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C7-1
CASE STUDY 7
DATA CENTER CONSOLIDATION AT GUARDIAN
LIFE
As one of the largest mutual life insurance firms in the United
States,
Guardian Life (www.guardianlife.com) has more than 5000
employees and
over 3000 financial representatives in 80 agencies. Guardian
and its
subsidiaries provide almost three million people with life and
disability
income insurance, retirement services, and investment products
such as
mutual funds, securities, variable life insurance, and variable
annuities. The
company also supplies employee benefits programs to six
million
participants, including life, health, and dental insurance, as well
as qualified
pension plans. In addition to regional home offices in New York
City;
Bethlehem, Pennsylvania; Spokane, Washington; and Appleton,
Wisconsin,
the company has 55 remote sales offices and 80 remote agency
offices.
Like other insurance companies, Guardian Life is an
information
intensive organization where data processing and
communications network
infrastructure have consistently been important contributors to
its success.
Guardian Life’s IT organization has earned numerous accolades
including
multiple CIO100 awards from CIO magazine [PRNE11].
According to Dennis
Callahan, Executive Vice President and Chief Information
Officer for,
Guardian Life, "A strong partnership between IT and the
businesses enables
http://www.guardianlife.com/
C7-2
Guardian to deliver cost-effective technology services that
facilitate world-
class customer service, product innovation, and operational
efficiency.”
Ensuring alignment between business and IT is important to
Guardian Life
and provides a consistent theme for many of the insurance
companies IT
projects including its data center consolidation initiatives
[CIOZ12].
Data center consolidation has been an ongoing concern at
Guardian for
more than a decade. Guardian’s IT governance structure is
team-oriented
and the company’s data center consolidation initiatives are
overseen by it
Infrastructure team. The Infrastructure team is primarily co-
located in New
York, and Bethlehem, Pennsylvania but it has key support teams
in Spokane,
Washington, Appleton, Wisconsin, and Pittsfield,
Massachusetts.
Guardian Life began taking a serious look at data center
consolidation in
2000, but in the aftermath of the September 11, 2001 terrorist
attack,
Guardian also became more concerned with business continuity
issues.
Guardian had four significant data centers, at its four home
offices, but the
primary data center was in New York City. After 9/11, Guardian
wanted
make infrastructure changes to ensure business continuity
across its existing
data centers and made plans to add two more data centers to the
mix.
Guardian performed an assessment of its data centers to
provide a basis
for planning on the location of data processing resources. One
surprising
outcome of this assessment had to do with utilization. The
assessment
revealed that the four data centers had about 1000 UNIX and
NT servers,
with an average capacity utilization of 10%. Even at peak
demand, only 25%
of the processing power of the servers was being used
[MUSI02]. Guardian
responded to this assessment with a plan that included the
following
objectives:
1. Move the primary data center from New York City to
Bethlehem,
Pennsylvania.
C7-3
2. Improve the efficiency of its data centers, including server
and storage
utilization. Specifically, Guardian set a goal of reducing the
number of
servers supporting Guardian's applications and databases by
40% and
reducing the server support staff by 60%.
3. Ensure a smooth transition to the new primary data center.
The company chose IT consulting firm Greenwich Technology
Partners
(GTP) to help it design and carry out the transition. GTP began
with an
assessment of the company's IT environment and looked at the
impact of
moving the data center from New York to Bethlehem. A major
issue related
to the move was that the largest number of data center users was
located in
the New York area. Thus, the new deployment needed to
provide sufficient
data transmission capacity to meet these users' needs.
Fortunately, the
network infrastructure already in place was fairly standard and
easily
scalable (Figure C7.1).
C7-4
An ATM WAN backbone linked the four regional home offices.
Frame
relay connections linked the remote sales offices and remote
agency offices,
and 100-Mbps and 1-Gbps Ethernet LANs provided connectivity
within
campuses. The Cisco Catalyst 5500 and 7200 series routers
provided
Ethernet support combined with a modular, easily scalable
design. The Cisco
IGX 8400 ATM switches could be scaled to support an ATM
network service
at any desired capacity. With three widely used networking
technologies in
place, some of the problems that might have been encountered
in a more
convoluted networking environment were avoided.
