Course Proposal: Calculus II

This document gives a detailed list of learning outcomes for Calculus II, the expanded syllabus and a tentative week-by-week teaching plan.

Learning Outcomes

On completion of this course students will be expected to:

1. be able to carry out the operations of addition, multiplication and division on complex numbers, know the meaning of conjugate, real part, imaginary part of complex numbers and be able to ﬁnd the argument and modulus of complex numbers;

2. understand the geometric representation of complex numbers in the Argand diagram, be able to represent geometrically sum, products and quotients of complex numbers;

3. be able to ﬁnd complex solutions of quadratic equations with real coefﬁcients;

4. be able to draw loci and regions in the Argand diagram, e.g.

|z − 1| = 1, |z − 1| ≤ 1, |z| = 2|z − 1|, arg(z − a) = π/4, |z| ≤ |z − 1|;

5. know the relation between trigonometric functions and hyperbolic functions;

6. know Euler’s relation eiθ = cos θ + i sin θ and De Moivre’s Theorem, be able to apply this theorem to express (i) sin nθ and cos nθ in powers of sin θ and cos θ, and (ii) powers of sin θ and cos θ in terms of sin and cos of multiple angles;

7. know the nth roots of unity and be able to ﬁnd n-th roots of complex numbers, e.g ﬁnding the three cubic roots of 8i;

8. understand the concepts of infinite sequence, converging sequence and diverging to infinity sequence, be able to find the limit of converging sequences, e.g.

sin n ln n n + 1n √ 1 a = , a = √ , a = , a = n n2, a = n sin ; n n n n n n − 1 n n n

9. understand the concept of converging series

1 10. know the n-th term test for divergence of series, the integral test, the ratio and root tests and the comparison tests, and be able to apply these with suitable hints e.g. use the ratio test to determine which of the following series converges and which diverges

∞ ∞ ∞ X 2n + 5 X (2n)! X n! , , ; 3n (n!)2 nn n=1 n=1 n=1

11. identify alternating series and know the alternating series test (Leibniz’s Theorem), be able to estimate the error introduced by truncation of converging alternating series;

12. understand the difference between absolute and conditional convergence, know the absolute convergence test;

13. recognize a power series and be able to calculate its radius of convergence;

14. recognize a geometric series and be able to expand simple algebraic fractions in powers of x, e.g. x2/(1 − 2x) or x/(1 + x2);

15. know Taylor formula with the remainder term in the Lagrange form and apply it to obtain power series and estimate error of approximation by Taylor polynomials;

16. know Taylor series for ex, ln(1 + x), sin x, cos x, (1 + x)α;

17. be aware about applications of Taylor series (Euler’s formula for complex numbers, series solution of differential equations, evaluating non-elementary integrals in terms of series, evaluating indeterminate forms of limits)

18. know what is meant for a function of two variables to be continuous, and be able to identify points of discontinuity;

19. be able to calculate partial derivatives, directional derivatives and estimate the rate of change in a given direction, be able to carry out implicit differentiation;

20. know the chain rule for functions of two and three variables and be able to use it for ﬁnding (i) the rate of change in a function’s values along a curve, and (ii) partial derivatives under transformation of variables;

21. be able to ﬁnd gradient vector and directions of maximal, minimal and zero change;

22. be able to ﬁnd tangents and normals to level curves, tangent planes and normal lines for surfaces z = f(x, y) and f(x, y, z) = 0;

23. be able to ﬁnd local extreme values and classify their type;

2 24. be able to use the method of Lagrange multipliers for ﬁnding maxima and minima of functions with one constraint;

25. know the properties of double integrals;

26. be able to reduce double integrals to repeated integrals, be able to reverse the order of integration in repeated integrals;

27. be able to calculate double integrals over rectangular and simple non-rectangular regions;

28. be able to ﬁnd the volume beneath a surface z = f(x, y);

29. be able to express and evaluate area as a double integral;

30. for simple coordinate transformations, be able to calculate Jacobians and ﬁnd the transformed regions;

31. be able to evaluate double and triple integrals by a given substitution;

32. know the Jacobian of transformation from cartesian to polar coordinates and be able to evaluate double integrals by changing to polar coordinates.

Syllabus and tentative weekly teaching plan

1. Complex numbers I: deﬁnition and their necessity for elementary operations, geometric representation, loci and regions in the complex plane, quadratic equations with real coefﬁcients.

2. Complex numbers II: Euler’s relation (the concept of series introduced but systematic treatment deferred to Calc III), DeMoivre’s Theorem and applications to trigonometric identities, square root and log functions. Application to integrals R eax cos(bx)dx.

3. Series I. Infinite sequences. Converging sequences. Diverging to infinity sequences. Calculating limits of sequences, including use of l’Hopital’s rule. Upper bounds and the least upper bound. Non-decreasing bounded sequences. Infinite series (n-th term, partial sum, convergence, sum). Examples of converging series. Examples of diverging series. The n-th term test for divergence. Operations on converging series.

4. Series II. Series of positive terms and the Integral Test for convergence (second look at improper integrals). The direct comparison and limit comparison tests. The ratio and root tests. Alternating series, absolute and conditional convergence (examples, Leibniz’s Theorem, estimation of the truncation error, the absolute convergence test).

3 5. Series III. Power series and convergence. Term-by-term differentiation and integration. Power series in the complex plane. Taylor and Maclaurin series and polynomials (via integration by parts). Various forms of the remainder term. Convergence of the Taylor series. Taylor series for common transcendental functions (exp, log, sin, cos, square root, cosh and sinh). Examples of applications of power series (power series solutions of differential equations, evaluating non-elementary integrals, evaluating indeterminant forms)

6. Derivatives IV: Functions of two variables. Domain, range and level curves. Graphing a function of two variables. Limits and continuity in the xy-plane. Partial derivatives. Statement and use of ”mixed derivatives theorem” without proof. Differentiability and continuity and related theorems without proof. Chain rule for functions of two and three variables. Implicit differentiation revisited (Implicit Function Theorem).

7. Revision and Test.

8. Derivatives V. Directional derivatives and gradient vector. Tangent planes and normal lines. Estimating change in a speciﬁc direction. Linearization and differentials. Extreme values and saddle points. Lagrange multipliers.

9. Integration IV. Double integrals as volumes. Properties of double integrals. Double integrals as repeated integrals, rectangular regions, simple non-rectangular regions.

10. Integration V. Double integrals over non-rectangular regions, bounded and unbounded. Determining limits of integration. Reversing the order of integration. Transformations of variables - maps, domains, one-to-one maps, inverse maps.

11. Integration VI. Change of variables in double integrals. Jacobians. Transformation to polar coordinates. Other transformations. Applications to normal distribution in probability. Triple and multiple integrals. Change of variables of integration. Applications of integrals.

12. Revision