I(t)) = 


(d) Use the result from part (c) to show that the tumor will grow in the long-term if, and only if,
ρ > δ.
If
ρ > δ,
 then the exponential term
e(ρ − δ)t
 will as t increases while the term
e−(α + β)t
 , so the tumor will grow in the long term. Conversely, if the tumor grows in the long term then
I(t)
 increases as t increases (and does not converge). This is possible only when the coefficient of t in the exponential term
e(ρ − δ)t
 is , that is when
ρ > δ.

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9. –/2 points SCalcLS1 8.2.013. My Notes
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–/2
 
Find the sum of the given vectors. (Simplify your answer completely.)
a = 
0, 3, 4
,    b = 
0, 0, −5
a + b =


Illustrate geometrically.

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10. –/1 points SCalcLS1 5.4.031. My Notes
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–/1
 
Evaluate the indefinite integral. (Use C for the constant of integration.)
 
x(4x + 7)8dx
 
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11. –/1 points SCalcLS1 A.F.020. My Notes
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–/1
 
Write the sum in sigma notation.
4 − 4x + 4x2 − 4x3 + · · · + (−1)n4xn
n
j = 0
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12. 0/1 points  |  Previous Answers SCalcLS1 DT.2.005e. My Notes
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0/1
 
Sketch the region in the xy-plane defined by the equation or inequalities.
x2 + y2 < 9
No Solution
Help
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Submission Data


If you have had difficulty with this problem, you may wish to consult the Review of Analytic Geometry in Appendix B of your text or YouBook.
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13. –/2 points SCalcLS1 A.A.022. My Notes
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–/2
 
Solve the inequality in terms of intervals and illustrate the solution set on the real number line.
(x + 5)(x − 4)(x + 8) ≥ 0

Use the tools to enter your answer.
Created with Raphaël 2.1.0-10-8-6-4-20246810
Created with Raphaël 2.1.0

NO SOLUTION

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14. –/2 points SCalcLS1 QP.6.008. My Notes
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(a) Sketch the line with slope
5
4
 that passes through the point
(−3, 1).
No Solution
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(b) Find an equation for this line.


Read more about Topic 6: Linear Functions
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