Submit Search
Upload
Introductory maths analysis chapter 11 official
•
Download as PPT, PDF
•
1 like
•
2,002 views
Evert Sandye Taasiringan
Follow
Matematika Bisnis
Read less
Read more
Education
Technology
Business
Report
Share
Report
Share
1 of 29
Download now
Recommended
Introductory maths analysis chapter 04 official
Introductory maths analysis chapter 04 official
Evert Sandye Taasiringan
Introductory maths analysis chapter 10 official
Introductory maths analysis chapter 10 official
Evert Sandye Taasiringan
Introductory maths analysis chapter 00 official
Introductory maths analysis chapter 00 official
Evert Sandye Taasiringan
Introductory maths analysis chapter 14 official
Introductory maths analysis chapter 14 official
Evert Sandye Taasiringan
Introductory maths analysis chapter 01 official
Introductory maths analysis chapter 01 official
Evert Sandye Taasiringan
Introductory maths analysis chapter 02 official
Introductory maths analysis chapter 02 official
Evert Sandye Taasiringan
Introductory maths analysis chapter 06 official
Introductory maths analysis chapter 06 official
Evert Sandye Taasiringan
Introductory maths analysis chapter 03 official
Introductory maths analysis chapter 03 official
Evert Sandye Taasiringan
Recommended
Introductory maths analysis chapter 04 official
Introductory maths analysis chapter 04 official
Evert Sandye Taasiringan
Introductory maths analysis chapter 10 official
Introductory maths analysis chapter 10 official
Evert Sandye Taasiringan
Introductory maths analysis chapter 00 official
Introductory maths analysis chapter 00 official
Evert Sandye Taasiringan
Introductory maths analysis chapter 14 official
Introductory maths analysis chapter 14 official
Evert Sandye Taasiringan
Introductory maths analysis chapter 01 official
Introductory maths analysis chapter 01 official
Evert Sandye Taasiringan
Introductory maths analysis chapter 02 official
Introductory maths analysis chapter 02 official
Evert Sandye Taasiringan
Introductory maths analysis chapter 06 official
Introductory maths analysis chapter 06 official
Evert Sandye Taasiringan
Introductory maths analysis chapter 03 official
Introductory maths analysis chapter 03 official
Evert Sandye Taasiringan
Introductory maths analysis chapter 05 official
Introductory maths analysis chapter 05 official
Evert Sandye Taasiringan
Introductory maths analysis chapter 13 official
Introductory maths analysis chapter 13 official
Evert Sandye Taasiringan
Chapter 4 - Exponential and Logarithmic Functions
Chapter 4 - Exponential and Logarithmic Functions
Muhammad Bilal Khairuddin
Introductory maths analysis chapter 12 official
Introductory maths analysis chapter 12 official
Evert Sandye Taasiringan
Chapter 10 - Limit and Continuity
Chapter 10 - Limit and Continuity
Muhammad Bilal Khairuddin
Chapter 1 - Applications and More Algebra
Chapter 1 - Applications and More Algebra
Muhammad Bilal Khairuddin
Chapter 2 - Functions and Graphs
Chapter 2 - Functions and Graphs
Muhammad Bilal Khairuddin
Chapter 14 - Integration
Chapter 14 - Integration
Muhammad Bilal Khairuddin
Introductory maths analysis chapter 17 official
Introductory maths analysis chapter 17 official
Evert Sandye Taasiringan
Chapter 6 - Matrix Algebra
Chapter 6 - Matrix Algebra
Muhammad Bilal Khairuddin
Chapter 3 - Lines , Parabolas and Systems
Chapter 3 - Lines , Parabolas and Systems
Muhammad Bilal Khairuddin
Chapter 13 - Curve Sketching
Chapter 13 - Curve Sketching
Muhammad Bilal Khairuddin
Chapter 12 - Additional Differentiation Topics
Chapter 12 - Additional Differentiation Topics
Muhammad Bilal Khairuddin
Chapter 11 - Differentiation
Chapter 11 - Differentiation
Muhammad Bilal Khairuddin
15.2 solving systems of equations by substitution
15.2 solving systems of equations by substitution
GlenSchlee
Chapter 17 - Multivariable Calculus
Chapter 17 - Multivariable Calculus
Muhammad Bilal Khairuddin
