MATH1042 Calculus student study guide 2024

MATH1042A Calculus Study Guide

Page 1: Course Overview

• Course: Engineering Mathematics MATH1042A

• Year: 2024

• Institution: School of Mathematics, University of the Witwatersrand

Page 2: Contents

1. Functions

• Introduction

• Piecewise-defined Functions

• Composite Functions

• Even and Odd Functions

• Trigonometric Functions

• Inverse Functions

• Tutorial answers

2. Introduction to Differentiation

• Slopes of Curves

• Limits

• Differentiation from First Principles

• Simple Rules for Differentiation

• Differentiation of Sine and Cosine

• Tangents and Normals

• Maxima and Minima

• Higher Derivatives and the 2nd Derivative Test

• Tutorial answers

3. Differentiation Completed

• The Product Rule

• The Quotient Rule

• The Chain Rule

• Derivatives of Other Basic Functions

• First Approximations

• Tutorial answers

4. Integration

• The Indefinite Integral

• Area as the Limit of a Sum

• The Definite Integral

• Area and Integration

• Volume and Integration

• Area and Polar Curves

• Tutorial questions

5. Further Applications of Differentiation

• Maxima and Minima Problems

• Curve Sketching

• Related Rates of Change

• Tutorial answers

Page 4: Functions (Chapter 1)

1.1 Introduction

• Definition of Function: A function, denoted by f from a set D to a set Y, assigns a UNIQUE element f(x) in Y to each element x in D.

• Domain: Set D of possible input values.

• Range: Set of all f(x) as x varies in D.

1.2 Piecewise-defined Functions

• Definition: Functions defined by different formulas on different parts of the domain.

• Example:

  • f(x) = { -2, x < 0

    • x², 0 ≤ x ≤ 3

    • x - 1, x > 3

  • }

Page 5: Function Examples

Example 1.1.3 to 1.1.5

• Example 1.1.3: f(x) = x + 1; find f(2) and f(a).

• Vertical Line Test: A function can only have ONE VALUE for f(x) for each x in its domain. No vertical line intersects the graph of a function more than ONCE.

• Tutorial Questions: Various function evaluations and showing properties of functions.

Page 6- 7: Composite Functions and Absolute Values

1.2.1 and 1.2.3 Absolute Values

• Definition: |x| = {x, x ≥ 0; -x, x < 0}

• Tutorial: Practice problems related to absolute values.

Page 8: Composite Functions

Definition

• Composite Function: (f ◦ g)(x) = f(g(x)).

• Example: Let f(x) = 2x + 1 and g(x) = x - 3; find f(g(1)), g(f(2)).

Page 10: Trigonometric Functions (Chapter 1.5)

1.5.1 Angles and Radian Measure

• Conversions: 1 degree = π/180 radians and vice versa.

Page 22: Introduction to Differentiation

2.1 Slopes of Curves

• Definition of Gradient: m = y2 − y1/x2 − x1 = ∆y/∆x.

• Derivative Definition: dy/dx = limit as ∆x approaches 0 of (f(x + ∆x) - f(x))/∆x.

Page 28: Simple Rules for Differentiation

• Rules:

  • Derivative of a constant is 0.

  • Sum Rule: d/dx [f(x) + g(x)] = f'(x) + g'(x).

  • Power Rule: d/dx [xⁿ] = nxⁿ⁻¹.

Page 38: Second Derivative Test

Page 40: Applications and Examples of Differentiation

Page 43: Product and Quotient Rule

Page 46: Implicit Differentiation

Page 53: Logarithmic Differentiation

Page 56: First Approximations

Page 59: Further Applications of Differentiation

Page 68: Fundamental Theorem of Calculus

And More...

(Determinants, volume areas, and curves of intersections, with various examples and tutorial questions throughout the guide.)

Note: This guide is an overview, focusing on areas for deeper study and review. For a detailed understanding and formulas, reference the lecture notes and textbooks provided in the course.