Calculus

Learning Outcomes

Learning Outcomes

Having successfully completed this module you will be able to:

• Be able to understand the concept of integration, and be able to apply different methods of integration to find areas under curves
• Be able to solve a range of first and second order ordinary differential equations
• Be able to accurately handle inequalities for real numbers
• Be able to understand the definition of limit, and be able to use the algebra of limits, l'Hopital's rule, etc., to determine limits of simple expressions
• Be able to understand the definitions of continuity and differentiability, and be able to differentiate continuous functions.
• Be able to state and apply the Intermediate Value Theorem, Rolle's Theorem and the Mean Value Theorem

Functions

• Domain, range.
• Inverse functions.
• Injection, surjection, bijection.

Limits

• Informal and formal concepts of limit.
• Examples of definition of limit in terms of epsilon and delta: linear and scalar cases.
• Algebra of limits.
• Definition of continuity; sums, products and compositions of continuous functions.
• Intermediate Value Theorem.

Differentiation

• Slopes and tangent lines, formal definition in terms of a limit.
• Rules of differentiation: product rule, quotient rule and chain rule.
• Differentiation of trigonometric and hyperbolic trigonometric functions, inverse trigonometric functions, the exponential and logarithmic functions.
• Rolles' Theorem and the Mean Value Theorem.
• L’Hôpital’s rule.
• Maxima and minima, curve sketching.
• Taylor series.

Integration

• The concept of integration via Riemann sums.
• Fundamental Theorem of Calculus.
• Methods of integration: substitution, parts, partial fractions.
• Improper integrals.

Differential Equations

• First order differential equations: separable, linear, homogeneous, Bernoulli, Clairaut.
• Second order constant coefficient differential equations: complementary functions and particular integrals.

Lectures, problem classes, workshops, private study

Formative

This is how we’ll give you feedback as you are learning. It is not a formal test or exam.

Class Test

• Assessment Type: Formative
• Feedback:
• Final Assessment: No
• Group Work: No

Summative

This is how we’ll formally assess what you have learned in this module.

Referral

This is how we’ll assess you if you don’t meet the criteria to pass this module.