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4.4 Analytical Connections F, F', F'' (part 1)ap Calculus

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Chapter 4

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Ċap chapter 4 answers 4.1 (day 1).pdf
190kv. 1 Jan 31, 2013, 8:26 PMDwayne Clouse
Ċap chapter 4 answers 4.1 (day 2).pdf
373kv. 1 Jan 31, 2013, 8:26 PMDwayne Clouse
Ċap chapter 4 answers 4.2.pdf
191kv. 1 Feb 27, 2014, 9:33 AMDwayne Clouse
Ċap chapter 4 answers 4.3.pdf
69kv. 1 Jan 31, 2013, 8:26 PMDwayne Clouse
Ċap chapter 4 answers 4.4 (day 1).pdf
44kv. 1 Jan 31, 2013, 8:26 PMDwayne Clouse
Ċap chapter 4 answers 4.4 (day 2).pdf
40kv. 1 Jan 31, 2013, 8:26 PMDwayne Clouse
Ċap chapter 4 answers 4.5.pdf
307kv. 1 Jan 31, 2013, 8:26 PMDwayne Clouse
Ċap chapter 4 answers 4.6 (day 1).pdf
255kv. 1 Jan 31, 2013, 8:27 PMDwayne Clouse
Ċap chapter 4 answers 4.6 (day 2).pdf
274kv. 1 Jan 31, 2013, 8:27 PMDwayne Clouse
Ċap chapter 4 answers 4.7 (day 1).pdf
246kv. 1 Jan 31, 2013, 8:27 PMDwayne Clouse
Ċap chapter 4 answers 4.7 (day 2).pdf
291kv. 1 Jan 31, 2013, 8:27 PMDwayne Clouse
Ċap chapter 4 answers 4.7 (day 3).pdf
164kv. 1 Jan 31, 2013, 8:27 PMDwayne Clouse
Ċap chapter 4 answers 4.7 quiz review.pdf
757kv. 2 Feb 5, 2019, 11:30 AMDwayne Clouse
Ċap chapter 4 answers 4.8.pdf
242kv. 1 Jan 31, 2013, 8:27 PMDwayne Clouse
Ċap chapter 4 answers 4.9.pdf
296kv. 1 Jan 31, 2013, 8:27 PMDwayne Clouse
Ċap chapter 4 answers area review.pdf
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Ċap chapter 4 answers FTC.pdf
166kv. 1 Feb 1, 2013, 12:41 PMDwayne Clouse
Ċap chapter 4 answers FTC review.pdf
240kv. 1 Feb 1, 2013, 12:41 PMDwayne Clouse
Ċap chapter 4 answers practice test.pdf
545kv. 1 Feb 1, 2013, 12:41 PMDwayne Clouse
Ċap chapter 4 answers quiz review 4.1 - 4.4.pdf
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346kv. 1 Feb 1, 2013, 12:43 PMDwayne Clouse

Published in 1991 by Wellesley-Cambridge Press, the book is a useful resource for educators and self-learners alike. It is well organized, covers single variable and multivariable calculus in depth, and is rich with applications. There is also an online Instructor's Manual and a student Study Guide.

  1. In part (b) the student only finds two of the three values, so the first point was not earned. The justification point was not earned because it is not true that f ′ changes from increasing to decreasing at x =−1. In part (d) the student earned the first 2 points since x =1 is identified as a candidate and the endpoints are considered.
  2. Enrolling in AP Calculus comes with the understanding that you will take the AP exam in May. The 2019 test will be given May 5, 2020. If you do not plan on taking the AP Exam, we must have a conversation about it first. My goal is for each of you to receive credit by passing the AP Exam. That means we strive for a five!!!
  3. This is the additional review for the first part of chapter 4. (The second of the 3 reviews).

The complete textbook is also available as a single file. (PDF - 38.5MB)

Highlights of Calculus

MIT Professor Gilbert Strang has created a series of videos to show ways in which calculus is important in our lives. The videos, which include real-life examples to illustrate the concepts, are ideal for high school students, college students, and anyone interested in learning the basics of calculus.

