# MH1811 Mathematics II

[Lecture: 26 hrs; Tutorial: 12 hrs; Co-requisite: MH1810 (only applicable to AY2017/18 intake onwards); Academic Unit: 3.0]

Learning Objective

This course extends the basic concepts of differentiation and integration learned in Mathematics 1 to the operations on functions of multiple variables. Advanced applications of differential and integral calculus are included, In addition, the course covers topics on sequences, series and ordinary differential equations.

Content

Partial Differentiation. Partial Differentiation. Sequences and Series. First Order Differential Equations. Second Order Differential Equations.

Course Outline

Partial Differentiation

Functions of several variables, plotting of 2-variable functions, introduction to cylindrical and spherical coordinates, chain rule, total differential, gradient, directional derivatives, normal lines and tangent planes, extreme of functions of two variables, applications.

Multiple Integrals

Double integrals, regions of integrations, triple integrals, meaning and applications.

Sequences and Series

Sequences and series, convergence and divergence of series, absolute convergence, conditional convergence, test for convergence and divergence. Power series for functions, interval of convergence, Taylor and Maclaurin series, Taylor’s Theorem with remainder.

First Order Differential Equations

Separable differential equations, homogeneous differential equations, exact differential equations, integrating factor, Bernoulli’s equation, applications.

Second Order Differential Equations

Second order homogeneous and non-homogeneous equations with constant coefficients, variation of parameters, method of undetermined coefficients, Operator D Methods, series solutions of differential equations, applications.

Learning Outcome

By the end of this course, you (as a student) would be able to:

 ​1. ​Express the general term of a sequence, determine and evaluate the limit of a convergent sequence, apply squeeze theorem for sequences, explain why a sequence is convergent or divergent. ​2. ​Manipulate basic series such as geometric series, telescoping series and determine its convergence and sum. Identify properties of convergent and divergent series. ​3. ​Determine whether a series is divergent by Divergence Test. Apply alternating test to determine an alternating series is convergent. Apply integral test, comparison test and limit comparison test, ratio test, and root test appropriately to series to determine convergence of series with positive terms. Determine absolute, conditional or divergence of a general series. ​4. ​Manipulate some power series related to geometric series; determine the radius and the interval of convergence for power series. Apply term-by-term differentiation and integration of power series. Apply power series to approximation. ​5. ​Evaluate Taylor's series (Maclaurin series) of functions, including exponential and sine functions. Determine Taylor series of some functions using known Taylor series. Use the Taylor's Remainder Theorem and Estimation theorem to determine the upper bound of the estimation error of a series. ​6. ​Determine the domain of simple 2-variable function and sketch the domain and level curve. ​7. ​Evaluate the limits of 2-variable and 3-variable functions; explain why the limit does not exist. ​8. ​Interpret the concept and meaning of partial derivative and evaluate partial derivatives. Apply Chain rule for partial differentiation. ​9. ​Evaluate the gradient vector and use its orthogonality to determine the equation of tangent plane to a level surface and the graph z=f(x,y). Determine the equation of normal line. Apply the equation of tangent plane to z=f(x,y) to linear approximation. Use total differential to approximate the errors or changes. ​10. ​Evaluate directional derivative, and use it to determine rate of change. ​11. ​Find stationary points and classify them as local maximum, minimum or saddle points. ​12. ​Use Lagrange Multiplier method to determine global maximum or minimum of a function subject to an equality constraint. Apply the method to solve optimization problems. ​13. ​Sketch the region of integration for 2-variable function and evaluate double integral, and apply it to some problems. Generalize the concept of double integral to triple integrals. ​14. ​Solve separable first order ordinary differential equations (ODEs), homogeneous first order ODEs, first order linear ODEs by integrating factors, Bernoulli's equations and exact ODEs. ​15. ​Solve linear homogeneous second order ODEs with constant real-number coefficients. Use undetermined coefficient method and method of variation of parameters to find a particular solution to non-homogeneous second order ODEs with constant real-number coefficients. Obtain the general solution for a non-homogeneous second order ODEs with constant real-number coefficients. ​16. ​Apply ODEs to model and solve simple practical problems.

Text

Thomas' Calculus, 13 th edt, Thomas, GB Jr., Weir MD and Hass J, Pearson-Addison-Wesley, 9781292089799, 2016

References

Calculus (International Student Edition), 6th edt, James Stewart, Thomson, 9780495482826 Brooks/Cole

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