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Stewart - Calculus: ET 9/e (Metric) (Homework)

James Finch

Math - College, section 1, Fall 2019

Instructor: Dr. Friendly

Current Score : 10 / 41

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 15
1/1 0/1 0/1 3/3 1/2 4/5 –/5 1/9 0/3 –/3 –/3 –/1 0/2 –/1 –/1
Total
10/41 (24.4%)

  • Instructions

    Calculus: Early Transcendentals, 9th edition by James Stewart, Daniel Clegg, and Saleem Watson, published by Cengage Learning, is widely renowned for its mathematical precision and accuracy, clarity of exposition, and outstanding examples and problem sets. The WebAssign enhancement to this textbook engages students with immediate feedback, rich tutorial content, video examples, interactive questions, and a media-rich eBook.

    Questions 1, 2, 4, 6, 9, 13, and 15 have Watch Its.

    Questions 1 and 2 have Master Its

    Question 3 includes Enhanced Feedback designed to help guide students to the correct answer.

    Question 4 is an Expanded Problem, which goes beyond the basic exercise and asks the student to show steps of their work or reasoning.

    Question 5 is an Explore It question, which is an interactive resource focusing on the real world applications of Calculus. Explore Its allow Calculus students to work with animations and video explanations to deepen their understanding of key concepts by helping them visualize the concepts they are learning.

    Questions 6 and 7 are questions that require the student to complete a proof.

    Question 8 is an application question focused on an application from engineering.

    Question 9 is a Just In Time exercise to remediate algebra skills required for the section.

    Question 10 is from the challenging set of Problems Plus to stretch students' problem solving skills.

    Question 11 is an application question focused on an application from biology.

    Question 12 uses 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 13 illustrates vector grading.

    Question 14 is from one of the four diagnostic tests covering Basic Algebra, Analytic Geometry, Functions, and Geometry.

    Question 15 is a question from the Quick Prep module on trigonometric identities. These modules can be used early in the course, or whenever the review is needed. 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.

    We have also turned on the display of the solutions after the first submission so you can view them. Normally, instructors elect to display these solutions only after the due date.

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1. 1/1 points  |  Previous Answers SCalcET9M 2.5.047.MI. My Notes
Question Part
Points
Submissions Used
1
1/1
6/50
Total
1/1
 
Let
f(x) = 
 cx2 + 4x    if  x < 3
x3 − cxif  x ≥ 3.
For what value of the constant c is the function f continuous on (−∞, ∞)?
c = Correct: Your answer is correct. seenKey

5/4


Solution or Explanation
f(x) = 
cx2 + 4x    if  x < 3
x3 − cxif  x ≥ 3
f is continuous on
(−∞, 3)
 and
(3, ∞).
 Now
lim x → 3− f(x) = lim x → 3− (cx2 + 4x) = 9c + 12 and lim x → 3+ f(x) = lim x → 3+ (x3 − cx) = 27 − 3c.
So f is continuous    ⇔   
9c + 12 = 27 − 3c    ⇔    12c = 15    ⇔    c =
5
4
.
Thus, for f to be continuous on
(−∞, ∞), c =
5
4
.

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2. 0/1 points  |  Previous Answers SCalcET9M 5.5.003.MI. My Notes
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Points
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1
0/1
1/50
Total
0/1
 
Evaluate the integral by making the given substitution. (Use C for the constant of integration.)
x2
x3 + 11
 dx,    u = x3 + 11
√21(21)
Incorrect: Your answer is incorrect. webMathematica generated answer key
Remember to use capital C.


Solution or Explanation
Let
u = x3 + 11.
 Then
du = 3x2 dx
 and
x2 dx = 
1
3
 du,
 so
x2
x3 + 11
 dx		 = 
u
 
1
3
 du
 = 
1
3
 
u3⁄2
3
2
 + C
 = 
1
3
 · 
2
3
 u3⁄2 + C = 
2
9
 (x3 + 11)3⁄2 + C.
Enhanced Feedback
Please try again. Recall the substitution rule for indefinite integrals: If
u = g(x)
 is a differentiable function whose range is an interval I and f is continuous on I, then
f(g(x))g'(x)dx
 = 
f(u)du.