Fundamental Theorem of Calculus
Fundamental Theorem of Calculus
gizemk
Ani agustina (a1 c011007) polynomial
Ani agustina (a1 c011007) polynomial
Ani_Agustina
Limits and continuity
Limits and continuity
Digvijaysinh Gohil
Chapter 3-2
Chapter 3-2
Dasrat goswami
31350052 introductory-mathematical-analysis-textbook-solution-manual
31350052 introductory-mathematical-analysis-textbook-solution-manual
Mahrukh Khalid
03 i-o
03 i-o
Evert Sandye Taasiringan
More Related Content
What's hot
Introductory maths analysis chapter 05 official
Introductory maths analysis chapter 05 official
Evert Sandye Taasiringan
Introductory maths analysis chapter 13 official
Introductory maths analysis chapter 13 official
Evert Sandye Taasiringan
Chapter 4 - Exponential and Logarithmic Functions
Chapter 4 - Exponential and Logarithmic Functions
Muhammad Bilal Khairuddin
Introductory maths analysis chapter 12 official
Introductory maths analysis chapter 12 official
Evert Sandye Taasiringan
Chapter 10 - Limit and Continuity
Chapter 10 - Limit and Continuity
Muhammad Bilal Khairuddin
Chapter 1 - Applications and More Algebra
Chapter 1 - Applications and More Algebra
Muhammad Bilal Khairuddin
Chapter 2 - Functions and Graphs
Chapter 2 - Functions and Graphs
Muhammad Bilal Khairuddin
Chapter 14 - Integration
Chapter 14 - Integration
Muhammad Bilal Khairuddin
Introductory maths analysis chapter 17 official
Introductory maths analysis chapter 17 official
Evert Sandye Taasiringan
Chapter 6 - Matrix Algebra
Chapter 6 - Matrix Algebra
Muhammad Bilal Khairuddin
Chapter 3 - Lines , Parabolas and Systems
Chapter 3 - Lines , Parabolas and Systems
Muhammad Bilal Khairuddin
Chapter 13 - Curve Sketching
Chapter 13 - Curve Sketching
Muhammad Bilal Khairuddin
Chapter 12 - Additional Differentiation Topics
Chapter 12 - Additional Differentiation Topics
Muhammad Bilal Khairuddin
Chapter 11 - Differentiation
Chapter 11 - Differentiation
Muhammad Bilal Khairuddin
15.2 solving systems of equations by substitution
15.2 solving systems of equations by substitution
GlenSchlee
Chapter 17 - Multivariable Calculus
Chapter 17 - Multivariable Calculus
Muhammad Bilal Khairuddin
Fundamental Theorem of Calculus
Fundamental Theorem of Calculus
gizemk
Ani agustina (a1 c011007) polynomial
Ani agustina (a1 c011007) polynomial
Ani_Agustina
Limits and continuity
Limits and continuity
Digvijaysinh Gohil
Chapter 3-2
Chapter 3-2
Dasrat goswami
What's hot
(20)
Introductory maths analysis chapter 05 official
Introductory maths analysis chapter 05 official
Introductory maths analysis chapter 13 official
Introductory maths analysis chapter 13 official
Chapter 4 - Exponential and Logarithmic Functions
Chapter 4 - Exponential and Logarithmic Functions
Introductory maths analysis chapter 12 official
Introductory maths analysis chapter 12 official
Chapter 10 - Limit and Continuity
Chapter 10 - Limit and Continuity
Chapter 1 - Applications and More Algebra
Chapter 1 - Applications and More Algebra
Chapter 2 - Functions and Graphs
Chapter 2 - Functions and Graphs
Chapter 14 - Integration
Chapter 14 - Integration
Introductory maths analysis chapter 17 official
Introductory maths analysis chapter 17 official
Chapter 6 - Matrix Algebra
Chapter 6 - Matrix Algebra
Chapter 3 - Lines , Parabolas and Systems
Chapter 3 - Lines , Parabolas and Systems
Chapter 13 - Curve Sketching
Chapter 13 - Curve Sketching
Chapter 12 - Additional Differentiation Topics
Chapter 12 - Additional Differentiation Topics
Chapter 11 - Differentiation
Chapter 11 - Differentiation
15.2 solving systems of equations by substitution
15.2 solving systems of equations by substitution
Chapter 17 - Multivariable Calculus
Chapter 17 - Multivariable Calculus
Fundamental Theorem of Calculus
Fundamental Theorem of Calculus
Ani agustina (a1 c011007) polynomial
Ani agustina (a1 c011007) polynomial
Limits and continuity
Limits and continuity
Chapter 3-2
Chapter 3-2
