954 math t [ppu] semester 2 topics-syllabus

[PPU] Semester 2 Topics-Syllabus

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954MATHEMATICS T

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SECOND TERM: CALCULUS

Topic Teaching

Period Learning Outcome

7 Limits and Continuity 12 Candidates should be able to:

7.1 Limits 6 (a) determine the existence and values of the left-

hand limit, right-hand limit and limit of a

function;

(b) use the properties of limits;

7.2 Continuity 6 (c) determine the continuity of a function at a

point and on an interval;

(d) use the intermediate value theorem.

8 Differentiation 28 Candidates should be able to:

8.1 Derivatives 12 (a) identify the derivative of a function as a limit;

(b) find the derivatives of xn (n ), e

x, ln x,

sin x, cos x, tan x, sin1

x, cos1

x, tan1

x, with

constant multiples, sums, differences,

products, quotients and composites;

(c) perform implicit differentiation;

(d) find the first derivatives of functions defined

parametrically;

8.2 Applications of

differentiation

16 (e) determine where a function is increasing,

decreasing, concave upward and concave

downward;

(f) determine the stationary points, extremum

points and points of inflexion;

(g) sketch the graphs of functions, including

asymptotes parallel to the coordinate axes;

(h) find the equations of tangents and normals to

curves, including parametric curves;

(i) solve problems concerning rates of change,

including related rates;

(j) solve optimisation problems.

9 Integration 28 Candidates should be able to:

9.1 Indefinite integrals 14 (a) identify integration as the reverse of

differentiation;

(b) integrate xn (n ), e

x, sin x, cos x, sec

2x, with

constant multiples, sums and differences;

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Topic Teaching


(c) integrate rational functions by means of

decomposition into partial fractions;

(d) use trigonometric identities to facilitate the

integration of trigonometric functions;

(e) use algebraic and trigonometric substitutions

to find integrals;

(f) perform integration by parts;

9.2 Definite integrals 14 (g) identify a definite integral as the area under a

curve;

(h) use the properties of definite integrals;

(i) evaluate definite integrals;

(j) calculate the area of a region bounded by a

curve (including a parametric curve) and lines

parallel to the coordinate axes, or between two

curves;

(k) calculate volumes of solids of revolution about

one of the coordinate axes.

10 Differential Equations 14 Candidates should be able to:

(a) find the general solution of a first order

differential equation with separable variables;

(b) find the general solution of a first order linear

differential equation by means of an integrating

factor;

(c) transform, by a given substitution, a first order

differential equation into one with separable

variables or one which is linear;

(d) use a boundary condition to find a particular

solution;

(e) solve problems, related to science and

technology, that can be modelled by differential

equations.

11 Maclaurin Series 12 Candidates should be able to:

(a) find the Maclaurin series for a function and the

interval of convergence;

(b) use standard series to find the series expansions

of the sums, differences, products, quotients

and composites of functions;


Topic Teaching


(c) perform differentiation and integration of a

power series;

(d) use series expansions to find the limit of a

function.

12 Numerical Methods 14 Candidates should be able to:

12.1 Numerical solution of

equations

10 (a) locate a root of an equation approximately by

means of graphical considerations and by

searching for a sign change;

(b) use an iterative formula of the form

1 f ( )n nx x to find a root of an equation to a

prescribed degree of accuracy;

(c) identify an iteration which converges or

diverges;

(d) use the Newton-Raphson method;

12.2 Numerical integration 4 (e) use the trapezium rule;

(f) use sketch graphs to determine whether the

trapezium rule gives an over-estimate or an

under-estimate in simple cases.
