618-171 Mathematics 1P

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Objectives:

On completion of this subject students should:

Comprehend:

• some of the nature of the different types of numbers they use;

• the intuitive nature of limits as used in continuity, differentiation and integration;

• the notion of integral as area;

• the extension of the notion of vectors in two or three dimensions to any finite number of dimensions;

• the theoretical treatment of systems of simultaneous linear equations;

• the classification and principles governing the solution of the basic first and second order differential equations;

• the range of calculus skills and techniques necessary for the solution of these differential equations, and the solution methods applicable to each type.

Have developed:

• an ability to manipulate complex numbers and to use them to solve problems;

• an ability to use differential calculus to solve extremal problems;

• an ability to compute a wide range of integrals;

• an ability to use integration to compute area, length and volume;

• an ability to solve arbitrary systems of simultaneous linear equations;

• the ability to classify and solve the basic differential equations of first and second order, and the integral and differential calculus skills to achieve these solutions with accuracy and confidence.

Appreciate:

• the role of proof and logical reasoning in mathematics;

• the use of complex numbers;

• the role of limits in both the differential and integral calculus;

• the practical uses of calculus;

• the use of the ideas of linear algebra in dealing with the solution of simultaneous linear equations;

• the role of differential equations in applied mathematics.

Content:

Foundations: sets, integers, mathematical induction; real numbers; complex numbers, polar form, de Moivre's theorem, complex exponential. Calculus: functions of one real variable (including limits and continuity), derivatives; curve sketching; maxima and minima, curvature; antiderivatives and the definite integral; trigonometric functions and their inverses, logarithm, exponential function, hyperbolic functions and their inverses; systematic integration; approximate integration; applications of integration, areas, arc length, surface areas and volumes of solids of revolution. Vectors and linear equations: vectors in three-dimensional space, dot and cross products, triple products, determinants; linear dependence; equations of lines and planes, geometrical applications; bases and coordinates, dimension; row reduction, rank, inverse, solution of linear equations, geometrical interpretation. Differential equations: gradient fields. first-order differential equations (linear via integrating factors, separable and homogeneous); linear differential equations with constant coefficients, particular integrals and complementary functions; applications to damped oscillators and resonance.

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