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 Academic Year: | 2018/9 |
| Owning Department/School: | Department of Mathematical Sciences |
| Credits: | 6 [equivalent to 12 CATS credits] |
| Notional Study Hours: | 120 |
| Level: | Certificate (FHEQ level 4) |
| Period: |
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| Assessment Summary: | EX 100% |
| Assessment Detail: |
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| Supplementary Assessment: |
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| Requisites: | In taking this module you cannot take MA10207 . You must have a familiarity with the material of MA10207 taught in Semester 1 to take this unit. |
| Description:
| Aims: To give rigorous proofs of the principal theorems on the analysis of real sequences and real functions of a real variable. Learning Outcomes: After taking this unit, the students should be able to: * State definitions and theorems in real analysis. * Present proofs of the main theorems. * Apply these definitions and theorems to simple examples. * Construct their own proofs of simple unseen results. Skills: Numeracy T/F A Problem Solving T/F A Written and Spoken Communication F (in tutorials) Content: Continuity and limits of functions of real variables. Inertia principle. Algebra of limits, continuous functions, polynomials. Composition of continuous functions. Relate continuity to convergence. Weierstrass's Theorem. Intermediate Value Theorem. Continuous inverse of strictly increasing (continuous) function on interval. Functions discontinuous on R or Q. Definition of derivative. Rules of derivation. Chain Rule. Inverse functions. Rolle's Theorem. Mean Value Theorem. Uniform continuity. Uniqueness of solution f to f ' = g. Sign of derivative; monotonicity. Derivative vanishes at minimum. Cauchy Mean Value Theorem. L'Hopital's Rule. Limits at infinity. Taylor's Theorem with Lagrange remainder. O and o notation. Sign of second derivative; maxima and minima, convexity. Power series. Uniform Convergence. Differentiation of power series. Series definitions of exponential and trigonometric functions. ex+y=ex ey by differentiation. Uniqueness for Cauchy problem for y'' = - y by differentiating (y')2+y2, trigonometric addition formulae. Sketch extension to complex arguments and exp(iz)=cos(z)+isin(z). Logarithmic function. ab for positive a, real b. Logarithmic series on (-1,1) from geometric series. Binomial Theorem for real exponent by differentiating quotient of series by (1+x)p. Density of Q in R. Fundamentals of Rn; norm, inner product, Cauchy-Schwarz inequality, convergent sequences, Bolzano-Weierstrass, open sets. |