2 Differential Calculus of Functions of one Variable
The derivative measures instantaneous rate of change, and this section develops it from the limit concept on which it rests.
The order is deliberate. Limits come first, since the derivative is defined as one, together with the techniques for evaluating them and the indeterminate forms where naive substitution fails. Continuity follows, being defined by a limit condition. The mean value theorems come next: they look modest but are the bridge from local information — the derivative at a point — to global conclusions about a function on an interval, and almost every later result depends on them.
The section then turns to infinite series, where the question is when adding infinitely many terms produces a finite answer, and to the representation of functions as power series. It closes with curvature, which applies the derivative to measure how sharply a curve bends and prepares the ground for the treatment of space curves in the section on vectors.
2.2 Limits
2.3 Properties of Limits
2.4 Limits at Infinity
2.5 Infinite Limits
2.6 Indeterminate Forms of Limits
2.6.1 Indeterminate types of the form \(0^0\), \(\infty ^0\) and \(1^{\infty }\)
2.7 Continuity of a Function
2.7.1 Right and Left \(-\) hand continuity
2.7.2 Continuity in an Interval
2.7.3 Theorems on Continuity
2.7.4 Piece-wise Continuity
2.8 Mean Value Theorems
2.8.1 Rolle’s Theorem
2.8.2 Mean Value Theorem
2.8.3 Generalized Mean Value Theorem
2.8.4 Extended Mean Value Theorem
2.9 Series
2.10 Infinite Sequences
2.11 Infinite Series
2.12 Tests for Convergence of Series
2.13 Power Series
2.14 Series Expansion of Functions
2.15 Curvature
2.16 Theorem (Curvature in Rectangular Coordinates)
2.17 Curvature when a Curve is given in Parametric Form
2.18 The Circle of Curvature
2.19 Intrinsic Coordinates
2.20 Practice Problems
Questions on this section
Stuck on something here? Ask below and it stays attached to this topic.