Viewers also liked
31350052 introductory-mathematical-analysis-textbook-solution-manual
31350052 introductory-mathematical-analysis-textbook-solution-manual
Mahrukh Khalid
03 i-o
03 i-o
Evert Sandye Taasiringan
Introductory maths analysis chapter 07 official
Introductory maths analysis chapter 07 official
Evert Sandye Taasiringan
Introductory maths analysis chapter 16 official
Introductory maths analysis chapter 16 official
Evert Sandye Taasiringan
Introductory maths analysis chapter 15 official
Introductory maths analysis chapter 15 official
Evert Sandye Taasiringan
Introduction to mathematical modelling
Introduction to mathematical modelling
Arup Kumar Paria
Introductory maths analysis chapter 09 official
Introductory maths analysis chapter 09 official
Evert Sandye Taasiringan
Viewers also liked
(7)
31350052 introductory-mathematical-analysis-textbook-solution-manual
31350052 introductory-mathematical-analysis-textbook-solution-manual
03 i-o
03 i-o
Introductory maths analysis chapter 07 official
Introductory maths analysis chapter 07 official
Introductory maths analysis chapter 16 official
Introductory maths analysis chapter 16 official
Introductory maths analysis chapter 15 official
Introductory maths analysis chapter 15 official
Introduction to mathematical modelling
Introduction to mathematical modelling
Introductory maths analysis chapter 09 official
Introductory maths analysis chapter 09 official
Similar to Introductory maths analysis chapter 11 official
Chapter12 additionaldifferentiationtopics-151003154510-lva1-app6891
Chapter12 additionaldifferentiationtopics-151003154510-lva1-app6891
Cleophas Rwemera
Chapter10 limitandcontinuity-151003153921-lva1-app6891
Chapter10 limitandcontinuity-151003153921-lva1-app6891
Cleophas Rwemera
Chapter17 multivariablecalculus-151007044001-lva1-app6891
Chapter17 multivariablecalculus-151007044001-lva1-app6891
Cleophas Rwemera
Chapter14 integration-151007043436-lva1-app6892
Chapter14 integration-151007043436-lva1-app6892
Cleophas Rwemera
Lesson 10 derivative of exponential functions
Lesson 10 derivative of exponential functions
Rnold Wilson
Chapter13 curvesketching-151007042831-lva1-app6891
Chapter13 curvesketching-151007042831-lva1-app6891
Cleophas Rwemera
Evaluating definite integrals
Evaluating definite integrals
منتدى الرياضيات المتقدمة
3 handouts section3-11
3 handouts section3-11
International advisers
Indefinite Integral
Indefinite Integral
JelaiAujero
Chapter0 reviewofalgebra-151003150137-lva1-app6891
Chapter0 reviewofalgebra-151003150137-lva1-app6891
Cleophas Rwemera
1519 differentiation-integration-02
1519 differentiation-integration-02
Dr Fereidoun Dejahang
Introduction to comp.physics ch 3.pdf
Introduction to comp.physics ch 3.pdf
JifarRaya
integration-131127090901-phpapp01.pptx
integration-131127090901-phpapp01.pptx
AlphaKoiSylvester
Derivatice Introduction.pptx
Derivatice Introduction.pptx
MuhammadToqeerAfzal
Derivatice Introduction.pptx
Derivatice Introduction.pptx
MuhammadToqeerAfzal
1520 differentiation-l1
1520 differentiation-l1
Dr Fereidoun Dejahang
Differential Calculus
Differential Calculus
OlooPundit
Similar to Introductory maths analysis chapter 11 official
(17)
Chapter12 additionaldifferentiationtopics-151003154510-lva1-app6891
Chapter12 additionaldifferentiationtopics-151003154510-lva1-app6891
Chapter10 limitandcontinuity-151003153921-lva1-app6891
Chapter10 limitandcontinuity-151003153921-lva1-app6891
Chapter17 multivariablecalculus-151007044001-lva1-app6891
Chapter17 multivariablecalculus-151007044001-lva1-app6891
Chapter14 integration-151007043436-lva1-app6892
Chapter14 integration-151007043436-lva1-app6892
Lesson 10 derivative of exponential functions
Lesson 10 derivative of exponential functions
Chapter13 curvesketching-151007042831-lva1-app6891
Chapter13 curvesketching-151007042831-lva1-app6891
Evaluating definite integrals
Evaluating definite integrals
3 handouts section3-11
