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Stewart - Calculus Concepts - 5/e (Homework)

James Finch

Math - College, section 1, Fall 2019

Instructor: Dr. Friendly

Current Score : 19 / 29

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
5/6 2/8 2/4 1/1 1/1 4/4 0/1 2/2 1/1 1/1
Total
19/29 (65.5%)

  • Instructions

    WebAssign provides a wide range of exercises that enable you to:
    • Develop Conceptual Understanding (#1-3: Read Its, Watch Its, Explore Its, Proof Problems, and Expanded Problems)
    • Build problem-solving skills (#4-7: (optional) Master Its, Master It Tutorials, Problems Plus)
    • Address Readiness Gaps (#8-10: Just in Time, QuickPrep, and Calculus Readiness Bootcamp Exercises)
    Please read the information in the box at the top of each exercise to learn more about the features, content, and/or grading. We encourage you to try correct and incorrect answers in any answer blanks.

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

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1. 5/6 points  |  Previous Answers SCalcCC5 6.1.EI.001. My Notes
Question Part
Points
Submissions Used
1 2 3 4 5 6
1/1 1/1 0/1 1/1 1/1 1/1
1/100 1/100 2/100 2/100 2/100 2/100
Total
5/6
 
  • This exercise will develop conceptual understanding.
  • Explore It exercises engage students with interactive learning modules that include video and explorations. The module is also available to use for studying in the eTextbook
Review the Explore It, then use it to complete the exercise below.
Click here to access the Explore It in a new window.
Select Function 1 under the Explore & Test section of the Explore It.
When integrating with respect to the x-axis, the formula for the approximate area, using n rectangles of equal width and left end points is
Area ≈ 
n
i = 1
(
xi − 1
 − x2i − 1) Δx.
(a)
Set the number of rectangles in the simulation to 5. What is the approximate area? (Round your answer to five decimal places.)
(b)
Set the number of rectangles in the simulation to 10. What is the approximate area? (Round your answer to five decimal places.)
(c)
What is the formula, in terms of n, for the width Δx of each rectangle? Before submitting your answer, verify your answer with various numbers of rectangles in the simulation.
Δx =
1
(d)
What is the formula, in terms of i and n, for the general left end point, xi − 1? Before submitting your answer, verify your answer with various numbers of rectangles in the simulation.
xi − 1 =
i−1n​
(e)
Using your answers to parts (c) and (d), give the general formula in terms of i and n for the approximate area when integrating with respect to the x-axis using n rectangles of equal width and left end points.
Area ≈ 
n
i = 1
(√i−1n​−(i−1n​)2)n​
(f)
Set the number of rectangles to 2. As you increase the number of rectangles to 5, 10, 20, and 40, which rectangles give the better approximation, vertical rectangles (integration with respect to the x-axis) or horizontal rectangles (integration with respect to the y-axis)?
    
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2. 2/8 points  |  Previous Answers SCalcCC5 2.4.054. My Notes
Question Part
Points
Submissions Used
1 2 3 4 5 6 7 8
1/1 0/1 0/1 0/1 0/1 0/1 0/1 1/1
1/100 2/100 1/100 1/100 1/100 1/100 1/100 1/100
Total
2/8
 
  • This exercise will develop conceptual understanding.
  • Automatically graded Proof Problems develop an understanding of the entire process of writing proofs as students get practice and immediate feedback.
  • Read It links are available as a learning tool under each question so students can quickly jump to the corresponding section of the eTextbook.
Suppose f is continuous on [1, 5] and the only solutions of the equation
f(x) = 6
 are
x = 1
 and
x = 4.
 If
f(2) = 8,
 show that
f(3) > 6.
Proof by contradiction: Select an appropriate statement to start the proof.
    
The only solutions of
f(x) = 6
 are
x = 1
 and
x = ,
 therefore
f(3) ≠ 6.
 By the applied to the continuous function f on the closed interval [2, 3], the fact that
f(2) = 8 6
 and
f(3) < 6
 implies that there is a number c in (2, 3) such that
f(c) = 6.
 This Incorrect: Your answer is incorrect. the fact that the only solutions of the equation
f(x) = 6
 are
x = 1
 and
x = .
 Hence, our supposition that
f(3) ≤ 6
 was Incorrect: Your answer is incorrect. . Therefore,
f(3) 6.
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3. 2/4 points  |  Previous Answers SCalcCC5 4.2.053.EP. My Notes
Question Part
Points
Submissions Used
1 2 3 4
1/1 0/1 0/1 1/1
1/100 4/100 1/100 1/100
Total
2/4
 
  • This exercise will develop conceptual understanding.
  • Expanded Problems go beyond a basic exercise to ask students to show the specific steps of their work or to explain the reasoning behind the answers they've provided.
Consider the following function on the given interval.
f(t) = t − 
3t
,    [−1, 6]
Find the derivative of the function.
f ′(t) =
1−13​t−(23​)
Correct: Your answer is correct.
Find any critical numbers of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)
t =
(13​)(32​)
Incorrect: Your answer is incorrect.
Find the absolute maximum and absolute minimum values of f on the given interval.
absolute minimum value
0
Incorrect: Your answer is incorrect. absolute maximum value
6−6(13​)
Correct: Your answer is correct.

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4. 1/1 points  |  Previous Answers SCalcCC5 4.5.014. My Notes
Question Part
Points
Submissions Used
1
1/1
1/100
Total
1/1
 
  • This exercise will build problem-solving skills.
  • Students get just-in-time learning support with Watch It videos that contain narrated and closed-captioned videos walking students through the proper steps to solve a similar problem.
Find the limit. Use l'Hospital's Rule where appropriate. If there is an applicable alternate method to l'Hospital's Rule, consider using it instead.
 lim x → (π⁄2)+ 
cos(x)
1 − sin(x)
 
−∞
Correct: Your answer is correct.

