Book contents
- Frontmatter
- Contents
- Preface to first edition
- Preface to second edition
- Glossary
- PART I NUMBERS
- PART II FUNCTIONS
- 6 Functions and continuity: neighbourhoods, limits of functions
- 7 Continuity and completeness: functions on intervals
- 8 Derivatives: tangents
- 9 Differentiation and completeness: Mean Value Theorems, Taylor's Theorem
- 10 Integration: the Fundamental Theorem of Calculus
- 11 Indices and circle functions
- 12 Sequences of functions
- Appendix 1 Properties of the real numbers
- Appendix 2 Geometry and intuition
- Appendix 3 Questions for student investigation and discussion
- Bibliography
- Index
8 - Derivatives: tangents
Published online by Cambridge University Press: 06 July 2010
Book contents
- Frontmatter
- Contents
- Preface to first edition
- Preface to second edition
- Glossary
- PART I NUMBERS
- PART II FUNCTIONS
- 6 Functions and continuity: neighbourhoods, limits of functions
- 7 Continuity and completeness: functions on intervals
- 8 Derivatives: tangents
- 9 Differentiation and completeness: Mean Value Theorems, Taylor's Theorem
- 10 Integration: the Fundamental Theorem of Calculus
- 11 Indices and circle functions
- 12 Sequences of functions
- Appendix 1 Properties of the real numbers
- Appendix 2 Geometry and intuition
- Appendix 3 Questions for student investigation and discussion
- Bibliography
- Index
Summary
Preliminary reading: Bryant ch. 4, Courant and John ch. 2.
Concurrent reading: Hart, Spivak ch. 9.
Further reading: Tall (1982).
It is easy enough to say when a straight line is a tangent to a circle or an ellipse. For these curves, a straight line – infinitely extended – meets the curve in 0, 1 or 2 points. Each line with a unique point of intersection is a tangent. However if we try to use such a test to identify tangents to other curves we are in for a disappointment, and on several counts.
1 At how many points does the line x = 1 intersect the parabola y = x2? Draw a sketch. Is this line a tangent to the curve?
2 At how many points does the line y = −2 intersect the cubic curve y = x3 − 3x? Draw a sketch. Is this line a tangent to the curve?
From qn 2 we learn that whether a line is a tangent to a curve or not is a local question, which must be asked relative to the particular point of intersection, that is, inside a sufficiently small neighbourhood of that point.
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- Numbers and FunctionsSteps into Analysis, pp. 203 - 223Publisher: Cambridge University PressPrint publication year: 2000
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