3 handouts section3-11
Indefinite Integral
Indefinite Integral
Chapter0 reviewofalgebra-151003150137-lva1-app6891
Chapter0 reviewofalgebra-151003150137-lva1-app6891
1519 differentiation-integration-02
1519 differentiation-integration-02
Introduction to comp.physics ch 3.pdf
Introduction to comp.physics ch 3.pdf
integration-131127090901-phpapp01.pptx
integration-131127090901-phpapp01.pptx
Derivatice Introduction.pptx
Derivatice Introduction.pptx
Derivatice Introduction.pptx
Derivatice Introduction.pptx
1520 differentiation-l1
1520 differentiation-l1
Differential Calculus
Differential Calculus
More from Evert Sandye Taasiringan
07 function 2
07 function 2
Evert Sandye Taasiringan
04 if-ifelse-switch-break
04 if-ifelse-switch-break
Evert Sandye Taasiringan
05 for-dowhile-while
05 for-dowhile-while
Evert Sandye Taasiringan
06 nested
06 nested
Evert Sandye Taasiringan
02 01-elemen
02 01-elemen
Evert Sandye Taasiringan
02 02-operasi
02 02-operasi
Evert Sandye Taasiringan
01 pseudocode
01 pseudocode
Evert Sandye Taasiringan
01 algoritma
01 algoritma
Evert Sandye Taasiringan
01 02-pseudocode
01 02-pseudocode
Evert Sandye Taasiringan
01 01-algoritma
01 01-algoritma
Evert Sandye Taasiringan
Introductory maths analysis chapter 08 official
Introductory maths analysis chapter 08 official
Evert Sandye Taasiringan
Pertemuan ke 1
Pertemuan ke 1
Evert Sandye Taasiringan
Manajemen operasional ringkasan uas
Manajemen operasional ringkasan uas
Evert Sandye Taasiringan
More from Evert Sandye Taasiringan
(13)
07 function 2
07 function 2
04 if-ifelse-switch-break
04 if-ifelse-switch-break
05 for-dowhile-while
05 for-dowhile-while
06 nested
06 nested
02 01-elemen
02 01-elemen
02 02-operasi
02 02-operasi
01 pseudocode
01 pseudocode
01 algoritma
01 algoritma
01 02-pseudocode
01 02-pseudocode
01 01-algoritma
01 01-algoritma
Introductory maths analysis chapter 08 official
Introductory maths analysis chapter 08 official
Pertemuan ke 1
Pertemuan ke 1
Manajemen operasional ringkasan uas
Manajemen operasional ringkasan uas
Recently uploaded
Activity 01 - Artificial Culture (1).pdf
Activity 01 - Artificial Culture (1).pdf
ciinovamais
Advanced Views - Calendar View in Odoo 17
Advanced Views - Calendar View in Odoo 17
Celine George
1029-Danh muc Sach Giao Khoa khoi 6.pdf
1029-Danh muc Sach Giao Khoa khoi 6.pdf
QucHHunhnh
Interactive Powerpoint_How to Master effective communication
Interactive Powerpoint_How to Master effective communication
nomboosow
Introduction to Nonprofit Accounting: The Basics
Introduction to Nonprofit Accounting: The Basics
TechSoup
INDIA QUIZ 2024 RLAC DELHI UNIVERSITY.pptx
INDIA QUIZ 2024 RLAC DELHI UNIVERSITY.pptx
RAM LAL ANAND COLLEGE, DELHI UNIVERSITY.
BAG TECHNIQUE Bag technique-a tool making use of public health bag through wh...
BAG TECHNIQUE Bag technique-a tool making use of public health bag through wh...
Sapna Thakur
Software Engineering Methodologies (overview)
Software Engineering Methodologies (overview)
eniolaolutunde
Ecosystem Interactions Class Discussion Presentation in Blue Green Lined Styl...
Ecosystem Interactions Class Discussion Presentation in Blue Green Lined Styl...
fonyou31
Mastering the Unannounced Regulatory Inspection
Mastering the Unannounced Regulatory Inspection
SafetyChain Software
Nutritional Needs Presentation - HLTH 104
Nutritional Needs Presentation - HLTH 104
misteraugie
BASLIQ CURRENT LOOKBOOK LOOKBOOK(1) (1).pdf
BASLIQ CURRENT LOOKBOOK LOOKBOOK(1) (1).pdf
SoniaTolstoy
Student login on Anyboli platform.helpin
Student login on Anyboli platform.helpin
RaunakKeshri1
Organic Name Reactions for the students and aspirants of Chemistry12th.pptx
Organic Name Reactions for the students and aspirants of Chemistry12th.pptx
VS Mahajan Coaching Centre
Kisan Call Centre - To harness potential of ICT in Agriculture by answer farm...
Kisan Call Centre - To harness potential of ICT in Agriculture by answer farm...
