Real Analysis
- Sciences
- 300 level
- 3 credit units
- 107 pages
- 3 units
This course introduces students to the fundamental concepts of real analysis, focusing on differentiability and mean value theorems. It covers derivatives, their geometrical interpretation, and the relationship between continuity and differentiability. Students will explore Rolle's theorem, Lagrange's mean value theorem, and Cauchy's mean value theorem. The course also delves into higher order derivatives, Taylor's theorem, Maclaurin's expansion, indeterminate forms, and extreme values of functions.
About this course
- Difficulty
- Intermediate
- Study hours
- 208 hours
- Maths
- Advanced
- Content
- Theoretical, problem solving
- Practical work
- No
- Assignments
- Tutor marked assignments
- Final examination
What you'll read
The real module and unit structure of MTH341, taken from the course material NOUN publishes.
One paragraph, so you can see how it reads
MTH341 · UNIT 1: DERIVATIVES
Let f be a real function defined on an open interval [a, b]. Let c be a point of this interval so that a < c < b. The function f is said to be differentiable at the point x = c if exists and is finite.
What you should be able to do
- Define and calculate derivatives of functions.
- Apply Mean Value Theorems to solve problems in real analysis.
- Use Taylor's and Maclaurin's theorems to expand functions.
- Evaluate limits of indeterminate forms using L'Hospital's Rule.
- Determine extreme values (maxima and minima) of functions.
- Understand the relationship between continuity and differentiability.
What it prepares you for
- Data Analyst
- Financial Analyst
- Statistician
- Research Scientist
- Mathematician
- Finance
- Engineering
- Data Science
- Research
- Academia
Where it gets hard
The units students slow down on, and what makes each one heavy.
- Module 1: Differentiability
Unit 1: Derivatives
The concepts of left-hand and right-hand derivatives require a solid understanding of limits and can be challenging to grasp initially.
- Module 1: Differentiability
Unit 2: Mean-Value Theorems
Applying Rolle's Theorem and other mean value theorems requires careful verification of conditions and can be difficult in complex functions.
A suggested way through it
13 weeks, about 52 hours in total. Yours will differ.
- Week 1Module 1: Differentiability
Unit 1: Derivatives · 4 hours
Understand the definition of the derivative of a function at a point.. Explore the geometrical interpretation of the derivative as the slope of a tangent.. Study the relationship between differentiability and continuity..
- Week 2Module 1: Differentiability
Unit 2: Mean-Value Theorems · 4 hours
Learn and apply Rolle's Theorem to solve problems.. Understand and apply Lagrange's Mean Value Theorem.. Explore Cauchy's Mean Value Theorem and its applications..
- Week 3Module 1: Differentiability
Unit 3: Higher Order Derivatives · 4 hours
Study higher order derivatives and their properties.. Learn Taylor's Theorem and Maclaurin's Expansion.. Practice finding derivatives of various functions..
- Week 4Module 1: Differentiability
Unit 1: Derivatives · 4 hours
Review derivatives of a function. Practice problems on geometrical meaning of derivatives. Work on tutor marked assignment.
- Week 5Module 1: Differentiability
Unit 2: Mean-Value Theorems · 4 hours
Review Rolle's Theorem. Practice problems on Lagrange's Mean Value Theorem. Work on tutor marked assignment.
- Week 6Module 1: Differentiability
Unit 3: Higher Order Derivatives · 4 hours
Review Taylor's Theorem. Practice problems on Maclaurin's Expansion. Work on tutor marked assignment.
- Week 7Module 1: Differentiability
Unit 1: Derivatives · 4 hours
Solve problems involving limits using L'Hopital's rule.. Practice converting indeterminate forms to apply L'Hopital's rule.. Review various indeterminate forms and their evaluation techniques..
- Week 8Module 1: Differentiability
Unit 2: Mean-Value Theorems · 4 hours
Apply first and second derivative tests to find local maxima and minima.. Solve optimization problems using calculus techniques.. Review conditions for local and global extrema..
- Week 9Module 1: Differentiability
Unit 3: Higher Order Derivatives · 4 hours
Review derivatives of a function. Practice problems on geometrical meaning of derivatives. Work on tutor marked assignment.
- Week 10Module 1: Differentiability
Unit 1: Derivatives · 4 hours
Solve problems involving limits using L'Hopital's rule.. Practice converting indeterminate forms to apply L'Hopital's rule.. Work on tutor marked assignment.
- Week 11Module 1: Differentiability
Unit 2: Mean-Value Theorems · 4 hours
Apply first and second derivative tests to find local maxima and minima.. Solve optimization problems using calculus techniques.. Work on tutor marked assignment.
- Week 12Module 1: Differentiability
Unit 3: Higher Order Derivatives · 4 hours
Review all units. Practice problems on all topics. Prepare for final examination.
- Week 13Module 1: Differentiability
Unit 3: Higher Order Derivatives · 4 hours
Final Revision. Work on pending assignments. Prepare for final examination.
Preparing for the exam
- Thoroughly review the definitions and theorems covered in each unit.
- Practice solving a variety of problems related to derivatives and mean value theorems.
- Focus on understanding the conditions required for applying each theorem.
- Create concept maps linking Taylor's and Maclaurin's theorems to specific function expansions.
- Practice applying L'Hospital's Rule to different indeterminate forms.
- Review all tutor-marked assignments and address any areas of weakness.
Questions students ask about this course
What is MTH341 about?
This course introduces students to the fundamental concepts of real analysis, focusing on differentiability and mean value theorems. It covers derivatives, their geometrical interpretation, and the relationship between continuity and differentiability. Students will explore Rolle's theorem, Lagrange's mean value theorem, and Cauchy's mean value theorem. The course also delves into higher order derivatives, Taylor's theorem, Maclaurin's expansion, indeterminate forms, and extreme values of functions.
How many units does MTH341 have?
MTH341, Real Analysis, has 3 units across 1 module, over 107 pages of course material. You can read it one unit at a time.
How many credit units is MTH341?
MTH341 carries 3 credit units, at 300 level in Sciences.
Is MTH341 hard?
MTH341 is rated intermediate level, with advanced mathematical content. It is mostly theoretical and problem solving work.
How long does MTH341 take to study?
About 208 hours of study, spread across its 3 units.
How is MTH341 assessed?
MTH341 is assessed by assignments, tutor marked assignments and final examination.
What can I do with MTH341?
Data Analyst, Financial Analyst, Statistician, Research Scientist and Mathematician.