STPM 2017 Term 1 MM Coursework Explained.

—00

The lower sum and upper sum are defined as follows:

Lower Sum = \sum_{i=1}^n f(m_i)\triangle x

Lower Sum = \sum_{i=1}^n f(M_i)\triangle x

You are required to investigate the area of the region bounded by the curve y=f(x)=x^2+1, the x-axis, and the lines x=0 and x=2.

“Only sample solution for mathematical part will be posted. Please ask your school teacher for introduction, methodology, and conclusion.”

—01

(a)
(i) Sketch the curve with 5 inscribed rectangles and evaluate the lower sum.
(ii) Sketch the curve with 5 circumscribed rectangles and evaluate the upper sum.
(iii) Comment on your results.

(b)
(i) Tabulate the lower sums and upper sums for n=10, 20, 40.
(ii) What do you observe about the lower sums and upper sums as n increases?

—02

Let A_n=\sum_{i=1}^n f(c_i)\triangle x.
(a)
(i) Show that A_n=\frac{4}{3}\left[\frac{(n+1)(2n+1)}{n^2}\right]+2 if c_i=a+i\triangle x,
(ii) Tabulate the values of A_i for n=1000, 5000, 10000, 50000, 100000, 500000.
(iii) Evaluate \lim_{n\to \infty}A_n.
(b) It is given that A_n=\frac{4}{3}\left[\frac{(n-1)(2n-1)}{n^2}\right]+2 if c_i=a+(i-1)\triangle x. Repeat steps (a)(ii) and (a)(iii).
(c) What happens to the value of \lim_{n\to\infty} A_n if a+(i-1)\triangle x < c_i < a+i\triangle x?

—03

(a) Use the Fundamental Theorem of Calculus to evaluate \int_0^2 f(x) dx.
(b) Suggest a definition of a definite integral.

“Scroll down for the sample solution!”

1 (a) (i)

Lower sum can be calculated by calculate the total area of the inscribed rectangles (refer the diagram).

For n=5, △ x=2/n=2/5=0.4. Area of rectangle is given by f(x) multiply △ x or f(x) multiply 0.4.

1 (a) (ii)

Lower sum can be calculated by calculate the total area of the circumscribed rectangles (refer the diagram).

For n=5, △ x=2/n=2/5=0.4. Area of rectangle is given by f(x) multiply △ x or f(x) multiply 0.4.

1 (a) (iii)

The upper sum is 40.8% larger than the lower sum. The exact area should be between 5.520 and 3.920.

1 (b) (i)

For n=10, △ x=2/n=2/10=0.2. Area of rectangle is given by f(x) multiply △ x or f(x) multiply 0.2.

1 (b) (i)

For n=20, △ x=2/n=2/20=0.1. Area of rectangle is given by f(x) multiply △ x or f(x) multiply 0.1.

1 (b) (i)

For n=40, △ x=2/n=2/40=0.05. Area of rectangle is given by f(x) multiply △ x or f(x) multiply 0.05.

1 (b) (ii)

As n increases the gap between the lower sum and upper sum is getting smaller.

2 (a) (i)

You need to find the proof for the sum \sum_{i=1}^n r^2=\frac{n(n+1)(2n+1)}{6}.

2 (a) (ii)

Prepare the table using the formula provided in part (i)

2 (a) (iii)

Find the limit.

2 (b) (ii)

Prepare the table using the formula provided in part (i)

2 (b) (iii)

Find the limit.

2 (c)

Same value.

3 (a)

Normal integration.

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