Krashi Coaching
A Critique of the Proposed National Education Policy Reform
A Critique of the Proposed National Education Policy Reform
Chameera Dedduwage
Mattingly "AI & Prompt Design: The Basics of Prompt Design"
Mattingly "AI & Prompt Design: The Basics of Prompt Design"
National Information Standards Organization (NISO)
Presentation by Andreas Schleicher Tackling the School Absenteeism Crisis 30 ...
Presentation by Andreas Schleicher Tackling the School Absenteeism Crisis 30 ...
EduSkills OECD
Código Creativo y Arte de Software | Unidad 1
Código Creativo y Arte de Software | Unidad 1
Maestría en Comunicación Digital Interactiva - UNR
Sports & Fitness Value Added Course FY..
Sports & Fitness Value Added Course FY..
Disha Kariya
Recently uploaded
(20)
Activity 01 - Artificial Culture (1).pdf
Activity 01 - Artificial Culture (1).pdf
Advanced Views - Calendar View in Odoo 17
Advanced Views - Calendar View in Odoo 17
1029-Danh muc Sach Giao Khoa khoi 6.pdf
1029-Danh muc Sach Giao Khoa khoi 6.pdf
Interactive Powerpoint_How to Master effective communication
Interactive Powerpoint_How to Master effective communication
Introduction to Nonprofit Accounting: The Basics
Introduction to Nonprofit Accounting: The Basics
INDIA QUIZ 2024 RLAC DELHI UNIVERSITY.pptx
INDIA QUIZ 2024 RLAC DELHI UNIVERSITY.pptx
BAG TECHNIQUE Bag technique-a tool making use of public health bag through wh...
BAG TECHNIQUE Bag technique-a tool making use of public health bag through wh...
Software Engineering Methodologies (overview)
Software Engineering Methodologies (overview)
Ecosystem Interactions Class Discussion Presentation in Blue Green Lined Styl...
Ecosystem Interactions Class Discussion Presentation in Blue Green Lined Styl...
Mastering the Unannounced Regulatory Inspection
Mastering the Unannounced Regulatory Inspection
Nutritional Needs Presentation - HLTH 104
Nutritional Needs Presentation - HLTH 104
BASLIQ CURRENT LOOKBOOK LOOKBOOK(1) (1).pdf
BASLIQ CURRENT LOOKBOOK LOOKBOOK(1) (1).pdf
Student login on Anyboli platform.helpin
Student login on Anyboli platform.helpin
Organic Name Reactions for the students and aspirants of Chemistry12th.pptx
Organic Name Reactions for the students and aspirants of Chemistry12th.pptx
Kisan Call Centre - To harness potential of ICT in Agriculture by answer farm...
Kisan Call Centre - To harness potential of ICT in Agriculture by answer farm...
A Critique of the Proposed National Education Policy Reform
A Critique of the Proposed National Education Policy Reform
Mattingly "AI & Prompt Design: The Basics of Prompt Design"
Mattingly "AI & Prompt Design: The Basics of Prompt Design"
Presentation by Andreas Schleicher Tackling the School Absenteeism Crisis 30 ...
Presentation by Andreas Schleicher Tackling the School Absenteeism Crisis 30 ...
Código Creativo y Arte de Software | Unidad 1
Código Creativo y Arte de Software | Unidad 1
Sports & Fitness Value Added Course FY..
Sports & Fitness Value Added Course FY..
Introductory maths analysis chapter 11 official
1.
INTRODUCTORY MATHEMATICALINTRODUCTORY MATHEMATICAL ANALYSISANALYSISFor
Business, Economics, and the Life and Social Sciences ©2007 Pearson Education Asia Chapter 11Chapter 11 DifferentiationDifferentiation
2.
©2007 Pearson Education
Asia INTRODUCTORY MATHEMATICAL ANALYSIS 0. Review of Algebra 1. Applications and More Algebra 2. Functions and Graphs 3. Lines, Parabolas, and Systems 4. Exponential and Logarithmic Functions 5. Mathematics of Finance 6. Matrix Algebra 7. Linear Programming 8. Introduction to Probability and Statistics
3.
©2007 Pearson Education
Asia 9. Additional Topics in Probability 10. Limits and Continuity 11. Differentiation 12. Additional Differentiation Topics 13. Curve Sketching 14. Integration 15. Methods and Applications of Integration 16. Continuous Random Variables 17. Multivariable Calculus INTRODUCTORY MATHEMATICAL ANALYSIS
4.
