General information for Maths students

This module provides a rigorous basis for differential and integral calculus, i.e. the theory behind differentiation and integration rather than their applications. The module will include some full proofs.

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MTH5104 Convergence and Continuity

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Normal assessment

Reassessment – synoptic

1. Differentiable functions: Definition of differentiability. Algebra of derivatives, chain rule. Derivative of inverse function. Rolle's Theorem, Mean Value Theorem and applications. Taylor's Theorem.
2. Integration: Darboux definition of Riemann integral, simple properties. Continuous functions are integrable (via uniform continuity). Fundamental Theorem of the calculus, integral form of Mean Value Theorem and of the remainder in Taylor's Theorem; applications to some well known series (log, arctan, binomial). Improper integrals.
3. Sequences of functions: pointwise and uniform convergence. Weierstrass M-test. Term-by-term integration of power series.

Academic Content

This module covers:

• Differentiable functions, the algebra of derivatives and key theorems.
• Integration involving the Riemann integral; the Fundamental Theorem of Calculus; applications.
• Sequences of functions; pointwise and uniform convergence; the Weierstrass M-test; term-by-term integration of power series.

Disciplinary Skills

At the end of this module, students should be able to:

• Define the derivative and state the properties of the derivative including the chain rule and inverse function rule.
• State and use key theorems concerning differentiable functions, such as Rolle's Theorem, the Mean Value Theorem and Taylor's Theorem.
• Define the Riemann integral, and state its properties.
• State the Fundamental Theorem of Calculus and apply it to the calculation of limits.
• Apply Taylor's Theorem to some well-known functions.
• Distinguish pointwise and uniform convergence.
• Apply the Weierstrass M-test to determine if an infinite series of functions converges uniformly.

Attributes

At the end of this module, students should have developed with respect to the following attributes:

• Grasp the principles and practices of their field of study.
• Acquire substantial bodies of new knowledge.
• Explain and argue clearly and concisely.
• Acquire and apply knowledge in a rigorous way.
• Connect information and ideas within their field of study.

Assessment Criteria

• Marking and grading criteria
• Discipline-specific level descriptors

Approved by Faculty Board 03/99. Assessment profile changed 05/05. Assessment in 2010–11 was 20% in-term test, 80% final exam.