The first aim of the module is to enable the student to master the idea of a function, both algebraically and graphically, and to become fully acquainted with the functions most likely to occur in engineering. This proficiency is required to support engineering subjects and forms the basis for further mathematical study in year 2. The second aim is to deepen the students understanding of the range of applicability of differentiation to engineering problems. The third aim is to introduce the student to vectors and matrices as an efficient language and means of working with engineering problems in 2D.
1. Functions
Basic concepts. Graphs of functions. Horizontal and vertical translation. Function inverse. Algebra of functions including composition.
a. Engineering functions and their inverse on a calculator. Polynomials, rational, power, exponential and natural logarithmic, cos, sin functions.
b. Sketching engineering functions e.g. ln(x). Solving equations involving logarithmic and exponential functions; growth and decay model problems.
c. Definition of periodic functions. Sketch trigonometric functions Asin(wx+p) and Acos(wx+p). Solving trigonometric equations of the form Asin(wx+p) = B and Acos(wx+p) =B.
d. Construction of simple 2 part piecewise defined functions.
2. Differentiation
Rate of change of a function, tangent slope of the graph of a function. Derivatives of x, e.g. ln(x), sin(x), cos(x), linear combinations of these and f(ax+b) for these, with engineering applications.
3. Vectors in 2D
Scalars and vectors. i, j notation. Addition and subtraction of vectors graphically (parallelogram law) and algebraically. Modulus of a vector (not just distance). Unit vectors. Resolution into component form. Scalar product. Angle between two vectors in 2-D. Engineering applications to statics, motion in 2D, electrical science…
4. 2 x 2 Matrices
a. Definition of a 2 by 2 matrix.
b. Transformation of vectors in 2D. Effect of a matrix transformation on a square in 2D (rotation, shear, boost, …). Determinant and inverse and their geometric meaning.
c. Find a matrix T from its action on unit vectors along the x and y axes
d. Addition subtraction and multiplication of 2 by 2 matrices.
e. Writing two simultaneous equations in matrix form. The augmented matrix and using row reduction to solve two simultaneous equations.