©2007 Pearson Education
Asia • To compute derivatives by using the limit definition. • To develop basic differentiation rules. • To interpret the derivative as an instantaneous rate of change. • To apply the product and quotient rules. • To apply the chain rule. Chapter 11: Differentiation Chapter ObjectivesChapter Objectives
5.
©2007 Pearson Education
Asia The Derivative Rules for Differentiation The Derivative as a Rate of Change The Product Rule and the Quotient Rule The Chain Rule and the Power Rule 11.1) 11.2) 11.3) Chapter 11: Differentiation Chapter OutlineChapter Outline 11.4) 11.5)
6.
©2007 Pearson Education
Asia Chapter 11: Differentiation 11.1 The Derivative11.1 The Derivative • Tangent line at a point: • The slope of a curve at P is the slope of the tangent line at P. • The slope of the tangent line at (a, f(a)) is ( ) ( ) ( ) ( ) h afhaf az afzf m haz −+ = − − = →→ 0 tan limlim
7.
©2007 Pearson Education
Asia Chapter 11: Differentiation 11.1 The Derivative Example 1 – Finding the Slope of a Tangent Line Find the slope of the tangent line to the curve y = f(x) = x2 at the point (1, 1). Solution: Slope = ( ) ( ) ( ) ( ) 2 11 lim 11 lim 22 00 = −+ = −+ →→ h h h fhf hh • The derivative of a function f is the function denoted f’ and defined by ( ) ( ) ( ) ( ) ( ) h xfhxf xz xfzf xf hxz −+ = − − = →→ 0 limlim'
8.
©2007 Pearson Education
Asia Chapter 11: Differentiation 11.1 The Derivative Example 3 – Finding an Equation of a Tangent Line If f (x) = 2x2 + 2x + 3, find an equation of the tangent line to the graph of f at (1, 7). Solution: Slope Equation ( ) ( ) ( ) ( ) ( )( ) ( ) 24 322322 limlim' 22 00 += ++−++++ = −+ = →→ x h xxhxhx h xfhxf xf hh ( ) 16 167 += −=− xy xy ( ) ( ) 62141' =+=f
9.
©2007 Pearson Education
Asia Chapter 11: Differentiation 11.1 The Derivative Example 5 – A Function with a Vertical Tangent Line Example 7 – Continuity and Differentiability Find . Solution: ( )x dx d ( ) xh xhx x dx d h 2 1 lim 0 = −+ = → a. For f(x) = x2 , it must be continuous for all x. b. For f(p) =(1/2)p, it is not continuous at p = 0, thus the derivative does not exist at p = 0.
10.
©2007 Pearson Education
Asia Chapter 11: Differentiation 11.2 Rules for Differentiation11.2 Rules for Differentiation • Rules for Differentiation: RULE 1 Derivative of a Constant: RULE 2 Derivative of xn : RULE 3 Constant Factor Rule: RULE 4 Sum or Difference Rule ( ) 0=c dx d ( ) 1− = nn nxx dx d ( )( ) ( )xcfxcf dx d '= ( ) ( )( ) ( ) ( )xgxfxgxf dx d '' ±=±=
11.
©2007 Pearson Education
Asia Chapter 11: Differentiation 11.2 Rules for Differentiation Example 1 – Derivatives of Constant Functions a. b. If , then . c. If , then . ( ) 03 = dx d ( ) 5=xg ( ) ( ) 4.807 623,938,1=ts ( ) 0' =xg 0=dt ds
12.
©2007 Pearson Education
Asia Chapter 11: Differentiation 11.2 Rules for Differentiation Example 3 – Rewriting Functions in the Form xn Differentiate the following functions: Solution: a. b. xy = ( ) x x dx dy 2 1 2 1 12/1 == − ( ) xx xh 1 = ( ) ( ) ( ) 2/512/32/3 2 3 2 3 ' −−−− −=−== xxx dx d xh
13.
©2007 Pearson Education
Asia Chapter 11: Differentiation 11.2 Rules for Differentiation Example 5 – Differentiating Sums and Differences of Functions Differentiate the following functions: ( ) xxxF += 5 3a. ( ) ( ) ( ) ( ) ( ) x xxx x dx d x dx d xF 2 1 15 2 1 53 3' 42/14 2/15 +=+= += − ( ) 3/1 4 5 4 b. z z zf −= ( ) ( ) 3/433/43 3/1 4 3 5 3 1 54 4 1 5 4 ' −− += −−= − = zzzz zdz dz dz d zf
14.