› Watch the videos

4.4 analytical connections f f

4.4 Analytical Connections f, f', f' (part 1) Video Notes 2nd Derivative Sign Diagram Ex 3 (day 2) Video Notes 2nd Derivative Test (day 2). View Homework Help - MathAssessmentSolutions.pdf from ECON 301 at Rutgers University. Calculus Assessment Test for Intermediate Microeconomics - Solutions Question 1.

Textbook Components

  • Table of Contents (PDF)
  • Answers to Odd-Numbered Problems (PDF - 2.4MB)
  • Equations (PDF)
4.4 Analytical Connections F, F
4.4
4.4 Analytical Connections F, F
ChapterSFILES
1: Introduction to Calculus, pp. 1-43
1.1 Velocity and Distance, pp. 1-7
1.2 Calculus Without Limits, pp. 8-15
1.3 The Velocity at an Instant, pp. 16-21
1.4 Circular Motion, pp. 22-28
1.5 A Review of Trigonometry, pp. 29-33
1.6 A Thousand Points of Light, pp. 34-35
1.7 Computing in Calculus, pp. 36-43

Chapter 1 - complete (PDF - 2.2MB)

Chapter 1 - sections:

1.1 - 1.4 (PDF - 1.6MB)
1.5 - 1.7 (PDF - 1.4MB)

2: Derivatives, pp. 44-90
2.1 The Derivative of a Function, pp. 44-49
2.2 Powers and Polynomials, pp. 50-57
2.3 The Slope and the Tangent Line, pp. 58-63
2.4 Derivative of the Sine and Cosine, pp. 64-70
2.5 The Product and Quotient and Power Rules, pp. 71-77
2.6 Limits, pp. 78-84
2.7 Continuous Functions, pp. 85-90

Chapter 2 - complete (PDF - 3.8MB)

Chapter 2 - sections:

2.1 - 2.4 (PDF - 2.3MB)
2.5 - 2.7 (PDF - 1.7MB)

3: Applications of the Derivative, pp. 91-153
3.1 Linear Approximation, pp. 91-95
3.2 Maximum and Minimum Problems, pp. 96-104
3.3 Second Derivatives: Minimum vs. Maximum, pp. 105-111
3.4 Graphs, pp. 112-120
3.5 Ellipses, Parabolas, and Hyperbolas, pp. 121-129
3.6 Iterations x[n+1] = F(x[n]), pp. 130-136
3.7 Newton's Method and Chaos, pp. 137-145
3.8 The Mean Value Theorem and l'Hôpital's Rule, pp. 146-153

Chapter 3 - complete (PDF - 3.3MB)

Chapter 3 - sections:

3.1 - 3.4 (PDF - 1.5MB)
3.5 - 3.8 (PDF - 2.0MB)

4: The Chain Rule, pp. 154-176
4.1 Derivatives by the Charin Rule, pp. 154-159
4.2 Implicit Differentiation and Related Rates, pp. 160-163
4.3 Inverse Functions and Their Derivatives, pp. 164-170
4.4 Inverses of Trigonometric Functions, pp. 171-176

Chapter 4 - complete (PDF - 1.1MB)

Chapter 4 - sections:

4.1 - 4.2 (PDF)
4.3 - 4.4 (PDF)

5: Integrals, pp. 177-227
5.1 The Idea of an Integral, pp. 177-181
5.2 Antiderivatives, pp. 182-186
5.3 Summation vs. Integration, pp. 187-194
5.4 Indefinite Integrals and Substitutions, pp. 195-200
5.5 The Definite Integral, pp. 201-205
5.6 Properties of the Integral and the Average Value, pp. 206-212
5.7 The Fundamental Theorem and Its Consequences, pp. 213-219
5.8 Numerical Integration, pp. 220-227

Chapter 5 - complete (PDF - 3.3MB)

Chapter 5 - sections:

5.1 - 5.4 (PDF - 1.1MB)
5.5 - 5.8 (PDF - 2.3MB)

