TMUA 2020 Paper 2 Q16 — Worked Solution

P2·Integration, Logic & Quantifiers·7.0Official · 2020 · Q16

proof-errorfundamental-theorem-of-calculustheorem-hypothesis

The Fundamental Theorem of Calculus (FTC) tells us that for any polynomial

f

\frac{d}{d x} \int_{0}^{x} f (t) d t = f (x) .

A student calculates

\frac{d}{d x} \int_{x}^{2 x} t^{2} d t

as follows.

(I)

\int_{x}^{2 x} t^{2} d t = \int_{0}^{2 x} t^{2} d t - \int_{0}^{x} t^{2} d t

(II)

By FTC,

\frac{d}{d x} \int_{0}^{x} t^{2} d t = x^{2}

(III)

By FTC,

\frac{d}{d x} \int_{0}^{2 x} t^{2} d t = (2 x)^{2} = 4 x^{2}

(IV)

\frac{d}{d x} \int_{x}^{2 x} t^{2} d t = 4 x^{2} - x^{2}

(V)

giving

\frac{d}{d x} \int_{x}^{2 x} t^{2} d t = 3 x^{2}

Which of the following best describes the student's calculation?

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