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Discussion – 

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STPM PBS Assignment

STPM 2017 Mathematics (M) Term 1 Assignment

By KK LEE

July 6, 2016
STPM 2017 Term 1 Assignment MM
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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KK LEE

KK LEE has been a STPM Mathematics tuition teacher since June 2006, but his love of Maths dates back to at least 1999 when he was Form 4. KK LEE started teaching in 2006 at Pusat Tuisyen Kasturi. He was known as "LK" when he was teaching in PTK. After teaching STPM Mathematics for 8 years in PTK, he joined Ai Tuition Centre in 2014. Over the years he has taught Mathematics (T), Mathematics (S), Mathematics (M), Additional Mathematics.

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