GTP also looked at the application and database patterns. They
determined that in addition to traditional applications such as
file and print
services, PeopleSoft, and Lotus Notes, Guardian also used a
collection of
applications to support its intranet. Then, as now, the site
included
marketing materials and sales tools for the firm's agents,
account profiling,
and customer data. The company had also invested in a number
of financial
services applications, including applications for supporting its
trading and
securities functions. Due to their complexity and the amount of
resource
required to support them, many of Guardian's applications have
historically
been supported by dedicated servers.
The transition team, consisting of Guardian and GTP
personnel, did
extensive validation work and benchmarking to make sure the
data they had
gathered during the initial assessment were accurate. They
measured
network utilization at granular levels and modeled various
consolidation
scenarios for reducing server hardware.
From this analysis, GTP proceeded to develop a plan for
consolidating
the servers, looking at both business and technology issues. For
example,
the team considered the criticality of the applications supported
by the
servers, as well as which business units they belonged to. Some
servers
were good candidates for consolidation; others were not; and
others were
out of warranty, which made them too expensive to keep. After
the
C7-5
migration plan was devised and the new architecture developed,
more
testing was conducted to ensure their viability.
The team devoted much thought and analysis to the migration
to the
new architecture, so as not to disrupt day-to-day operations. The
plan
targeted the least complex opportunities first, starting with file
and print
services. Guardian initially had more than 30 servers providing
file and print
services. These were consolidated into just two servers
clustered in a high-
availability, fault-tolerant configuration. For more complex
parts of the plan,
the team opted to do some of the consolidation in New York,
and only then
move the servers to Bethlehem after the consolidation had
settled down.
The initial consolidation and relocation project yielded tangible
benefits
to Guardian in terms of reduced hardware and personnel
requirements. But
the benefits extended well beyond these initial objectives. The
mindset of
solving new problems efficiently and in the context of the
existing
infrastructure had taken hold. Guardian no longer automatically
takes orders
for new servers to support applications as it did throughout the
1990s.
Instead, Guardian analyzes each new application requirement
and attempts
to support it with the existing hardware/software suite or with
minimal
upgrades and extensions.
The total cost of the data centered consolidation project was
$4.5
million, but the company saved more than $3 million in 2002,
offsetting
much of that cost. Even greater savings were realized during
each of the
following two years.
Further Consolidation
In 2010, Guardian embarked on a second major data centered
consolidation
initiative. The maturation of virtualization technologies and
evolution of high-
bandwidth WAN connections encouraged Guardian to
consolidate six data
centers into two [MITC11]. This is illustrated in Figure C7.2.
Guardian will
C7-6
consolidate its mission critical data processing infrastructure
into one
primary data center that it will own. The company plans to lease
a second
modular pod for use as its second data center. Pod data centers
can be
thought of as data centers in a box. These can be configured by
vendors to
customer specifications and delivered as a container that looks
similar to
refrigerated box car or multi-modal shipping container stacked
on ships or
carried by tractor trailer trucks. Pods are energy efficient but
can be a
cramped for human movement.
C7-7
In addition to moving to a much leaner data center
infrastructure,
Guardian is migrating from Unix to Linux. It has also started to
move some
of its applications to the cloud [MITC11]. By the beginning of
2012, Guardian
had moved 18 back-end applications to SaaS and had begun
transitioning e-
mail, HRIS, and IT services into the cloud. The breadth of
Guardian's move
to the cloud put the company on the leading edge among
Fortune 250
organizations [MITC12].
Guardian is using the IDEAS Advantage services from Ideas
International to help it make decisions about remodeling its data
center
infrastructure [GOLI11]. These sophisticated tools in
combination with
Guardian’s commitment to cloud services indicate that the
insurance
company’s data center consolidation initiatives are far from
over. Guardian
has not yet moved its core ERP systems to the cloud, but even
this is under
consideration.
Discussion Points
1. Do some Internet research on the reasons why businesses
generally
invest in data center consolidation projects. What benefits do
they
commonly hope to realize? How do Guardian’s rationale for
consolidating data centers compare to those of other businesses?