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3. 0/1 points  |  Previous Answers SCalcET9M 2.3.046. My Notes
Question Part
Points
Submissions Used
1
0/1
1/50
Total
0/1
 
Find the limit, if it exists. (If an answer does not exist, enter DNE.)
 lim x→−7 
7 − |x|
7 + x
 
25 √35 (56)
Incorrect: Your answer is incorrect. webMathematica generated answer key


Solution or Explanation
Since
|x| = −x
 for
x < 0,
 we have
lim x→−7 
7 − |x|
7 + x
 = lim x→−7 
7 − (−x)
7 + x
 = lim x→−7 
7 + x
7 + x
 = lim x→−7 1 = 1.
Enhanced Feedback
Please try again. For evaluating limits involving the absolute value, remember to evaluate the left and right limit separately. You can write
|x|
 as
x
 if
x ≥ 0,
 and as
− x
 if
x < 0.
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4. 3/3 points  |  Previous Answers SCalcET9M 2.2.029.EP. My Notes
Question Part
Points
Submissions Used
1 2 3
1/1 1/1 1/1
1/50 1/50 3/50
Total
3/3
 
Consider the following.
lim x→2+ 
x + 5
x − 2
If x is chosen to be sufficiently close to 2 with x greater than 2, then f(x) Correct: Your answer is correct. seenKey

>

 0 and f(x) can be made Correct: Your answer is correct. seenKey

arbitrarily large positive

 .
Determine the infinite limit.
     Correct: Your answer is correct.


Solution or Explanation
lim x→2+ 
x + 5
x − 2
 = ∞
 since the numerator is positive and the denominator approaches 0 from the positive side as
x → 2+.

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5. 1/2 points  |  Previous Answers SCalcET9M 6.2.EI.001. My Notes
Question Part
Points
Submissions Used
1 2
1/1 0/1
2/50 2/50
Total
1/2
 
Review the Explore It, then use it to complete the exercise below.
Click here to access the Explore It in a new window.
(a)
For which of the three regions (given on the Explore & Test page of the Explore It) can the disk method alone be used to find the volume of the solid generated by revolving the region around the x-axis? (Select all that apply.)
Correct: Your answer is correct.

(b)
For which of the three regions (given on the Explore & Test page of the Explore It) can the disk method alone be used to find the volume of the solid generated by revolving the region around the y-axis? (Select all that apply.)
Incorrect: Your answer is incorrect.

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6. 4/5 points  |  Previous Answers SCalcET9M 2.4.015. My Notes
Question Part
Points
Submissions Used
1 2 3 4 5
1/1 1/1 1/1 1/1 0/1
1/50 1/50 1/50 1/50 2/50
Total
4/5
 
Prove the statement using the ε, δ definition of a limit.
 lim x → 5 
1
5
x − 4
 = 5 
Given ε > 0, we need δ Correct: Your answer is correct. seenKey

> 0

 such that if
0 < |x − 5| < δ,
 then
4 + 
1
5
x
 − 5
  Correct: Your answer is correct. seenKey

< ε

 .
 But
4 + 
1
5
x
 − 5
 < ε   ⇔   
1
5
x − 1
 < ε   ⇔   
1
5
|x − 5| < ε   ⇔   |x − 5| < Correct: Your answer is correct. seenKey

5ε

 .
 So if we choose
δ = Correct: Your answer is correct. seenKey

5ε
then
0 < |x − 5| < δ
   right double arrow implies  
4 + 
1
5
x
 − 5
 < ε.
 Thus,
lim x → 5 
4 + 
1
5
x
 = 5
 by the definition of a limit.
Illustrate with a diagram.

Incorrect: Your answer is incorrect.


Solution or Explanation
Given ε > 0, we need δ > 0 such that if
0 < |x − 5| < δ,
 then
1
5
x − 4
 − 5
 < ε.
 But
1
5
x − 4
 − 5
 < ε   ⇔   
1
5
x − 2
 < ε   ⇔   
1
5
|x − 5| < ε   ⇔   |x − 5| < 5ε.
 So if we choose
δ = 5ε,
 then
0 < |x − 5| < δ   right double arrow implies   
1
5
x − 4
 − 5
 < ε.
 Thus,
lim x → 5 
1
5
x − 4
 = 5
 by the definition of a limit.
The x y coordinate plane is given. The line starts at a point on the positive y-axis, goes up and right, passes through the point (5 − δ, 5 − ε), passes through the point (5, 5), passes through the point (5 + δ, 5 + ε), and exits the window in the first quadrant.

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7. –/5 points SCalcET9M 7.1.057. My Notes
Question Part
Points
Submissions Used
1 2 3 4 5
–/1 –/1 –/1 –/1 –/1
0/50 0/50 0/50 0/50 0/50
Total
–/5
 
Use integration by parts to prove the reduction formula.
(ln(x))n dx = x(ln(x))n − n
(ln(x))n − 1 dx
Let u = (ln(x))n, then dv =
||||||||
. Then
du = 
dx
 and v =
. By Equation 2, which states that
u dv = uv − 
v du,
 the integration by parts gives the following.
(ln(x))n dx		 = 
(ln(x))n − 
x 
 dx
 = x(ln(x))n − n
(ln(x))n − 1 dx