6: Exponentials and Logarithms, pp. 228-282
6.1 An Overview, pp. 228-235
6.2 The Exponential e^x, pp. 236-241
6.3 Growth and Decay in Science and Economics, pp. 242-251
6.4 Logarithms, pp. 252-258
6.5 Separable Equations Including the Logistic Equation, pp. 259-266
6.6 Powers Instead of Exponentials, pp. 267-276
6.7 Hyperbolic Functions, pp. 277-282

Chapter 6 - complete (PDF - 3.1MB)

Chapter 6 - sections:

6.1 - 6.4 (PDF - 2.1MB)
6.5 - 6.7 (PDF - 1.2MB)

7: Techniques of Integration, pp. 283-310
7.1 Integration by Parts, pp. 283-287
7.2 Trigonometric Integrals, pp. 288-293
7.3 Trigonometric Substitutions, pp. 294-299
7.4 Partial Fractions, pp. 300-304
7.5 Improper Integrals, pp. 305-310

Chapter 7 - complete (PDF - 1.7MB)

Chapter 7 - sections:

7.1 - 7.3 (PDF - 1.2MB)
7.4 - 7.5 (PDF)

8: Applications of the Integral, pp. 311-347
8.1 Areas and Volumes by Slices, pp. 311-319
8.2 Length of a Plane Curve, pp. 320-324
8.3 Area of a Surface of Revolution, pp. 325-327
8.4 Probability and Calculus, pp. 328-335
8.5 Masses and Moments, pp. 336-341
8.6 Force, Work, and Energy, pp. 342-347

Chapter 8 - complete (PDF - 2.1MB)

Chapter 8 - sections:

8.1 - 8.3 (PDF - 1.1MB)
8.4 - 8.6 (PDF - 1.1MB)

9: Polar Coordinates and Complex Numbers, pp. 348-367
9.1 Polar Coordinates, pp. 348-350
9.2 Polar Equations and Graphs, pp. 351-355
9.3 Slope, Length, and Area for Polar Curves, pp. 356-359
9.4 Complex Numbers, pp. 360-367

Chapter 9 - complete (PDF)

Chapter 9 - sections:

9.1 - 9.2 (PDF)
9.3 - 9.4 (PDF)

10: Infinite Series, pp. 368-391
10.1 The Geometric Series, pp. 368-373
10.2 Convergence Tests: Positive Series, pp. 374-380
10.3 Convergence Tests: All Series, pp. 325-327
10.4 The Taylor Series for e^x, sin x, and cos x, pp. 385-390
10.5 Power Series, pp. 391-397

Chapter 10 - complete (PDF - 2.0MB)

Chapter 10 - sections:

10.1 - 10.3 (PDF - 1.3MB)
10.4 - 10.5 (PDF)

11: Vectors and Matrices, pp. 398-445
11.1 Vectors and Dot Products, pp. 398-406
11.2 Planes and Projections, pp. 407-415
11.3 Cross Products and Determinants, pp. 416-424
11.4 Matrices and Linear Equations, pp. 425-434
11.5 Linear Algebra in Three Dimensions, pp. 435-445

Chapter 11 - complete (PDF - 3.3MB)

Chapter 11 - sections:

11.1 - 11.3 (PDF - 2.2MB)
11.4 - 11.5 (PDF - 1.2MB)

12: Motion along a Curve, pp. 446-471
12.1 The Position Vector, pp. 446-452
12.2 Plane Motion: Projectiles and Cycloids, pp. 453-458
12.3 Tangent Vector and Normal Vector, pp. 459-463
12.4 Polar Coordinates and Planetary Motion, pp. 464-471

Chapter 12 - complete (PDF - 1.2MB)

Chapter 12 - sections:

12.1 - 12.2 (PDF)
12.3 - 12.4 (PDF)

13: Partial Derivatives, pp. 472-520
13.1 Surface and Level Curves, pp. 472-474
13.2 Partial Derivatives, pp. 475-479
13.3 Tangent Planes and Linear Approximations, pp. 480-489
13.4 Directional Derivatives and Gradients, pp. 490-496
13.5 The Chain Rule, pp. 497-503
13.6 Maxima, Minima, and Saddle Points, pp. 504-513
13.7 Constraints and Lagrange Multipliers, pp. 514-520

