Calculus 2012 Student Edition (by Finney/Demana/Waits/Kennedy)
Calculus 2012 Student Edition (by Finney/Demana/Waits/Kennedy)
4th Edition
ISBN: 9780133178579
Author: Ross L. Finney
Publisher: PEARSON
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Chapter 5.3, Problem 32E

a.

To determine

To find: the logistic regression for the data.

a.

Expert Solution
Check Mark

Answer to Problem 32E

  S=28984386.596891+49.25205e0.85116t .

Explanation of Solution

Given information: Table shows the cumulative number of subscriber added to tat.

baseline number from 1978 to 1985.

    Year since 1977Added Subscribers Since 1977
    11,391,910
    22,814,380
    35,671,490
    411,219,200
    517,340,570
    622,113,790
    725,290,870
    827,872,520

Calculation:

The logistic regression curve, for number of subscriber S at t years after 1977 as produced by graphing utility:

  S=28984386.596891+49.25205e0.85116t .

b.

To determine

To graph: the data in a scatter plot and super impose the regression curve.

b.

Expert Solution
Check Mark

Answer to Problem 32E

Explanation of Solution

Given information: S=28984386.596891+49.25205e0.85116t

Calculation:

Scatter plot of the table data with S=28984386.596891+49.25205e0.85116t shown below.

  Calculus 2012 Student Edition (by Finney/Demana/Waits/Kennedy), Chapter 5.3, Problem 32E , additional homework tip  1

c.

To determine

To find: in what year between 1977 and 1985 were basic cable TV subscriptions growing the fastest and what significant behavior does the graph of the regression equation exhibit at that point .

c.

Expert Solution
Check Mark

Answer to Problem 32E

Between 1981 and 1982.

Explanation of Solution

Given information: S=28984386.596891+49.25205e0.85116t

Calculation:

Scatter plot of the table data with S=28984386.596891+49.25205e0.85116t shown below.

  Calculus 2012 Student Edition (by Finney/Demana/Waits/Kennedy), Chapter 5.3, Problem 32E , additional homework tip  2

From the graph, the number of subscribers is increasing the fastest around t = 4 and t = 5 or between the year 1981 and 1982.

The slope of the tangent line at that point is greatest there, in other words, the change in the number of subscribers over the change in time is largest.

The concavity also changes at that point, so it is a point of inflection.

d.

To determine

To find: what does the regression equation indicate about the number of basic cable television subscribers in the long run.

d.

Expert Solution
Check Mark

Answer to Problem 32E

In the long term, the equation says the number of subscribers will stabilize at around

29 million.

Explanation of Solution

Given information: S=28984386.596891+49.25205e0.85116t .

Calculation:

  limt28984386.596891+49.25205e0.85116t=28984386.596891+49.25205e0.85116(0)=28984386.5968929million.

So, in the long term, the equation says the number of subscribers will stabilize at around 29 million.

e.

To determine

e.

Expert Solution
Check Mark

Answer to Problem 32E

To find: what circumstances changes to render the earlier model so ineffective.

Explanation of Solution

Given information: S=28984386.596891+49.25205e0.85116t .

Calculation:

In the latter half of the 1990s, the cable companies heavily invested in and upgraded their networks, spreading out to more areas, installing fiber optics and stuff, increasing their ability to provide broadband internet and other services. These fitted curves don't usually work too much beyond the given data set anyway.

Chapter 5 Solutions

Calculus 2012 Student Edition (by Finney/Demana/Waits/Kennedy)

Ch. 5.1 - Prob. 11QRCh. 5.1 - Prob. 12QRCh. 5.1 - Prob. 1ECh. 5.1 - Prob. 2ECh. 5.1 - Prob. 3ECh. 5.1 - Prob. 4ECh. 5.1 - Prob. 5ECh. 5.1 - Prob. 6ECh. 5.1 - Prob. 7ECh. 5.1 - Prob. 8ECh. 5.1 - Prob. 9ECh. 5.1 - Prob. 10ECh. 5.1 - Prob. 11ECh. 5.1 - Prob. 12ECh. 5.1 - Prob. 13ECh. 5.1 - Prob. 14ECh. 5.1 - Prob. 15ECh. 5.1 - Prob. 16ECh. 5.1 - Prob. 17ECh. 5.1 - Prob. 18ECh. 5.1 - Prob. 19ECh. 5.1 - Prob. 20ECh. 5.1 - Prob. 21ECh. 5.1 - Prob. 22ECh. 5.1 - Prob. 23ECh. 5.1 - Prob. 24ECh. 5.1 - Prob. 25ECh. 5.1 - Prob. 26ECh. 5.1 - Prob. 27ECh. 5.1 - Prob. 28ECh. 5.1 - Prob. 29ECh. 5.1 - Prob. 30ECh. 5.1 - Prob. 31ECh. 5.1 - Prob. 32ECh. 5.1 - Prob. 33ECh. 5.1 - Prob. 34ECh. 5.1 - Prob. 35ECh. 5.1 - Prob. 36ECh. 5.1 - Prob. 37ECh. 5.1 - Prob. 38ECh. 5.1 - Prob. 39ECh. 5.1 - Prob. 40ECh. 5.1 - Prob. 41ECh. 5.1 - Prob. 42ECh. 5.1 - Prob. 43ECh. 5.1 - Prob. 44ECh. 5.1 - Prob. 45ECh. 5.1 - Prob. 46ECh. 5.1 - Prob. 47ECh. 5.1 - Prob. 48ECh. 5.1 - Prob. 49ECh. 5.1 - Prob. 50ECh. 5.1 - Prob. 51ECh. 5.1 - 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