MAT101 Calculus I - Assignment 1 - Solutions

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MAT 101 - Calculus I - Assignment 2

Please start on the Exercises from Sections 1.6 to 2.4 right away. Your first midterm on Sep 14 will be up to Sec 2.4.

Assignment 2

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Part A

The following exercises from Stewart's Essential Calculus are to be solved for presentation and discussion in the tutorials:

• [Section 1.6] 1, 3, 13, 26, 41, 47
• [Section 2.1] 3, 7, 10, 16, 26, 31, 47
• [Section 2.2] 1, 3, 6, 9, 21, 33, 43
• [Section 2.3] 10, 18, 21, 22, 33, 57, 63
• [Section 2.4] 6 to 10, 25, 27, 43, 48, 54
• [Section 2.5] 4, 23, 47, 55, 65, 66
• [Section 2.6] 8, 16, 19, 32
• [Section 2.7] 2, 9, 13, 24
• [Section 2.8] 7, 21
• [Section 3.1] 13, 23, 24
• [Section 3.2] 22, 32, 35, 66
• [Section 3.3] 7, 29, 39, 58
• [Section 3.4] 9, 20

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Part B

Submit written solutions to the following to your tutor by September 23:

• [Section 1.6] 19, 23, 49
• [Section 2.1] 28, 48
• [Section 2.2] 4, 23, 39
• [Section 2.3] 23, 40, 67
• [Section 2.4] 16, 30, 51
• [Section 2.5] 49, 57, 69
• [Section 2.6] 11, 17, 44
• [Section 2.7] 11, 28, 38
• [Section 2.8] 13, 24
• [Section 3.1] 25, 31
• [Section 3.2] 23, 38, 78
• [Section 3.3] 23, 62, 68
• [Section 3.4] 17

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Extra Credit

Submit the following to Prof Habib by September 16: Consider the cubic function \(f(x)=x^3+ax^2+bx+c\). Calculate \(\lim\limits_{x\to\pm\infty}\frac{f(x)}{x^3}\)and use this to show that there is a real number \(c\) such that \(f(c)=0\).

Edit on Sep 16: Last date of submission of Extra Credit problem is changed to Friday, Sep  20. Submission must be to Prof Habib. Also note that there are two uses of \(c\) in the problem which is an oversight. Change the second \(c\) to \(d\).

MAT101 Calculus I - Assignment 1

Department of Mathematics, Shiv Nadar University
Monsoon Semester 2013-14
MAT 101 Calculus I

Assignment 1

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Part A

The following exercises from Stewart's Essential Calculus are to be solved for presentation & discussion in the tutorials:

• [Section 1.1] 1, 3, 4, 17, 23, 25, 28, 35, 43, 51, 57, 59, 61
• [Section 1.2] 2, 4, 5, 8, 13, 17, 23, 26, 35, 52, 53
• [Section 1.3] 3, 7, 11, 23, 29
• [Section 1.4] 10, 13, 21, 22, 30, 43, 46
• [Section 1.5] 3, 10, 13, 15, 26, 35, 37

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Part B

Submit written solutions of the following to your Tutor by September 2:

• [Section 1.1] 5, 6, 18, 44, 62
• [Section 1.2] 18, 49, 58, 62
• [Section 1.3] 9, 27, 43
• [Section 1.4] 7, 23, 31
• [Section 1.5] 16, 31, 40

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Extra Credit

• [Section 1.5] 47

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MAT 101 - Calculus I

Syllabus for MAT 101 – Calculus I

This is a compulsory course for B.Tech. students in their 1^st semester.

Credits (Lec:Tut:Lab)= 3:1:0 (3 lectures and 1 tutorial weekly)

Prerequisites: Class XII mathematics or MAT 100 (Precalculus)

Brief Description:

This course covers one variable calculus and applications. It forms the base for subsequent courses in advanced vector calculus and real analysis as well as for applications in probability, differential equations, optimization, etc.

Detailed Syllabus:

1. Differentiation: Functions, limits, sandwich theorem, continuity, intermediate value theorem, tangent line, rates of change, derivative as function, algebra of derivatives, implicit differentiation, related rates, linear approximation, differentiation of inverse functions, derivatives of standard functions (polynomials, rational functions, trigonometric and inverse trigonometric functions, hyperbolic and inverse hyperbolic functions).
2. Applications of Differentiation: Indeterminate forms and L'Hopital's rule, absolute and local extrema, first derivative test, Rolle's theorem, mean value theorem, concavity, 2^nd derivative test, curve sketching.
3. Integration: Area under a curve, Riemann sums, integrability, fundamental theorem, mean value theorem for integrals, substitution, integration by parts, trigonometric integrals, partial fractions, improper integrals.
4. Applications of Integration: Area between curves, volume, arc length, applications to physics (work, center of mass).
5. Ordinary Differential Equations: 1^st order and separable, logistic growth, 1^st order and linear, 2^nd order linear with constant coefficients, method of undetermined coefficients, method of variation of parameters.