©2007 Pearson Education
Asia Chapter 11: Differentiation 11.2 Rules for Differentiation Example 5 – Differentiating Sums and Differences of Functions 8726c. 23 −+−= xxxy 7418 )8()(7)(2)(6 2 23 +−= −+−= xx dx d x dx d x dx d x dx d dx dy
15.
©2007 Pearson Education
Asia Chapter 11: Differentiation 11.2 Rules for Differentiation Example 7 – Finding an Equation of a Tangent Line Find an equation of the tangent line to the curve when x = 1. Solution: The slope equation is When x = 1, The equation is x x y 23 2 − = 2 1 2 23 23 23 − − += −=−= x dx dy xx xx x y ( ) 5123 2 1 =+= − =xdx dy ( ) 45 151 −= −=− xy xy
16.
©2007 Pearson Education
Asia Chapter 11: Differentiation 11.3 The Derivative as a Rate of Change11.3 The Derivative as a Rate of Change Example 1 – Finding Average Velocity and Velocity • Average velocity is given by • Velocity at time t is given by ( ) ( ) t tfttf t s vave ∆ −∆+ = ∆ ∆ = ( ) ( ) t tfttf v t ∆ −∆+ = →∆ 0 lim Suppose the position function of an object moving along a number line is given by s = f(t) = 3t2 + 5, where t is in seconds and s is in meters. a.Find the average velocity over the interval [10, 10.1]. b. Find the velocity when t = 10.
17.
©2007 Pearson Education
Asia Chapter 11: Differentiation 11.3 The Derivative as a Rate of Change Example 1 – Finding Average Velocity and Velocity Solution: a. When t = 10, b. Velocity at time t is given by When t = 10, the velocity is ( ) ( ) ( ) ( ) ( ) ( ) m/s3.60 1.0 30503.311 1.0 101.10 1.0 101.010 = = = − = −+ = ∆ −∆+ = ∆ ∆ = ffff t tfttf t s vave t dt ds v 6== ( ) m/s60106 10 == =tdt ds
18.
©2007 Pearson Education
Asia Chapter 11: Differentiation 11.3 The Derivative as a Rate of Change Example 3 – Finding a Rate of Change • If y = f(x), then ( ) ( ) ∆+ = ∆ −∆+ = ∆ ∆ xxx x xfxxf x y tofrominterval theoverxtorespectwith yofchangeofrateaverage = ∆ ∆ = →∆ xrespect toy with ofchangeofrateousinstantane lim 0 x y dx dy x Find the rate of change of y = x4 with respect to x, and evaluate it when x = 2 and when x = −1. Solution: The rate of change is . 3 4x dx dy =
19.
©2007 Pearson Education
Asia Chapter 11: Differentiation 11.3 The Derivative as a Rate of Change Example 5 – Rate of Change of Volume A spherical balloon is being filled with air. Find the rate of change of the volume of air in the balloon with respect to its radius. Evaluate this rate of change when the radius is 2 ft. Solution: Rate of change of V with respect to r is When r = 2 ft, ( ) 22 43 3 4 rr dr dV ππ == ( ) ft ft 1624 3 2 2 ππ == =rdr dV
20.
©2007 Pearson Education
Asia Chapter 11: Differentiation 11.3 The Derivative as a Rate of Change Applications of Rate of Change to Economics • Total-cost function is c = f(q). • Marginal cost is defined as . • Total-revenue function is r = f(q). • Marginal revenue is defined as . dq dc dq dr Relative and Percentage Rates of Change • The relative rate of change of f(x) is . • The percentage rate of change of f (x) is ( ) ( )xf xf ' ( ) ( ) ( )%100 ' xf xf
21.
©2007 Pearson Education
Asia Chapter 11: Differentiation 11.3 The Derivative as a Rate of Change Example 7 – Marginal Cost If a manufacturer’s average-cost equation is find the marginal-cost function. What is the marginal cost when 50 units are produced? Solution: The cost is Marginal cost when q = 50, q qqc 5000 502.00001.0 2 ++−= 5000502.00001.0 5000 502.00001.0 23 2 ++−= ++−== qqq q qqqcqc 504.00003.0 2 +−= qq dq dc ( ) ( ) 75.355004.0500003.0 2 50 =+−= =q dq dc
22.
©2007 Pearson Education
Asia Chapter 11: Differentiation 11.3 The Derivative as a Rate of Change Example 9 – Relative and Percentage Rates of Change 11.4 The Product Rule and the Quotient Rule11.4 The Product Rule and the Quotient Rule Determine the relative and percentage rates of change of when x = 5. Solution: ( ) 2553 2 +−== xxxfy ( ) 56' −= xxf ( ) ( ) ( ) ( ) %3.33333.0 75 25 5 5' change% 255565' =≈== =−= f f f The Product Rule ( ) ( )( ) ( ) ( ) ( ) ( )xgxfxgxfxgxf dx d '' +=
23.