2. Getting outside consultants to manage data center
consolidation
projects is a common practice. Discuss the pros and cons of
using
consultants to manage data center consolidation projects?
3. How/why has virtualization fueled business interest in data
center
consolidation?
4. Why is the availability of high-speed, high bandwidth
communications
an important consideration in data center consolidation plans
and
decision-making?
5. Do some Internet research on modular (pod) data centers.
Summarize
the advantages of disadvantages of modular data centers.
C7-8
6. Guardian is consolidating from six to two data centers
supplemented
with SaaS cloud services. As the company continues to move
applications to the cloud, could it consolidate to zero data
centers?
Why or why not?
7. Do you think that winning multiple CIO 100 awards has
encouraged
Guardian’s IT executives to recommend risky IT projects? Why
or why
not?
Sources
[CIOZ12] CIO Zone.”Guardian CIO on Aligning IT, Business.”
Retrieved
online at:
http://www.ciozone.com/index.php/Management/Guardian-CIO-
On-Aligning-IT-Business.html.
[GOLI11] Golia, N. “Guardian Life Using Analytics to Assist
Data Center
Planning. Insurance Technology, September 1, 2011. Retrieved
online at:
http://www.insurancetech.com/architecture-
infrastructure/231600651
[MITC11] Mitchell, R. “Cloud to Lower Bar, Intensify
Competition.”
Computerworld, March 7, 2011. Retrieved online at:
http://blogs.computerworld.com/17935/cloud_to_lower_bar_inte
nsify_comp
etition
[MITC11] Mitchell, R. “Best Practices for Scaling up SaaS to
the Cloud.
Techworld, February 14, 2012. Retrieved online at:
http://features.techworld.com/sme/3337405/best-practices-for-
scaling-up-
saas-in-the-cloud/
[MUSI02] Musich, P. “Project Gets Helping Hand.” eWeek,
November 25,
2002. www.eweek.com.
[PRNE11] PR Newswire. “Guardian Life Recognized by CIO
Magazine for
Third Consecutive Year in CIO 100 Awards.” Retrieved online
at:
http://www.prnewswire.com/news-releases/guardian-life-
recognized-by-cio-
magazine-for-third-consecutive-year-in-cio-100-awards-
54985272.html
http://www.ciozone.com/index.php/Management/Guardian-CIO-
On-Aligning-IT-Business.html
http://www.ciozone.com/index.php/Management/Guardian-CIO-
On-Aligning-IT-Business.html
http://www.insurancetech.com/architecture-
infrastructure/231600651
http://blogs.computerworld.com/17935/cloud_to_lower_bar_inte
nsify_competition
http://blogs.computerworld.com/17935/cloud_to_lower_bar_inte
nsify_competition
http://features.techworld.com/sme/3337405/best-practices-for-
scaling-up-saas-in-the-cloud/
http://features.techworld.com/sme/3337405/best-practices-for-
scaling-up-saas-in-the-cloud/
http://www.prnewswire.com/news-releases/guardian-life-
recognized-by-cio-magazine-for-third-consecutive-year-in-cio-
100-awards-54985272.html
http://www.prnewswire.com/news-releases/guardian-life-
recognized-by-cio-magazine-for-third-consecutive-year-in-cio-
100-awards-54985272.htmlCASE STUDY 7Further
ConsolidationDiscussion PointsSources
MAT 1339B (Winter 2018)
Assignment 1
Professor: Rachid Bentoumi
Deadline: Friday, February 09, 2018 Before 15:00
(585 King Eduard math-stat department drop boxes)
Student Name
Student Number
By signing below, you declare that this work was your own and
that you
have not copied from any other individual or other source.
Signature
Late assignments will NOT be accepted; nor will unstapled
assignments.
Professors in the math department will not lend you a stapler.
You should complete ALL the questions in the assignments. It
is possible,
however, that not all the questions will be marked. In that case,
the same
questions will be marked in all assignments. You will not be
informed be-
forehand which questions will be marked.
