This module equips students with mathematical tools essential for biomedical design and future mathematics learning, focusing on integral calculus, linear algebra and numerical methods. Learners will see the wide variety of engineering problems that can be tackled with integration and linear algebra techniques. The module also emphasizes the use of numerical techniques to solve linear and non-linear engineering problems, highlighting both their applications and limitations. Students will develop and implement algorithms in a modern programming language to support biomedical design challenges.
1. Review of differentiation. Definite integration as area under a curve. Integration of standard engineering functions using a table. Linear substitution and integration by parts. Application of integration to finding surface and volume of revolution, mean values, moments of inertia and centres of gravity.
2. Approximate definite integrals using numerical methods. The Trapezoid and Simpsons Methods. Calculations using spreadsheets. Estimation of surface areas and volumes of surfaces of revolution.
3. Solve first order initial value problems by separating the variables. Introduction of terms: first order, linear, separable, dependent and independent variables. General and Particular solutions, implicit solution form.
4. Solving the eigenvalue and eigenvector equations for 2 × 2 matrices. Verify that a vector is an eigenvector in higher dimensions by showing is satisfies Ax = ax. Diagonalize 2 by 2 matrices with distinct eigenvalues. Use Matlab to find eigenvalues and eigenvectors of higher dimensional matrices.
5. Solve Ax=b using Gaussian elimination for 3 variable and higher. Use Matlab solve A\b. Formulate and solve engineering problems of the form Ax = b, including using software to fit cubic splines between data points. Plot surfaces using appropriate software, including surfaces of revolution.