©2007 Pearson Education
Asia Chapter 11: Differentiation 11.4 The Product and Quotient Rule Example 1 – Applying the Product Rule Example 3 – Differentiating a Product of Three Factors Find F’(x). ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )( ) 153412435432 543543' 543 22 22 2 ++=++++= ++++ += ++= xxxxxx x dx d xxxxx dx d xF xxxxF Find y’. ( )( ) ( ) ( )( )( ) ( ) 26183 432432' )4)(3)(2( 2 ++= +++++ ++= +++= xx x dx d xxxxx dx d y xxxy
24.
©2007 Pearson Education
Asia Chapter 11: Differentiation 11.4 The Product and Quotient Rule Example 5 – Applying the Quotient Rule If , find F’(x). Solution: The Quotient Rule ( ) ( ) ( ) ( ) ( ) ( ) ( )( )2 '' xg xgxfxfxg xg xf dx d − = ( ) 12 34 2 − + = x x xF ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( )22 2 2 22 12 32122 12 234812 12 12343412 ' − −+ = − +−− = − −+−+− = x xx x xxx x x dx d xx dx d x xF
25.
©2007 Pearson Education
Asia Chapter 11: Differentiation 11.4 The Product and Quotient Rule Example 7 – Differentiating Quotients without Using the Quotient Rule Differentiate the following functions. ( ) ( ) ( ) 5 6 3 5 2 ' 5 2 a. 2 2 3 x xxf x xf == = ( ) ( ) ( ) ( ) 4 4 3 3 7 12 3 7 4 ' 7 4 7 4 b. x xxf x x xf −=−= == − − ( ) ( ) ( ) ( ) 4 5 5 4 1 ' 35 4 1 4 35 c. 2 == −= − = xf x x xx xf
26.
©2007 Pearson Education
Asia Chapter 11: Differentiation 11.4 The Product and Quotient Rule Example 9 – Finding Marginal Propensities to Consume and to Save If the consumption function is given by determine the marginal propensity to consume and the marginal propensity to save when I = 100. Solution: Consumption Function dI dC consumetopropensityMarginal = consumetopropensityMarginal-1savetopropensityMarginal = ( ) 10 325 3 + + = I I C ( ) ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( ) + +−++ = + ++−++ = 2 32/1 2 32/3 10 1323310 5 10 10323210 5 I III I I dI d II dI d I dI dC 536.0 12100 1297 5 100 ≈ = =IdI dC
27.
©2007 Pearson Education
Asia Chapter 11: Differentiation 11.5 The Chain Rule and the Power Rule11.5 The Chain Rule and the Power Rule Example 1 – Using the Chain Rule a. If y = 2u2 − 3u − 2 and u = x2 + 4, find dy/dx. Solution: Chain Rule: Power Rule: dx du du dy dx dy ⋅= ( ) dx du nuu dx d nn 1− = ( ) ( ) ( )( )xu x dx d uu du d dx du du dy dx dy 234 4232 22 −= +⋅−−=⋅= ( )[ ]( ) ( )[ ]( ) xxxxxx dx dy 26821342344 322 +=+=−+=
28.
©2007 Pearson Education
Asia Chapter 11: Differentiation 11.5 The Chain Rule and the Power Rule Example 1 – Using the Chain Rule Example 3 – Using the Power Rule b. If y = √w and w = 7 − t3 , find dy/dt. Solution: ( ) ( ) 3 22 32/1 72 3 2 3 7 t t w t t dt d w dw d dt dy − −=−= −⋅= If y = (x3 − 1)7 , find y’. Solution: ( ) ( ) ( ) ( ) ( )622263 3173 121317 117' −=−= −−= − xxxx x dx d xy
29.
©2007 Pearson Education
Asia Chapter 11: Differentiation 11.5 The Chain Rule and the Power Rule Example 5 – Using the Power Rule Example 7 – Differentiating a Product of Powers If , find dy/dx. Solution: 2 1 2 − = x y ( )( ) ( ) ( )22 2112 2 2 221 − −=−−−= − x x x dx d x dx dy If , find y’. Solution: ( ) ( )452 534 +−= xxy ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2425215342 4531053412 453534' 2342 424352 524452 −++−= −+++−= −+++−= xxxx xxxxx x dx d xx dx d xy
Download now