Chapter 13 - complete (PDF - 3.9MB)

Chapter 13 - sections:

13.1 - 13.4 (PDF - 2.3MB)
13.5 - 13.7 (PDF - 1.5MB)

14: Multiple Integrals, pp. 521-548
14.1 Double Integrals, pp. 521-526
14.2 Changing to Better Coordinates, pp. 527-535
14.3 Triple Integrals, pp. 536-540
14.4 Cylindrical and Spherical Coordinates, pp. 541-548

Chapter 14 - complete (PDF - 1.9MB)

Chapter 14 - sections:

14.1 - 14.2 (PDF - 1.0MB)
14.3 - 14.4 (PDF)

15: Vector Calculus, pp. 549-598
15.1 Vector Fields, pp. 549-554
15.2 Line Integrals, pp. 555-562
15.3 Green's Theorem, pp. 563-572
15.4 Surface Integrals, pp. 573-581
15.5 The Divergence Theorem, pp. 582-588
15.6 Stokes' Theorem and the Curl of F, pp. 589-598

Chapter 15 - complete (PDF - 3.1MB)

Chapter 15 - sections:

15.1 - 15.3 (PDF - 1.5MB)
15.4 - 15.6 (PDF - 1.6MB)

16: Mathematics after Calculus, pp. 599-615
16.1 Linear Algebra, pp. 599-602
16.2 Differential Equations, pp. 603-610
16.3 Discrete Mathematics, pp. 611-615

Chapter 16 - complete (PDF)

Chapter 16 - sections:

16.1 - 16.2 (PDF)
16.3 (PDF)

Welcome!

This OCW supplemental resource provides material from outside the official MIT curriculum.

MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum.

No enrollment or registration. Freely browse and use OCW materials at your own pace. There's no signup, and no start or end dates.

Knowledge is your reward. Use OCW to guide your own life-long learning, or to teach others. We don't offer credit or certification for using OCW.

1)ap

4.4 Analytical Connections f, f', f' (part 1) Video Notes 2nd Derivative Sign Diagram Ex 3 (day 2) Video Notes 2nd Derivative Test (day 2). View Homework Help - MathAssessmentSolutions.pdf from ECON 301 at Rutgers University. Calculus Assessment Test for Intermediate Microeconomics - Solutions Question 1.

Textbook Components

  • Table of Contents (PDF)
  • Answers to Odd-Numbered Problems (PDF - 2.4MB)
  • Equations (PDF)
ChapterSFILES
1: Introduction to Calculus, pp. 1-43
1.1 Velocity and Distance, pp. 1-7
1.2 Calculus Without Limits, pp. 8-15
1.3 The Velocity at an Instant, pp. 16-21
1.4 Circular Motion, pp. 22-28
1.5 A Review of Trigonometry, pp. 29-33
1.6 A Thousand Points of Light, pp. 34-35
1.7 Computing in Calculus, pp. 36-43

Chapter 1 - complete (PDF - 2.2MB)

Chapter 1 - sections:

1.1 - 1.4 (PDF - 1.6MB)
1.5 - 1.7 (PDF - 1.4MB)

2: Derivatives, pp. 44-90
2.1 The Derivative of a Function, pp. 44-49
2.2 Powers and Polynomials, pp. 50-57
2.3 The Slope and the Tangent Line, pp. 58-63
2.4 Derivative of the Sine and Cosine, pp. 64-70
2.5 The Product and Quotient and Power Rules, pp. 71-77
2.6 Limits, pp. 78-84
2.7 Continuous Functions, pp. 85-90

Chapter 2 - complete (PDF - 3.8MB)

Chapter 2 - sections:

2.1 - 2.4 (PDF - 2.3MB)
2.5 - 2.7 (PDF - 1.7MB)