Main References:

• Essential Calculus – Early Transcendentals, by James Stewart. Cengage, India Edition.
• Advanced Engineering Mathematics, Erwin Kreyszig, 9^th edition, Wiley India, 2011.

Supplementary References:

• Advanced Engineering Mathematics, Dennis G Zill and Warren S Wright, 4^th edition, Jones and Bartlett.
• The Calculus Lifesaver, by A Banner, Princeton, 2007.
• Calculus and Analytic Geometry by G B Thomas and R L Finney, 9^th edition, Pearson.

MAT 102 - Calculus II

Syllabus for MAT 102 – Calculus II

This is a compulsory course for BS Mathematics students in their 3^rd semester.

Credits (Lec:Tut:Lab)= 3:1:0 (3 lectures and 1 tutorial weekly)

Prerequisites: MAT 101 (Calculus I) or equivalent

Brief Description: The first part deals with series of numbers and functions. The second part is an introduction to multivariable calculus, finishing with the various versions of Stokes' theorem. The concepts and techniques covered here are used extensively in the social and natural sciences as well as in engineering to study systems with many dimensions.

Detailed Syllabus:

1. Sequences and Series: Limits of sequences, algebra of limits, series, divergence test, comparison and limit comparison tests, integral test, alternating series test, absolute convergence, root & ratio tests, power series, Taylor series

2. Vectors: Dot and cross product, equations of lines and planes, quadric surfaces, space curves, arc length and curvature

3. Differential calculus in several variables: Functions of several variables, level curves and surfaces, limits and continuity, partial derivatives, tangent planes, chain rule, directional derivatives, gradient, Lagrange multipliers, extreme values and saddle points, 2^nd derivative test

4. Double and triple integrals: Double integrals over rectangles, double integrals over general regions, double integrals in polar coordinates, center of mass, triple integrals, triple integrals in cylindrical coordinates, triple integrals in spherical coordinates, change of variables

5. Vector Integration: Vector fields, line integrals, fundamental theorem, independence of path, Green's theorem, divergence, curl, parametric surfaces, area of a parametric surface, surface integrals, Stokes' theorem, Gauss' divergence theorem

Main Reference:

• Essential Calculus – Early Transcendentals, by James Stewart. Cengage, India Edition.

Supplementary References:

• Calculus and Analytic Geometry by G B Thomas and R L Finney, 9^th edition, Pearson.
• Basic Multivariable Calculus by J E Marsden, A J Tromba and A Weinstein, 1^st edition, Springer (India), 2011.
• Calculus by Ken Binmore and Joan Davies, 1^st edition, Cambridge, 2010.

BS Mathematics - Calculus I - Course Instructions

• The main text for the course is Essential Calculus: Early Transcendentals by James Stewart, 1^st edition, Cengage, 2011. Please immediately issue a copy from the Library. You will be required to bring this book to each class.
• There will be three lectures and two tutorials every week.
• There will be fortnightly assignments. Part A of each assignment will be discussed in tutorials, and students will be allotted problems for presentation. Part B is meant for written submission. The problems will mainly be assigned from the exercises in Stewart.
• It is expected that about 10 assignments will be distributed during the course. You will then be allowed to drop your worst two assignment marks.
• No late submissions will be accepted. The only leeway for taking care of special circumstances is provided by the above-mentioned dropping of your worst two assignment marks.
• Attendance will be taken in each class. Be aware that SNU requires a minimum of 70% attendance separately in lectures and tutorials. No further waiver is given beyond this 30%, even for illness.

Content:

1. Functions & Graphs, Limits, Continuity, Derivatives, L’Hôpital’s Rule

2.  Higher derivatives, Maxima/ Minima, Curve sketching

3. Area & Integration, Fundamental Theorem of Calculus, Techniques of integration, Improper integrals

4. Applications to Area, Volume, Arc-Length

5. Parametric Equations and Polar Coordinates

6. Several variables: Level curves, limits & continuity, partial derivatives, tangent planes, chain rule, directional derivative, gradient, Maxima/Minima, Lagrange multipliers

Further References:

1.       The Calculus Lifesaver by A Banner, Princeton, 2007.

2.       Calculus and Analytic Geometry by G B Thomas and R L Finney, 9^th edition, Pearson.

3.       Basic Multivariable Calculus by J E Marsden, A J Tromba and A Weinstein, 1^st edition, Springer (India), 2011.

4.       Calculus by Ken Binmore and Joan Davies, 1^st edition, Cambridge, 2010.

Assessment:

Final Exam	50%
Midterm Exam	25%
Assignments	15%
Class Performance	10%