All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions.

(i)

Graph of 2i
Graph of 2i

Width of each rectangle = \frac{1}{n}
Total area of rectangles = \frac{1}{n} \sum_{r=1}^n f(\frac{r}{n})

When n \rightarrow \infty, we have the exact area instead.

\therefore, \lim_{n\to\infty} \frac{1}{n} \sum_{r=1}^n f(\frac{r}{n}) = \int_0^1 f(x) dx

(ii)
\lim_{n\to\infty} \frac{1}{n} [(\frac{1}{n})^{\frac{1}{3}} + (\frac{2}{n})^{\frac{1}{3}} + \ldots + (\frac{n}{n})^{\frac{1}{3}}]

= \lim_{n\to\infty} \frac{1}{n} \sum_{r=1}^n f(\frac{r}{n}) where f(x) = x^{\frac{1}{3}}

= \int_0^1 f(x) dx

= \frac{3x^{\frac{4}{3}}}{4} \bigl|_0^1

= \frac{3}{4}

Back to 2015 A-level H2 Mathematics (9740) Paper 1 Suggested Solutions

KS Comments:

(i) needs some explanations to be done and there are many ways to present it, just make sure you use a graph. They are looking for students to be able to write out the definition of integration here. This question should be one of those “high A” questions to set students apart. You don’t have to use summation signs here too. (ii), we can check answer with GC.

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