3: Applications of the Derivative, pp. 91-153
3.1 Linear Approximation, pp. 91-95
3.2 Maximum and Minimum Problems, pp. 96-104
3.3 Second Derivatives: Minimum vs. Maximum, pp. 105-111
3.4 Graphs, pp. 112-120
3.5 Ellipses, Parabolas, and Hyperbolas, pp. 121-129
3.6 Iterations x[n+1] = F(x[n]), pp. 130-136
3.7 Newton's Method and Chaos, pp. 137-145
3.8 The Mean Value Theorem and l'Hôpital's Rule, pp. 146-153

Chapter 3 - complete (PDF - 3.3MB)

Chapter 3 - sections:

3.1 - 3.4 (PDF - 1.5MB)
3.5 - 3.8 (PDF - 2.0MB)

4: The Chain Rule, pp. 154-176
4.1 Derivatives by the Charin Rule, pp. 154-159
4.2 Implicit Differentiation and Related Rates, pp. 160-163
4.3 Inverse Functions and Their Derivatives, pp. 164-170
4.4 Inverses of Trigonometric Functions, pp. 171-176

Chapter 4 - complete (PDF - 1.1MB)

Chapter 4 - sections:

4.1 - 4.2 (PDF)
4.3 - 4.4 (PDF)

5: Integrals, pp. 177-227
5.1 The Idea of an Integral, pp. 177-181
5.2 Antiderivatives, pp. 182-186
5.3 Summation vs. Integration, pp. 187-194
5.4 Indefinite Integrals and Substitutions, pp. 195-200
5.5 The Definite Integral, pp. 201-205
5.6 Properties of the Integral and the Average Value, pp. 206-212
5.7 The Fundamental Theorem and Its Consequences, pp. 213-219
5.8 Numerical Integration, pp. 220-227

Chapter 5 - complete (PDF - 3.3MB)

Chapter 5 - sections:

5.1 - 5.4 (PDF - 1.1MB)
5.5 - 5.8 (PDF - 2.3MB)

6: Exponentials and Logarithms, pp. 228-282
6.1 An Overview, pp. 228-235
6.2 The Exponential e^x, pp. 236-241
6.3 Growth and Decay in Science and Economics, pp. 242-251
6.4 Logarithms, pp. 252-258
6.5 Separable Equations Including the Logistic Equation, pp. 259-266
6.6 Powers Instead of Exponentials, pp. 267-276
6.7 Hyperbolic Functions, pp. 277-282

Chapter 6 - complete (PDF - 3.1MB)

Chapter 6 - sections:

6.1 - 6.4 (PDF - 2.1MB)
6.5 - 6.7 (PDF - 1.2MB)

7: Techniques of Integration, pp. 283-310
7.1 Integration by Parts, pp. 283-287
7.2 Trigonometric Integrals, pp. 288-293
7.3 Trigonometric Substitutions, pp. 294-299
7.4 Partial Fractions, pp. 300-304
7.5 Improper Integrals, pp. 305-310

Chapter 7 - complete (PDF - 1.7MB)

Chapter 7 - sections:

7.1 - 7.3 (PDF - 1.2MB)
7.4 - 7.5 (PDF)

8: Applications of the Integral, pp. 311-347
8.1 Areas and Volumes by Slices, pp. 311-319
8.2 Length of a Plane Curve, pp. 320-324
8.3 Area of a Surface of Revolution, pp. 325-327
8.4 Probability and Calculus, pp. 328-335
8.5 Masses and Moments, pp. 336-341
8.6 Force, Work, and Energy, pp. 342-347

Chapter 8 - complete (PDF - 2.1MB)

Chapter 8 - sections:

8.1 - 8.3 (PDF - 1.1MB)
8.4 - 8.6 (PDF - 1.1MB)

9: Polar Coordinates and Complex Numbers, pp. 348-367
9.1 Polar Coordinates, pp. 348-350
9.2 Polar Equations and Graphs, pp. 351-355
9.3 Slope, Length, and Area for Polar Curves, pp. 356-359
9.4 Complex Numbers, pp. 360-367

Chapter 9 - complete (PDF)

Chapter 9 - sections:

9.1 - 9.2 (PDF)
9.3 - 9.4 (PDF)