1
Question 1. Solve the following inequalities:
(a) |2x− 5| > 2
(b) |3 − 2x| ≤ 7
2
Question 2. Find the domain of the functions defined below
(a) f(x) =
1
x2 − 8x
(b) g(x) =
x
|x| + 4
(c) h(x) =
√
x− 2
x− 7
3
Question 3. Calculate the following limits (show all steps of
your calcula-
tion).
(a) lim
x→0
√
19 − 3x−x
x
(b) lim
x→−∞
√
x2 − 1
2x
(c) lim
x→∞
cos (x)
x4 + 1
4
Question 4. Let f be a function defined as
f(x) =
x2 − 16
2x− 8
, if x < 4
−4, si x = 4
x− 4
√
x− 2
if x > 4
(a) Determine lim
x→4
f(x) ?
(b) Is the function f continuous on R? Justify your answer.
5
Question 5. Determine, if it is possible, the value of k that
makes the
functions below continuous on R.
(a) h(x) =
x + 2 if x < 1
k if x = 1
x2 + 3x− 1 if x > 1
(b) f(t) =
t2 − 25
t− 5
if t < 5
kt if t ≥ 5
6
Question 6. Consider the following function
f(x) =
√
25 − 2x
(a) Use the definition of the derivative to find f
′
(x)
(b) Determine the tangent line to f at x = 0.
7
Question 7. Use the derivative rules to calculate the derivatives
of the
following functions:
(a) y =
√
2x2 + 5x + 7
(b) y = (x2 + 3)4(x3 − 5)3
8
Question 8. Consider a closed rectangular box with the
following
dimensions: length= x cm, width= 2x cm and height= x + 1 cm.
(a) According to x, determine
i. the function A(x) giving the total area of the rectangular box.
ii. the function V (x) giving the volume of the rectangular box.
(b) Calculate average rates of change of the area and the volume
when
x passes from 3 cm to 6 cm.
9
Question 9. A manufacturer estimates that his weekly revenue
R(x)
can be expressed by
R(x) = 1008x− 12x2 − 8x3
where x is the number of units sold, R(x) is expressed in
dollars. Under
certain constraints, one cannot produce more than 10 units per
week.
(a) Using the variation table (or test points), determine on
which
interval(s) the revenue is increasing.
(b) Determine the quantity of units that gives a maximum
revenue
and evaluate this maximum revenue.
10
Question 10. Determine the concavity intervals and, if it is
possible,
the inflection points of the function h(x) = (x− 1)2(x + 1)2.
11

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C7-1 CASE STUDY 7 DATA CENTER CONSOLIDATION AT GUARDI.docx

  • 1. C7-1 CASE STUDY 7 DATA CENTER CONSOLIDATION AT GUARDIAN LIFE As one of the largest mutual life insurance firms in the United States, Guardian Life (www.guardianlife.com) has more than 5000 employees and over 3000 financial representatives in 80 agencies. Guardian and its subsidiaries provide almost three million people with life and disability income insurance, retirement services, and investment products such as mutual funds, securities, variable life insurance, and variable annuities. The company also supplies employee benefits programs to six million participants, including life, health, and dental insurance, as well as qualified
  • 2. pension plans. In addition to regional home offices in New York City; Bethlehem, Pennsylvania; Spokane, Washington; and Appleton, Wisconsin, the company has 55 remote sales offices and 80 remote agency offices. Like other insurance companies, Guardian Life is an information intensive organization where data processing and communications network infrastructure have consistently been important contributors to its success. Guardian Life’s IT organization has earned numerous accolades including multiple CIO100 awards from CIO magazine [PRNE11]. According to Dennis Callahan, Executive Vice President and Chief Information Officer for, Guardian Life, "A strong partnership between IT and the businesses enables http://www.guardianlife.com/ C7-2 Guardian to deliver cost-effective technology services that
  • 3. facilitate world- class customer service, product innovation, and operational efficiency.” Ensuring alignment between business and IT is important to Guardian Life and provides a consistent theme for many of the insurance companies IT projects including its data center consolidation initiatives [CIOZ12]. Data center consolidation has been an ongoing concern at Guardian for more than a decade. Guardian’s IT governance structure is team-oriented and the company’s data center consolidation initiatives are overseen by it Infrastructure team. The Infrastructure team is primarily co- located in New York, and Bethlehem, Pennsylvania but it has key support teams in Spokane, Washington, Appleton, Wisconsin, and Pittsfield, Massachusetts. Guardian Life began taking a serious look at data center consolidation in 2000, but in the aftermath of the September 11, 2001 terrorist
  • 4. attack, Guardian also became more concerned with business continuity issues. Guardian had four significant data centers, at its four home offices, but the primary data center was in New York City. After 9/11, Guardian wanted make infrastructure changes to ensure business continuity across its existing data centers and made plans to add two more data centers to the mix. Guardian performed an assessment of its data centers to provide a basis for planning on the location of data processing resources. One surprising outcome of this assessment had to do with utilization. The assessment revealed that the four data centers had about 1000 UNIX and NT servers, with an average capacity utilization of 10%. Even at peak demand, only 25% of the processing power of the servers was being used [MUSI02]. Guardian responded to this assessment with a plan that included the
