the Further Mathematics network
the Further Mathematics network
www.fmnetwork.org.uk
www.fmnetwork.org.uk
Teaching Topics: Calculus
and Maclaurin Series in FP2
Let Maths take you Further…
FP2 in General
Calculus in FP2
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For many students, FP2 is the end point for
their school career in pure mathematics.
It’s excellent for improving students fluency in
lots and lots of topics and send them off to
their university or career destination as a
confident mathematician.
Don’t worry about setting routine exercises, in
my experience students seem to like them.
Calculus in FP2
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Students starting FP2 are hopefully about to
become very fluent, confident users of
calculus.
My interest is in applications of calculus in
computing.
Good tip - start FP2 lessons with quickfire
calculus.
For a very good mark in FP2 a very high level
of fluency with calculus and algebra is
needed.
Calculus in FP2
In FP2 students find out how to differentiate
arcsinx, arccos x, arctan x, sinh x, cosh x,
tanh x, arsinh x, arcosh x, artanx x.
This enables them to extend the class of
functions that they can integrate enormously.
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Revisiting and developing
understanding of the chain rule
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In C3 and C4 calculus it’s possible for
students to learn types of derivative and
almost begin to forget about the chain rule.
Graphs and Derivatives
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e.g the derivative of sin(f(x)) is f`(x)cos(f(x)).
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It’s good to discuss with students the graphs
of the inverse trig functions and their
derivatives.
Deriving the derivatives of arcsinx and
arccosx using the inverse function theorem is
recommended, students will enjoy this!
In FP2 test students out by asking them to
differentiate e.g.
f(x) = arctan (4x)
Integration
Proving these results
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Care with constants
A nice way to have students prove these
results is by differentiation.
e.g.
Examiners Report
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Past Paper Question
Examiners Report
Using the standard formulae
More difficult examples,
completion of square
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Move any constant off the x2 term
(Complete the square and substitute if there
is an x term)
Use the standard formula
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Ask students to give you any quadratic, f(x),
with no real roots, then integrate 1/f(x)
Some examples require completion of square
for a negative x2 term.
Even though this isn’t examined very often,
doing this with students is thoroughly
recommended.
Integrating rational functions
This deals with ANY rational
functions
Consider integration of 1/(ax2 + bx + c).
„ If b2 – 4ac >= 0 then this can be done using
partial fractions or by inspection.
„ If b2 – 4ac < 0, then, as seen, this can be
done by completing the square, then using
the derivative of arctan.
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If f(x) = p(x)/q(x), then by using the
fundamental theorem of algebra, long division
and partial fractions it will be possible to
integrate f(x).
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Example
Substitution requiring some
thought
Use of integration by parts
Solution
Examiner’s Report
Past Paper Question
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Examiners Report
Past Paper Question
Examiner’s Report
Past Paper Questions
Examiners Report
Past Paper Question
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Examiners Report
Maclaurin Series
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Graphical Demonstration
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f(x) = sin x
Get students to do this by hand up to the x2
and x3 term.
Ask them where the curves are cutting the xaxis.
Get them to substitute values in to see how
good the approximation is and how it is
improving.
Maclaurin Series - Advice
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A wonderful topic for discussion
Motivates entirely the years that students
have spent studying polynomials,
‘polynomials are everything’.
Graphical approach is great.
Idea that the whole function is captured in the
information about it at 0.
Delving a bit deeper
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Define
f1(x) = x – sin x,
f2(x) = cos x – 1 + x2/2
f3(x) = sin x – x + x3/6
f4(x) = 1 – x2/2 + x4/24 – cos x
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Differentiating these shows that they are
strictly increasing for x > 0.
It follows that for x > 0
x > sin x > x – x3/6
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Past Paper Question
Remember the range of examples that
students are expected to deal with
Make sure that students know the short cuts
for getting series from other series
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Examiners Report
Past Paper Questions
Examiners Report
The Story of Sine
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From opposite over hypotenuse
..to the Maclaurin Series.
First Definition
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sin x is defined to be the ratio of the opposite
side to angle x and the hypotenuse in a right
angled triangle.
Extending this to all the reals
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sin x is now defined as in the diagram below
sin x = a
(a, b)
x
cos x = b
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To Maclaurin Series
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Continuity
Differentiability
Rolle’s Theorem
The Mean Value Theorem
Taylor’s Theorem
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