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8. 1/9 points  |  Previous Answers SCalcET9M 2.4.012. My Notes
Question Part
Points
Submissions Used
1 2 3 4 5 6 7 8 9
–/1 –/1 –/1 –/1 0/1 –/1 1/1 –/1 –/1
0/50 0/50 0/50 0/50 1/50 0/50 1/50 0/50 0/50
Total
1/9
 
Crystal growth furnaces are used in research to determine how best to manufacture crystals used in electric components. For proper growth of a crystal, the temperature must be controlled accurately by adjusting the input power. Suppose the relationship is given by
T(w) = 0.1w2 + 2.154w + 20,
where T is the temperature in degrees Celsius and w is the power input in watts.
(a)
How much power is needed to maintain the temperature at 199°C? (Round your answer to two decimal places.)
watts
(b)
If the temperature is allowed to vary from 199°C by up to ±1°C, what range of wattage is allowed for the input power? (Round your answers to two decimal places.)
watts < w < watts
(c)
In terms of the ε, δ definition of
lim x→a f(x) = L,
 what is x?
    
What is f(x)?
     Incorrect: Your answer is incorrect.
What is a?
    
What is L?
     Correct: Your answer is correct.
What value of ε is given?
°C
What is the corresponding value of δ? (Round your answer to two decimal places.)
watts
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9. 0/3 points  |  Previous Answers SCalcET9M 2.3.JIT.003. My Notes
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1 2 3
0/1 –/1 –/1
1/50 0/50 0/50
Total
0/3
 
Find
f(a), f(a + h),
 and the difference quotient
f(a + h) − f(a)
h
,
 where h ≠ 0.
f(x) = 4 − 3x + 5x2
f(a)
 =
5√7
Incorrect: Your answer is incorrect. webMathematica generated answer key
f(a + h)
 =
f(a + h) − f(a)
h
 =

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10. –/3 points SCalcET9M 2.PP.013. My Notes
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Points
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1 2 3
–/1 –/1 –/1
0/50 0/50 0/50
Total
–/3
 
Suppose f is a function that satisfies the equation
f(x + y) = f(x) + f(y) + x2y + xy2
for all real numbers x and y. Suppose also that
lim x → 0 
f(x)
x
 = 6.
(a)
Find
f(0).
(b)
Find
f '(0).
(c)
Find
f '(x).
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11. –/3 points SCalcET9M 11.2.072. My Notes
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Points
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1 2 3
–/1 –/1 –/1
0/50 0/50 0/50
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–/3
 
A patient is injected with a drug every 12 hours. Immediately before each injection the concentration of the drug has been reduced by 90% and the new dose increases the concentration by 1.4 mg/L.
(a)
What is the concentration after three doses?
mg/L
(b)
If Cn is the concentration after the nth dose, find a formula for Cn as a function of n.
Cn =
(c)
What is the limiting value of the concentration?
mg/L
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12. –/1 points SCalcET9M 9.3.010. My Notes
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–/1
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–/1
 
Solve the differential equation.
dz
dt
 + et + z = 0
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13. 0/2 points  |  Previous Answers SCalcET9M 12.4.019. My Notes
Question Part
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1 2
0/1 0/1
1/50 1/50
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0/2
 
Find two unit vectors orthogonal to both
9, 4, 1
 and
−1, 1, 0
.
(smaller i-value)
4
Incorrect: Your answer is incorrect. webMathematica generated answer key (larger i-value)
5
Incorrect: Your answer is incorrect. webMathematica generated answer key


Solution or Explanation
A theorem states that the vector a ✕ b is orthogonal to both a and b. By this theorem, the cross product of two vectors is orthogonal to both vectors. So we calculate
9, 4, 1
 × 
−1, 1, 0
 = 
i  j  k
941
−110
 = 
4  1
1 0
i − 
9  1
−1  0
j + 
9  4
−1  1
k = −i − j + 13k.
So two unit vectors orthogonal to both are
±
−1, −1, 13
1 + 1 + 169
 = ±
−1, −1, 13
3 
19
,
 that is,
− 
1
3 
19
, − 
1
3 
19
, 
13
3 
19
 and
1
3 
19
, 
1
3 
19
, − 
13
3 
19
.

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14. –/1 points SCalcET9M DT.1.005b. My Notes
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1
–/1
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–/1
 
Simplify the rational expression.
4x2 − 3x − 1
x2 − 25
 · 
x + 5
4x + 1
If you have had difficulty with this problem, you may wish to consult the Review of Algebra on the website StewartCalculus.com.
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15. –/1 points SCalcET9M QP.20.009. My Notes
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1
–/1
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–/1
 
Write the expression in terms of sine only.
9(sin(2x) − cos(2x))
Read more about Topic 20: Trigonometric Identities

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