有一條高中數..唔識呀 ! 請救兵！詳細D吾該！

a)Sketch the x – y graph;

The graph should be very simple, it is a parabola x^2 at the range between 1 and 5.

b. For y(x) = x^2
f(1) = 1
f(2) = 4
f(3) = 9
f(4) = 16
f(5) = 25

c)Use Trapezoidal rule to find the area under the curve using 4 strips of equal width;

OK, the trapezoidal rule is: Area under the graph = (b - a)/2n (f(x0) + 2f(x1) + 2f(x2) + ... + f(xn))

For a is the start of the range
b is the end of the range

Now, let's apply the rule to our problem,

Area under the graph = (b - a)/2n (f(x0) + 2f(x1) + 2f(x2) + 2f(x3) + f(x4))

a = 1
b = 5
x0 = 1
x1 = 2
x2 = 3
x3 = 4
x4 = 5
n = number of strips = 4

Area under the graph = (5 - 1)/(2 x 4) x (1 + 2x4 + 2x9 + 2x16 + 25) = 1/2 x (1 + 8 + 18 + 32 +25) = 84/2 = 42

d)Use Simpson’s rule to find the area under the curve using 4 strips of equal width;

Now, the Simpson’s rule is: Area under the graph = (b - a)/3n (f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + ... + f(xn))


For a is the start of the range
b is the end of the range

Let's apply the rule to our problem

Area under the graph = (b - a)/3n (f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + f(x4))

= (5 - 1)/(3 x 4) x (1 + 4x4 + 2x9 + 4x16 + 25) = 1/3 (1 + 16 + 18 + 64 +25) = 124/3

e)Calculate the percentage differences of the answers obtained using the Trapezoidal AND the Simpson’s rules with respect to the exact answer, given that the exact answer is 49.33.

Ok, the exact answer here is WRONG. If you know how to do integration to find out the area under the curve, you should know that the exact answer is [(5^3)/3 - (1^3)/3] = (125 - 1)/3 = 124/3

If we use this as the exact answer,

Percentage differences of the answers obtained using the Trapezoidal rule compare to the exact answer
= (42 - 124/3)/(124/3) = 1.613%

Percentage differences of the answers obtained using the Simpson's rule compare to the exact answer
= (124/3 - 124/3)/(124/3) = 0/(124/3) = 0%