 Mathematics
Te Tari Pāngarau me te Tatauranga
Department of Mathematics & Statistics

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## Resources for Math201

Recent additions are indicated like this

### 2022 S1 Assignments

Assignment 1277 K
Assignment 1: Solutions309 K
Assignment 2305 K
Assignment 2: Solutions310 K
Assignment 3281 K
Assignment 3: Solutions294 K

### 2022 S1 Course Information

Course Outline151 K
MATH201 Zoom Details75 K

### 2022 S1 Lecture Notes

Chapter 1: Introduction, Logic, and Proof232 K
Chapter 2: The Real Numbers373 K
Chapter 3: Sequences and Series447 K
Chapter 4: Functions, Limits, and Continuity363 K
Chapter 5: The Derivative393 K

### 2022 S1 Lectures

Lecture 1: Introduction (What is real analysis? + motivating examples)159131 K
Lecture 2: Logic (mathematical statements, compound statements, truth tables)147336 K
Lecture 3: Quantifiers (the quantifiers 'for all' and 'there exists', negating quantifiers)74238 K
Lecture 4: Proof (common proof strategies and examples)108725 K
Lecture 5: The Real Numbers (what is a number? set of rational numbers have holes, axiomatic definition of real numbers)135717 K
Lecture 6: The Ordering Axiom (ordering on natural numbers, intervals, bounded sets)83483 K
Lecture 7: The Completeness Axiom (sup and inf, completeness axiom, Archimedean property)119382 K
Lecture 8: Applications of Completeness Axiom (density of rationals, existence of square root, nested intervals theorem)109925 K
Lecture 9: The Real Numbers are Uncountable (rationals are countable but reals are not, max and min, examples)76940 K
Lecture 10: Open Sets (open sets, open intervals are open, examples, closed sets)122495 K
Lecture 11: Closed Sets (examples of closed sets, the absolute value, the triangle inequality)119536 K
Lecture 12: Convergence (Introduction to limits, definition of convergence, how to prove convergence)90945 K
Lecture 13: The Algebra of Limits (examples of limits, uniqueness of limits, statement of algebra of limits theorem)109281 K
Lecture 14: The Squeeze Theorem (proof of Algebra of limits, statement of Squeeze theorem, examples)116855 K
Lecture 15: Divergence (proof of Squeeze Theorem, divergence, examples of diverging sequences)78729 K
Lecture 16: Monotone Convergence (bounded sequences, monotone sequences, monotone convergence theorem, subsequences)120409 K
Lecture 17: Subsequences (subsequences and convergence, applications of monotone convergence)111942 K
Lecture 18: The Bolzano-Weierstrass Theorem (Bounded sequences have convergent subsequences, definition of Cauchy sequences, Cauchy sequences and convergence)118580 K
Lecture 19: Cauchy Sequences (proof of convergence of Cauchy sequence, convergence of series, the geometric series)120308 K
Lecture 20: Series (convergence of series vs convergence of sequences, harmonic series, alternating series test)112122 K
Lecture 21: Ratio Test (absolute convergence, comparison test, ratio test, Riemann rearrangement theorem)76167 K
Lecture 22: Limits and functions (outline of ch4, definition of limit, examples, limits of functions via limits of sequences)106517 K
Lecture 23: Functional Limits via Sequences (functional limits vs sequential limits, algebra of limits and squeeze theorem for functions, non-existence of limits)108002 K
Lecture 24: Continuous Functions (Definition, algebra of limits, composition, polynomials are continuous, IVT)100541 K
Lecture 25: Applications of IVT (flipped IVT, examples, Nth root, irrational roots)108401 K
Lecture 26: The Extreme Value Theorem (motivation, proof of EVT, pointwise convergence for sequences of functions, examples)107928 K
Lecture 27: Uniform Convergence79168 K
Lecture 28: The Derivative (Definition, examples, continuous vs differentiable, product and quotient rules)91650 K
Lecture 29: Manipulating Derivatives (Chain rule, derivatives are local, applications, intro to MVT)94001 K
Lecture 30: The Mean Value Theorem (MVT, derivative zero at max/min, Rolle's Theorem, applications)105972 K
Lecture 31: Taylor's Theorem (higher order derivatives, generalised Rolle's Theorem, Taylor's Theorem, examples)88772 K
Lecture 32: Power Series (exponential function, power series, basic properties, radius of convergence)93217 K
Lecture 33: Power Series II (differentiability of power series, swapping derivatives and limits)99500 K
Lecture 34: The Integral (partitions, upper and lower Riemann sums, Riemann integrable functions, Fundamental Theorem of Calculus)96334 K
Lecture 35: Review Lecture84043 K

### 2022 S1 Practice Exam

2022 MATH201 practice exam289 K
MATH201 exam front page 2022255 K

### 2022 S1 Test

Information on Test81 K
MATH201 Test 2022255 K
MATH201 Test 2022: Solutions251 K
Practise test: Solution to Part B, Q11320 K
Practise Test122 K

### 2022 S1 Tutorial Problems

Tutorial 1 (sets, truth tables, negation, proof)134 K
Tutorial 1: Solutions131 K
Tutorial 2 (sup and inf)111 K
Tutorial 2: Solutions161 K
Tutorial 3 (max and min, open and closed sets)125 K
Tutorial 3: Solutions142 K
Tutorial 4 (triangle inequality, limits of sequences)159 K
Tutorial 4: Solutions189 K
Tutorial 5 (diverging sequences, increasing and decreasing sequences, subsequences)137 K
Tutorial 5: Solutions161 K
Tutorial 6 (sequences, subsequences, monotone convergence)131 K
Tutorial 6: Solutions145 K
Tutorial 7 (convergence of series)155 K
Tutorial 7: Solutions178 K
Tutorial 8 (Limits for functions)121 K
Tutorial 8: Solutions146 K
Tutorial 9 (continuous functions, IVT, EVT))136 K
Tutorial 9: Solutions156 K
Tutorial 10 (differentiable functions)148 K
Tutorial 10: Solutions163 K
Tutorial 11 (higher order derivatives, MVT, Taylor's theorem)129 K
Tutorial 11: Solutions149 K

### Private Tutoring for 2022

• assignments
• tests and exam