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5. 1/1 points  |  Previous Answers SCalcCC5 2.4.046.MI. My Notes
Question Part
Points
Submissions Used
1
1/1
1/100
Total
1/1
 
  • This exercise will build problem-solving skills.
  • Master It tutorials are an optional student-help tool available within select questions for just-in-time support. Students can use the tutorial to guide them through the problem-solving process step-by-step using different numbers.
Let
f(x) = 
 cx2 + 8x    if  x < 3
x3 − cxif  x ≥ 3.
Find the value of the constant c such that f is continuous on (−∞, ∞).
c = Correct: Your answer is correct.

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6. 4/4 points  |  Previous Answers SCalcCC5 5.5.004.MI.SA. My Notes
Question Part
Points
Submissions Used
1 2 3 4
1/1 1/1 1/1 1/1
1/100 1/100 2/100 2/100
Total
4/4
 
  • This exercise will build problem-solving skills.
  • Master It tutorials—Standalone are embedded, step-by-step tutorials used to help students understand each step required to solve the problem, before inputting their final answer.

This question has several parts that must be completed sequentially. If you skip a part of the question, you will not receive any points for the skipped part, and you will not be able to come back to the skipped part.

Tutorial Exercise
Evaluate the indefinite integral by making the indicated substitution.
x2
x3 + 16
 dx,    u = x3 + 16
Step 1
We know that if
u = f(x),
 then
du = f ′(x) dx.
 Therefore, if
u = x3 + 16,
 then
du =
$$3x2
Correct: Your answer is correct. webMathematica generated answer key dx.
Step 2
If
u = x3 + 16
 is substituted into
x2
x3 + 16
 dx
,
 then we have
x2 (u)1⁄2 dx
 = 
u1⁄2 x2 dx
.
We must also convert
x2 dx
 into an expression involving u.
We know that
du = 3x2 dx,
 so
x2 dx =
$$13​
Correct: Your answer is correct. webMathematica generated answer key du.
Step 3
Now, if
u = x3 + 16,
 then
x2
x3 + 16
 dx = 
u1⁄2 
1
3
 du
 = 
1
3
u1⁄2 du.
This evaluates as follows. (Enter your answer in terms of u.)
1
3
u1⁄2 du
 =
$$29​u(32​)
Correct: Your answer is correct. webMathematica generated answer key + C
Step 4
Since
u = x3 + 16,
 then converting back to an expression in x we get the following.
2
9
 u3⁄2 + C =
$$29​(x3 + 16)(32​) + C
Correct: Your answer is correct. webMathematica generated answer key
You have now completed the Master It.

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7. 0/1 points  |  Previous Answers SCalcCC5 3.PP.001. My Notes
Question Part
Points
Submissions Used
1
0/1
2/100
Total
0/1
 
  • This exercise will build problem-solving skills.
  • Problems Plus exercises (PP) feature examples of how to tackle challenging calculus problems and encourage students to recognize which problem-solving principles are relevant.
The figure shows a circle of radius 1 inscribed in the parabola
y = x2.
 Find the center of the circle.
The x y-coordinate plane is given. There is 1 curve and 1 circle on the graph.
  • The curve labeled y = x2 enters the window in the second quadrant, goes down and right becoming less steep, changes direction at the origin, goes up and right becoming more steep, and exits the window in the first quadrant.
  • The circle is centered on the positive y-axis such that its edge touches the curve labeled y = x2 twice, once in the second quadrant and again in the first quadrant. The points where the circle and curve touch are approximately equidistant from the origin. Two radii of length 1 are shown each connecting the center of the circle to a unique point where the circle and curve touch.
(x, y) = 
0,√(2)
Incorrect: Your answer is incorrect.
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8. 2/2 points  |  Previous Answers SCalcCC5 3.4.JIT.001.MI. My Notes
Question Part
Points
Submissions Used
1 2
1/1 1/1
1/100 1/100
Total
2/2
 
  • This exercise will address readiness gaps.
  • Just In Time exercises provide timely support by reviewing prerequisites within the context of new concepts throughout the course.
Use
f(x) = 4x − 3
 and
g(x) = 5 − x2
 to evaluate the expression.
(a)
(f ∘ g)(x)
(f ∘ g)(x) =
17 −4x2
Correct: Your answer is correct.
(b)
(g ∘ f)(x)
(g ∘ f)(x) =
−16x2 + 24x − 4
Correct: Your answer is correct.

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9. 1/1 points  |  Previous Answers SCalcCC5 QP.12.003. My Notes
Question Part
Points
Submissions Used
1
1/1
1/100
Total
1/1
 
  • This exercise will address readiness gaps.
  • Quick Prep exercises address readiness gaps by reviewing prerequisite concepts and skills. They can be used early in the course or whenever needed.
Two polynomials P and D are given. Use either synthetic or long division to divide
P(x)
 by
D(x)
 and express the quotient
P(x)
D(x)
 in the form
P(x)
D(x)
 = Q(x) + 
R(x)
D(x)
.
P(x) = x2 + 6x − 12,    D(x) = x + 5
P(x)
D(x)
 =
x + 1 −17x + 5​
Correct: Your answer is correct.

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10. 1/1 points  |  Previous Answers calculustp1 2.3.010.defective My Notes
Question Part
Points
Submissions Used
1
1/1
1/100
Total
1/1
 
  • This exercise will address readiness gaps.
  • The Calculus Readiness Bootcamp contains a formative student assessment on prerequisite topics needed for success and targeted learning modules for areas where students struggled.

Evaluate
(f ∘ g)(−3).
f(x) = x + 9;    g(x) = 4x2 − 4
41
Correct: Your answer is correct.
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