  • 5. following objectives: 1. Move the primary data center from New York City to Bethlehem, Pennsylvania. C7-3 2. Improve the efficiency of its data centers, including server and storage utilization. Specifically, Guardian set a goal of reducing the number of servers supporting Guardian's applications and databases by 40% and reducing the server support staff by 60%. 3. Ensure a smooth transition to the new primary data center. The company chose IT consulting firm Greenwich Technology Partners (GTP) to help it design and carry out the transition. GTP began with an assessment of the company's IT environment and looked at the impact of moving the data center from New York to Bethlehem. A major
  • 6. issue related to the move was that the largest number of data center users was located in the New York area. Thus, the new deployment needed to provide sufficient data transmission capacity to meet these users' needs. Fortunately, the network infrastructure already in place was fairly standard and easily scalable (Figure C7.1). C7-4 An ATM WAN backbone linked the four regional home offices. Frame relay connections linked the remote sales offices and remote agency offices, and 100-Mbps and 1-Gbps Ethernet LANs provided connectivity within campuses. The Cisco Catalyst 5500 and 7200 series routers provided Ethernet support combined with a modular, easily scalable
  • 7. design. The Cisco IGX 8400 ATM switches could be scaled to support an ATM network service at any desired capacity. With three widely used networking technologies in place, some of the problems that might have been encountered in a more convoluted networking environment were avoided. GTP also looked at the application and database patterns. They determined that in addition to traditional applications such as file and print services, PeopleSoft, and Lotus Notes, Guardian also used a collection of applications to support its intranet. Then, as now, the site included marketing materials and sales tools for the firm's agents, account profiling, and customer data. The company had also invested in a number of financial services applications, including applications for supporting its trading and securities functions. Due to their complexity and the amount of resource
  • 8. required to support them, many of Guardian's applications have historically been supported by dedicated servers. The transition team, consisting of Guardian and GTP personnel, did extensive validation work and benchmarking to make sure the data they had gathered during the initial assessment were accurate. They measured network utilization at granular levels and modeled various consolidation scenarios for reducing server hardware. From this analysis, GTP proceeded to develop a plan for consolidating the servers, looking at both business and technology issues. For example, the team considered the criticality of the applications supported by the servers, as well as which business units they belonged to. Some servers were good candidates for consolidation; others were not; and others were out of warranty, which made them too expensive to keep. After the
  • 9. C7-5 migration plan was devised and the new architecture developed, more testing was conducted to ensure their viability. The team devoted much thought and analysis to the migration to the new architecture, so as not to disrupt day-to-day operations. The plan targeted the least complex opportunities first, starting with file and print services. Guardian initially had more than 30 servers providing file and print services. These were consolidated into just two servers clustered in a high- availability, fault-tolerant configuration. For more complex parts of the plan, the team opted to do some of the consolidation in New York, and only then move the servers to Bethlehem after the consolidation had settled down. The initial consolidation and relocation project yielded tangible benefits
  • 10. to Guardian in terms of reduced hardware and personnel requirements. But the benefits extended well beyond these initial objectives. The mindset of solving new problems efficiently and in the context of the existing infrastructure had taken hold. Guardian no longer automatically takes orders for new servers to support applications as it did throughout the 1990s. Instead, Guardian analyzes each new application requirement and attempts to support it with the existing hardware/software suite or with minimal upgrades and extensions. The total cost of the data centered consolidation project was $4.5 million, but the company saved more than $3 million in 2002, offsetting much of that cost. Even greater savings were realized during each of the following two years.