10: Infinite Series, pp. 368-391
10.1 The Geometric Series, pp. 368-373
10.2 Convergence Tests: Positive Series, pp. 374-380
10.3 Convergence Tests: All Series, pp. 325-327
10.4 The Taylor Series for e^x, sin x, and cos x, pp. 385-390
10.5 Power Series, pp. 391-397

Chapter 10 - complete (PDF - 2.0MB)

Chapter 10 - sections:

10.1 - 10.3 (PDF - 1.3MB)
10.4 - 10.5 (PDF)

11: Vectors and Matrices, pp. 398-445
11.1 Vectors and Dot Products, pp. 398-406
11.2 Planes and Projections, pp. 407-415
11.3 Cross Products and Determinants, pp. 416-424
11.4 Matrices and Linear Equations, pp. 425-434
11.5 Linear Algebra in Three Dimensions, pp. 435-445

Chapter 11 - complete (PDF - 3.3MB)

Chapter 11 - sections:

11.1 - 11.3 (PDF - 2.2MB)
11.4 - 11.5 (PDF - 1.2MB)

12: Motion along a Curve, pp. 446-471
12.1 The Position Vector, pp. 446-452
12.2 Plane Motion: Projectiles and Cycloids, pp. 453-458
12.3 Tangent Vector and Normal Vector, pp. 459-463
12.4 Polar Coordinates and Planetary Motion, pp. 464-471

Chapter 12 - complete (PDF - 1.2MB)

Chapter 12 - sections:

12.1 - 12.2 (PDF)
12.3 - 12.4 (PDF)

13: Partial Derivatives, pp. 472-520
13.1 Surface and Level Curves, pp. 472-474
13.2 Partial Derivatives, pp. 475-479
13.3 Tangent Planes and Linear Approximations, pp. 480-489
13.4 Directional Derivatives and Gradients, pp. 490-496
13.5 The Chain Rule, pp. 497-503
13.6 Maxima, Minima, and Saddle Points, pp. 504-513
13.7 Constraints and Lagrange Multipliers, pp. 514-520

Chapter 13 - complete (PDF - 3.9MB)

Chapter 13 - sections:

13.1 - 13.4 (PDF - 2.3MB)
13.5 - 13.7 (PDF - 1.5MB)

14: Multiple Integrals, pp. 521-548
14.1 Double Integrals, pp. 521-526
14.2 Changing to Better Coordinates, pp. 527-535
14.3 Triple Integrals, pp. 536-540
14.4 Cylindrical and Spherical Coordinates, pp. 541-548

Chapter 14 - complete (PDF - 1.9MB)

Chapter 14 - sections:

14.1 - 14.2 (PDF - 1.0MB)
14.3 - 14.4 (PDF)

15: Vector Calculus, pp. 549-598
15.1 Vector Fields, pp. 549-554
15.2 Line Integrals, pp. 555-562
15.3 Green's Theorem, pp. 563-572
15.4 Surface Integrals, pp. 573-581
15.5 The Divergence Theorem, pp. 582-588
15.6 Stokes' Theorem and the Curl of F, pp. 589-598

Chapter 15 - complete (PDF - 3.1MB)

Chapter 15 - sections:

15.1 - 15.3 (PDF - 1.5MB)
15.4 - 15.6 (PDF - 1.6MB)

16: Mathematics after Calculus, pp. 599-615
16.1 Linear Algebra, pp. 599-602
16.2 Differential Equations, pp. 603-610
16.3 Discrete Mathematics, pp. 611-615

Chapter 16 - complete (PDF)

Chapter 16 - sections:

16.1 - 16.2 (PDF)
16.3 (PDF)

Welcome!

This OCW supplemental resource provides material from outside the official MIT curriculum.

MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum.

No enrollment or registration. Freely browse and use OCW materials at your own pace. There's no signup, and no start or end dates.

Knowledge is your reward. Use OCW to guide your own life-long learning, or to teach others. We don't offer credit or certification for using OCW.

Made for sharing. Download files for later. Send to friends and colleagues. Modify, remix, and reuse (just remember to cite OCW as the source.)

4.4 Analytical Connections F F' F' (part 1)ap Calculus 14th Edition

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