ACARA ACMMM122, A specific anti-derivative from an initial condition

Year 11-12 · Mathematical Methods

Determine f(x) given its derivative f'(x) and an initial condition f(a)=b, by anti-differentiating and solving for the constant.

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Teaching guide

What students learn

Determine f(x) given its derivative f'(x) and an initial condition f(a)=b, by anti-differentiating and solving for the constant.

How to teach it

First antidifferentiate to get the family f(x) = F(x) + C, then use the given point: substitute its coordinates to make an equation in C and solve. Emphasise that the point pins down one member of the family of parallel curves.

Worked example

The gradient function of a curve is f'(x) = 2x - 1, and the curve passes through the point (-1, 7). Find f(x).

Answer: f(x) = x^2 - x + 5

Why: Antidifferentiate: f(x) = x^2 - x + C. Substitute (-1, 7): 2 + C = 7, so C = 5. Therefore f(x) = x^2 - x + 5.

Generated and checked in code, so it is correct.

Watch out for

Substituting the point before antidifferentiating, or leaving the answer as F(x) + C without solving for C once a point is given.

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Year 11-12 Math

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More Year 11-12 Mathematical Methods codes: ACMMM114 · ACMMM116 · ACMMM117 · ACMMM121 · ACMMM125 · ACMMM131 · ACMMM132

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Australian Curriculum content descriptions (c) ACARA, used under CC BY 4.0. Descriptions are paraphrased. See australiancurriculum.edu.au.