  • 11. Further Consolidation In 2010, Guardian embarked on a second major data centered consolidation initiative. The maturation of virtualization technologies and evolution of high- bandwidth WAN connections encouraged Guardian to consolidate six data centers into two [MITC11]. This is illustrated in Figure C7.2. Guardian will C7-6 consolidate its mission critical data processing infrastructure into one primary data center that it will own. The company plans to lease a second modular pod for use as its second data center. Pod data centers can be thought of as data centers in a box. These can be configured by vendors to customer specifications and delivered as a container that looks similar to refrigerated box car or multi-modal shipping container stacked on ships or carried by tractor trailer trucks. Pods are energy efficient but
  • 12. can be a cramped for human movement. C7-7 In addition to moving to a much leaner data center infrastructure, Guardian is migrating from Unix to Linux. It has also started to move some of its applications to the cloud [MITC11]. By the beginning of 2012, Guardian had moved 18 back-end applications to SaaS and had begun transitioning e- mail, HRIS, and IT services into the cloud. The breadth of Guardian's move to the cloud put the company on the leading edge among Fortune 250 organizations [MITC12]. Guardian is using the IDEAS Advantage services from Ideas International to help it make decisions about remodeling its data center
  • 13. infrastructure [GOLI11]. These sophisticated tools in combination with Guardian’s commitment to cloud services indicate that the insurance company’s data center consolidation initiatives are far from over. Guardian has not yet moved its core ERP systems to the cloud, but even this is under consideration. Discussion Points 1. Do some Internet research on the reasons why businesses generally invest in data center consolidation projects. What benefits do they commonly hope to realize? How do Guardian’s rationale for consolidating data centers compare to those of other businesses? 2. Getting outside consultants to manage data center consolidation projects is a common practice. Discuss the pros and cons of using consultants to manage data center consolidation projects? 3. How/why has virtualization fueled business interest in data center
  • 14. consolidation? 4. Why is the availability of high-speed, high bandwidth communications an important consideration in data center consolidation plans and decision-making? 5. Do some Internet research on modular (pod) data centers. Summarize the advantages of disadvantages of modular data centers. C7-8 6. Guardian is consolidating from six to two data centers supplemented with SaaS cloud services. As the company continues to move applications to the cloud, could it consolidate to zero data centers? Why or why not? 7. Do you think that winning multiple CIO 100 awards has encouraged Guardian’s IT executives to recommend risky IT projects? Why or why not?
  • 15. Sources [CIOZ12] CIO Zone.”Guardian CIO on Aligning IT, Business.” Retrieved online at: http://www.ciozone.com/index.php/Management/Guardian-CIO- On-Aligning-IT-Business.html. [GOLI11] Golia, N. “Guardian Life Using Analytics to Assist Data Center Planning. Insurance Technology, September 1, 2011. Retrieved online at: http://www.insurancetech.com/architecture- infrastructure/231600651 [MITC11] Mitchell, R. “Cloud to Lower Bar, Intensify Competition.” Computerworld, March 7, 2011. Retrieved online at: http://blogs.computerworld.com/17935/cloud_to_lower_bar_inte nsify_comp etition [MITC11] Mitchell, R. “Best Practices for Scaling up SaaS to the Cloud. Techworld, February 14, 2012. Retrieved online at: http://features.techworld.com/sme/3337405/best-practices-for- scaling-up- saas-in-the-cloud/ [MUSI02] Musich, P. “Project Gets Helping Hand.” eWeek, November 25, 2002. www.eweek.com. [PRNE11] PR Newswire. “Guardian Life Recognized by CIO Magazine for Third Consecutive Year in CIO 100 Awards.” Retrieved online at:
