Calculus and Applications

Overview

This unit provides students with the knowledge and skills to apply mathematical and statistical techniques to a variety of engineering calculations and decisions; and to provide students with a thorough grounding in mathematics, laying a foundation for further studies in engineering mathematics.

Unit learning outcomes

Students who successfully complete this unit will be able to:

1. Apply general concepts of functions and graphs to polynomial, rational, exponential, logarithmic, trigonometric, hyperbolic functions of one variable, their inverses and compositions
2. Apply the induction principle and basic inequalities to verify important relations
3. Determine the convergence or divergence of sequences. Determine limits of functions of one variable
4. Determine first and higher order derivatives of functions of one variable. Determine the derivatives of inverse functions of one variable and apply implicit differentiation. Use differentiation for detailed graph drawing of relevant functions. Apply differentiation to determine rates of change, derivation of Taylor polynomials and correct use of de l’Hopital’s rule
5. Determine indefinite and definite integrals of basic trigonometric, hyperbolic, rational and other functions of one variable, using partial fractions, substitutions and integration by parts. Apply these concepts to evaluate the area under and between curves, arc lengths, volumes of solids of revolution and other examples
6. Determine the solution to first order separable differential equations and linear differential equations using an integrating factor
7. Determine the solution to second order homogeneous and non-homogeneous linear differential equations with constant coefficients. Apply these methods to simple, fundamental equations
8. Determine partial derivatives of functions of more than one variable and stationary points

Teaching methods

Hawthorn

Assessment

Content

• Fundamental properties of functions. Domain, image, composition, inversion and graph. Inverse trigonometric functions, hyperbolic functions and their inverses.
• The induction principle and some basic inequalities
• Introduction to sequences, convergence and divergence. Limits of sequences and functions: definition, meaning and properties. Fundamental limits and indeterminate forms.
• Continuity: definition, properties, graphing and examples. Differentiation of functions of one variable: rules, properties, inverse functions, implicit differentiation, applications to graphing of functions. Differentials, higher derivatives, rates of change, Taylor polynomials, l’Hopital rule.
• Integration of functions of one variable: anti-differentiation, properties, substitutions, integration by parts and partial fractions. Application to areas, volumes, arc lengths and other examples.
• Differential equations: first order separable differential equations, first order linear differential equations, orthogonal trajectories, second order linear differential equations with constant coefficients and simple right-hand sides. Applications to relevant, simple models.
• Functions of two and more variables. Differentiation: partial and directional derivatives, higher derivatives, gradients and differentials. Properties and stationary points of simple, important surfaces