MAT1512
Calculus A

MAT1512 Study Guide

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1. Introduction
1.1. How to Use the Study Guide
1.2. Keys to Success in Studying Mathematics
1.3. Preparing for the Examination
2. Functions and Models
2.1. Background
2.2. Learning Outcomes
2.3. Principles of Problem Solving
2.4. Summary
2.5. The Way Forward
3. Limits and Derivatives
3.1. Background
3.2. Learning Outcomes
3.3. Prescribed Reading
3.4. Limit
3.4.1. Introduction to the Limit Concept
3.4.2. Definition of a Limit: Left-hand and Right-hand Limits
3.5. Worked Examples
3.5.1. Limits as x → c (c ∈ ℝ)
3.5.2. Limits as x → ±∞
3.5.3. Limits Involving |x| (Absolute Values)
3.5.4. Left-hand and Right-hand Limits
3.5.5. Limits Involving Trigonometric Functions
3.5.6. The Squeeze Theorem
3.5.7. The ε-δ Definition of a Limit (Read only for other modules eg MAT2615)
3.5.8. Continuity
4. Differentiation Rules
4.1. Background
4.2. Learning Outcomes
4.3. Prescribed Reading
4.4. The Derivative
4.4.1. Introducing the Derivative
4.4.2. Definition of the Derivative
4.5. Worked Examples
4.5.1. Differentiation from First Principles (derivative as a Function)
4.5.2. Basic Differentiation Formulas
4.5.3. Derivatives of Trigonometric Functions and Inverse Trigonometric Functions
4.5.4. Derivatives of Exponential and Logarithmic Functions
4.5.5. Logarithmic Functions
4.5.6. Implicit Differentiation
4.5.7. Tangents and Normal Lines
4.5.8. The Mean Value Theorem
5. Integrals
5.1. Background
5.2. Learning Outcomes
5.3. Prescribed Reading
5.4. Worked Examples
5.4.1. Antiderivatives
5.4.2. The Definite Integral and the Fundamental Theorem of Calculus - Part II
5.4.3. The Definite Integral and the Area Between the Curve and the x-axis
5.4.4. The Definite Integral and Area Under the Curve
5.5. The Mean Value Theorem for Definite Integrals
5.6. The Fundamental Theorem of Calculus - Part I
5.7. Integration in General
5.8. Indefinite Integrals ∫f(x)dx
5.9. The Substitution Rule
5.10. Integration of Exponential (e^x) and Logarithmic (ln(x)) Functions
5.11. Review of Formulas and Techniques of Integration
6. Differential Equations, Growth and Decay and Partial Derivatives/Chain Rule
6.1. Background
6.2. Learning Outcomes
6.3. Prescribed Reading
6.4. Worked Examples
6.4.1. Differential Equations
6.4.2. Growth and Decay
6.4.3. Partial Derivatives
6.4.4. The Chain Rule
A. Sequence and Summation Notation
A.1. Definition of a sequence
A.2. Recursive sequences
A.3. Partial sums of sequences
A.4. Sigma notation
A.5. Finding terms of a sequence
A.6. Finding the nth term of a sequence
A.7. Finding partial sums of sequences
A.8. Using sigma notation
B. Mathematical Induction
B.1. The principle of mathematical induction
B.2. Proving statements P(n) using mathematical induction