Question:
(i). Approximate the integral, $\int_0^1 \frac{1}{1+x^4} dx$, using trapezoidal rule by dividing the interval $[0, 1]$ into 4 intervals of equal length.
(ii). Let $f$, $g$ be quadratic polynomials. Let $x_0$, $x_1$, $x_2$ be points such that $x_0 < x_1 < x_2$, $x_1 – x_0 = x_2 – x_1$, $f(x_i) = g(x_i)$ for $i = 0, 1$ and $\int_{x_0}^{x_2} f(x) dx = \int_{x_0}^{x_2} g(x) dx$. Use Simpson's rule to prove that $f(x) = g(x)$ for each $x \in \mathbb{R}$.
My answer:
Using Simpson's Rule,
$$h [ f(x_0) + 4f(x_1) + f(x_2) ] = h [ g(x_0) + 4g(x_1) + g(x_2) ]$$
From this, $f(x_2) = g(x_2) $
Since $f(x_0) = g(x_0)$, $f(x_1) = g(x_1)$ , $f(x_2) = g(x_2)$ and this is a quadratic polynomial equation; $f(x) = g(x)$.
Is my answer correct and is it worth for 12 points?
Best Answer
Your objective is always to convince the examiner that you know why your analysis is correct. This requires an explicit reference to the relevant theorems and manual verification that the assumptions are satisfied.
I shall demonstrate. Since Simpson's rule is exact for all polynomials of degree at most 3, and $f$ and $g$ are polynomials of degree 2 with $$\int_{x_1}^{x_2} f(x)dx = \int_{x_1}^{x_2} g(x)dx, \quad h = x_1 - x_0 = x_2 - x_1$$ we conclude that $$h \left( f(x_0) + 4 f(x_1) + f(x_2) \right) = h \left(g(x_0) + 4 g(x_1) + g(x_2)\right).$$ Since $f$ and $g$ agree on $x_0$ and $x_2$ we can immediately deduce that $$f(x_1) = g(x_1).$$ We shall now show that $k=f-g$ is the zero polynomial. Because $f$ and $g$ are quadratic polynomials it is clear that $k$ is a polynomial of degree at most 2. Since $k$ has three distinct roots, namely $x_0$, $x_1$ and $x_2$, the fundamental theorem of algebra implies that $k$ must be the zero polynomial. We conclude that $f(x) = g(x)$ for all $x \in \mathbb{R}$. This completes that proof.
By listing all the salient facts you eliminate any doubt and the examiner has no choice but to award you the maximum number of points. Never shy away from writing more words than the average student. As long as you remain on point no objection can be made.