  • 16. http://www.prnewswire.com/news-releases/guardian-life- recognized-by-cio- magazine-for-third-consecutive-year-in-cio-100-awards- 54985272.html http://www.ciozone.com/index.php/Management/Guardian-CIO- On-Aligning-IT-Business.html http://www.ciozone.com/index.php/Management/Guardian-CIO- On-Aligning-IT-Business.html http://www.insurancetech.com/architecture- infrastructure/231600651 http://blogs.computerworld.com/17935/cloud_to_lower_bar_inte nsify_competition http://blogs.computerworld.com/17935/cloud_to_lower_bar_inte nsify_competition http://features.techworld.com/sme/3337405/best-practices-for- scaling-up-saas-in-the-cloud/ http://features.techworld.com/sme/3337405/best-practices-for- scaling-up-saas-in-the-cloud/ http://www.prnewswire.com/news-releases/guardian-life- recognized-by-cio-magazine-for-third-consecutive-year-in-cio- 100-awards-54985272.html http://www.prnewswire.com/news-releases/guardian-life- recognized-by-cio-magazine-for-third-consecutive-year-in-cio- 100-awards-54985272.htmlCASE STUDY 7Further ConsolidationDiscussion PointsSources MAT 1339B (Winter 2018) Assignment 1 Professor: Rachid Bentoumi Deadline: Friday, February 09, 2018 Before 15:00
  • 17. (585 King Eduard math-stat department drop boxes) Student Name Student Number By signing below, you declare that this work was your own and that you have not copied from any other individual or other source. Signature Late assignments will NOT be accepted; nor will unstapled assignments. Professors in the math department will not lend you a stapler. You should complete ALL the questions in the assignments. It is possible, however, that not all the questions will be marked. In that case, the same questions will be marked in all assignments. You will not be informed be- forehand which questions will be marked. 1 Question 1. Solve the following inequalities: (a) |2x− 5| > 2 (b) |3 − 2x| ≤ 7 2
  • 18. Question 2. Find the domain of the functions defined below (a) f(x) = 1 x2 − 8x (b) g(x) = x |x| + 4 (c) h(x) = √ x− 2 x− 7 3 Question 3. Calculate the following limits (show all steps of your calcula- tion). (a) lim x→0 √ 19 − 3x−x x
  • 19. (b) lim x→−∞ √ x2 − 1 2x (c) lim x→∞ cos (x) x4 + 1 4 Question 4. Let f be a function defined as f(x) = x2 − 16 2x− 8 , if x < 4 −4, si x = 4 x− 4 √ x− 2 if x > 4
  • 20. (a) Determine lim x→4 f(x) ? (b) Is the function f continuous on R? Justify your answer. 5 Question 5. Determine, if it is possible, the value of k that makes the functions below continuous on R. (a) h(x) = x + 2 if x < 1 k if x = 1 x2 + 3x− 1 if x > 1 (b) f(t) = t2 − 25 t− 5 if t < 5 kt if t ≥ 5
  • 21. 6 Question 6. Consider the following function f(x) = √ 25 − 2x (a) Use the definition of the derivative to find f ′ (x) (b) Determine the tangent line to f at x = 0. 7 Question 7. Use the derivative rules to calculate the derivatives of the following functions: (a) y = √ 2x2 + 5x + 7 (b) y = (x2 + 3)4(x3 − 5)3 8
  • 22. Question 8. Consider a closed rectangular box with the following dimensions: length= x cm, width= 2x cm and height= x + 1 cm. (a) According to x, determine i. the function A(x) giving the total area of the rectangular box. ii. the function V (x) giving the volume of the rectangular box. (b) Calculate average rates of change of the area and the volume when x passes from 3 cm to 6 cm. 9 Question 9. A manufacturer estimates that his weekly revenue R(x) can be expressed by R(x) = 1008x− 12x2 − 8x3 where x is the number of units sold, R(x) is expressed in dollars. Under certain constraints, one cannot produce more than 10 units per week. (a) Using the variation table (or test points), determine on which interval(s) the revenue is increasing. (b) Determine the quantity of units that gives a maximum revenue and evaluate this maximum revenue.
  • 23. 10 Question 10. Determine the concavity intervals and, if it is possible, the inflection points of the function h(x) = (x− 1)2(x + 1)2. 11