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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.

(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)

Find the limit.

##### 2 (b) (ii)

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

Find the limit.

Same value.

##### 3 (a)

Normal integration.

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