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Week
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Topics
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Core
Learning Outcomes
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1-2
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Indefinite integration
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Students should be able to:
reverse the process of
differentiation to obtain an indefinite integral for simple
functions
understand the role of the
arbitrary constant
use a table of indefinite
integrals of simple functions
understand and use the
notation for indefinite integrals
use the constant multiple
rule and the sum rule
use
indefinite integration to solve practical problems such as
obtaining velocity from a formula for acceleration or
displacement from a formula for velocity..
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3-4
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Definite integration,
applications to areas and volumes
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Students should be able to:
realize the importance of the
Fundamental Theorem of the Calculus
obtain definite integrals of
simple functions
use the main properties of
definite integrals
calculate the area under a
graph and recognize the meaning of a negative value
calculate the area between
two curves
calculate the volume of a
solid of revolution
use trapezium and
Simpson's rules to approximate the value of a definite integral.
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5-6
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Methods of integration
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Students
should be able to:
obtain
definite and indefinite integrals of rational functions in
partial fraction form
apply
the method of integration by parts to indefinite and definite
integrals
use
the method of substitution on indefinite and definite integrals
solve
practical problems which require the evaluation of an integral
recognise
simple examples of improper integrals
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7-9
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Sequences and series
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Students
should be able to:
understand
convergence and divergence of a sequence
know
what is meant by a partial sum
understand
the concept of a power series
apply
simple tests for convergence of a series
understand
the idea of radius of convergence of a power series
recognise
Maclaurin series for standard functions
understand
how Maclaurin series generalise to Taylor series
use
Taylor series to obtain approximate percentage changes in a
function.
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10-11
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Solution of simultaneous
linear equations
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Students
should be able to:
represent
a system of linear equations in matrix form
understand
how the general solution of an inhomogeneous linear system of
m equations in n unknowns is obtained from the solution of
the homogeneous system and a particular solution
recognise
the different possibilities for the solution of a system of
linear equations
give
a geometrical interpretation of the solution of a system of
linear equations
understand
how and why the rank of the coefficient matrix and the augmented
matrix of a linear system can be used to analyse its solution
use
the inverse matrix to find the solution of 3 simultaneous linear
equations
understand
the term ‘ill-conditioned’
apply
the Gauss elimination method and recognise when it fails
understand
the Gauss-Jordan variation
use
appropriate software to solve simultaneous linear equations
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12-13
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Matrices
and determinants
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Students
should be able to:
understand
what is meant by a matrix
recall
the basic terms associated with matrices (for example, diagonal,
trace, square, triangular, identity)
obtain
the transpose of a matrix
determine
any scalar multiple of a matrix
recognise
when two matrices can be added and find, where possible, their
sum
recognise
when two matrices can be multiplied and find, where possible,
their product
calculate
the determinant of 2 x 2 and 3 x 3 matrices
understand
the geometric interpretation of 2 x 2 and 3 x 3 determinants
use
the elementary properties of determinants in their evaluation
state
the criterion for a square matrix to have an inverse
write
down the inverse of a 2 x 2 matrix when it exists
determine
the inverse of a matrix, when it exists, using row operations
calculate
the rank of a matrix
use
appropriate software to determine inverse matrices.
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