UG code: 
MTH5105

Module name: 

Description: 
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. 
Organiser: 
Dr Huy Nguyen 
Details: 
Level
Semester B15 credits

Links: 

Subject area: 
Pure mathematics 
Overlaps: 
None 
Essential prerequisites: 
MTH5104 Convergence and Continuity 
Helpful prerequisites: 
None 
Restrictions: 
None 
Assessment: 
Normal assessment
Description  Duration / Length  Weighting 

Midterm test 
40 minutes 
10% 
Final examination 
2 hours 
90% 
Reassessment – synoptic
Description  Duration / Length  Weighting 

Resit examination 
2 hours 
100% 

Comments: 

Syllabus: 
 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.
 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.
 Sequences of functions: pointwise and uniform convergence. Weierstrass Mtest. Termbyterm integration of power series.

Learning outcomes: 
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 Mtest; termbyterm 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 wellknown functions.
 Distinguish pointwise and uniform convergence.
 Apply the Weierstrass Mtest 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

History: 
Approved by Faculty Board 03/99. Assessment profile changed 05/05. Assessment in 2010–11 was 20% interm test, 80% final exam. 
Last modified: Friday, 24 March 2017, 4:42 PM
