Achievement objective M811
In a range of meaningful contexts, students will be engaged in thinking mathematically and statistically. They will solve problems and model situations that require them to:
 choose and apply a variety of differentiation, integration, and antidifferentiation techniques to functions and relations, using both analytical and numerical methods.
Indicators
 Understands and uses a range of differentiation techniques including:
 first principles (only for polynomials of degree ≤ 3)
 product rule, quotient rule, chain rule
 other techniques including implicit or parametric differentiation, for example, conic relations.
Functions include:
 polynomials
 rational
 e^{x}
 ln(x)
 trig functions
 x^{n}, n ∈ R .
 Develops an understanding of integration by:
 using numerical methods for finding areas under curves (rectangle rule, trapezium rule, and Simpson’s rule)
 relating the definite integral to the limit of a sum (Riemann Integral)
 linking integration and antidifferentiation using the fundamental theorem of calculus, including:
 Part 1: If f is continuous on [a, b] and F(x) =
for [a, b] then F'(x) = f(x) (explores using technology) (Note: this guarantees that an antiderivative does exist for Part 2.)
 Part 2: If F' = f then
 Uses a variety of integration techniques for functions including:
 polynomials
 x^{n}, n ∈ R
 e^{x}
 sin(x)
 cos(x)
 f ' ( g ( x ) ) ( g ' ( x ) )
 f'(x)/f(x)
Applications include:
 finding areas under and between curves
 (related) rates of change
 kinematics
 optimisation
 points of inflection
 concavity.
 Makes links with differential equations
M812.
 See
key mathematical ideas on NZmaths.
Progression
M811 links from
M79,
M710.
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Possible context elaborations
 Applications for differentiation techniques include:
 rates of change
 tangents and normals to curves
 features of graphs including turning points, points of inflection and concavity
 optimisation.
 Applications for integration techniques include:
 finding areas under and between curves
 rates of change
 kinematics.

Activity: 28 Days later – Zombies
Assessment for qualifications
NCEA achievement standards at levels 1, 2 and 3 have been aligned to the New Zealand Curriculum. Please ensure that you are using the correct version of the standards by going to the
NZQA website.
The NZQA subjectspecific resources pages are very helpful. From there, you can find all the achievement standards and links to assessment resources, both internal and external.
The following achievement standard(s) could assess learning outcomes from this AO:
Last updated September 17, 2018
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