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Assignment...
Assignment B: Mathematical Investigation
Numerical methods offer an approximation of solutions to mathematical problems where the analytical methods of solutions may not exist and the information available does not admit direct use of the existing analytical methods.
One of the advantages of a numerical method is that it allows problems with limited analytical solution to be solved. However, the limitations for those methods to work properly require certain condition(s).
In computational mathematics, an iterative method is a mathematical procedure that uses initial value(s) to generate a sequence of improving approximate solutions, in which the $n^{\text {म }}$ approximation is derived from the previous ones. The method is said to be convergent if the corresponding sequence converges for given initial approximation(s). One of the applications of iterative methods is to determine the point of intersection(s) for intersecting curves, which can sometimes be difficult to obtain. Determining the point of intersection(s) is important in the process of evaluating the area and volume of a bounded region. Hence, the aim of this assignment is to illustrate the use of iterative methods in finding the point of intersection(s).
Task 1:
Given that $2 x^{2}-x+e^{2 x}-3=0$.
(a) Rearrange the above equation as $\mathrm{f}(x)=\mathrm{g}(x)$, where $\mathrm{f}$ and $\mathrm{g}$ are functions of $x$. Sketch the graphs of the functions on the same coordinates axes. Hence, show that the equation has two real roots.
(b) Repeat (a) with two other different combinations of functions.
(c) State the intervals where those roots exist.
(d) Verify the existence of the real roots in the intervals.
Task 2:
(a) Use suitable initial approximation(s) for each of the following iterative method to find one of the real roots of the equation $2 x^{2}-x+\mathrm{e}^{2 x}-3=0$ correct to four significant figures.
(i) Newton-Raphson method,
(ii) Fixed-point iteration method,
(iii) Secant method,
(iv) Bisection method.
STPM 2023 - 954/4
7
(b) Based on your findings, complete the following fable.
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Answer from Sia

Posted 4 months ago

Solution by Steps

step 1

Given the equation $2x^2 - x + e^{2x} - 3 = 0$, we need to rearrange it as $f(x) = g(x)$. One possible rearrangement is $f(x) = 2x^2 - x$ and $g(x) = 3 - e^{2x}$

step 2

To sketch the graphs of $f(x) = 2x^2 - x$ and $g(x) = 3 - e^{2x}$, plot both functions on the same coordinate axes

step 3

By plotting the graphs, we observe that the curves intersect at two points, indicating that the equation has two real roots

step 4

Another possible rearrangement is $f(x) = x$ and $g(x) = \frac{3 - e^{2x}}{2x}$

step 5

Sketch the graphs of $f(x) = x$ and $g(x) = \frac{3 - e^{2x}}{2x}$ on the same coordinate axes

step 6

The graphs intersect at two points, confirming the existence of two real roots

step 7

A third rearrangement could be $f(x) = e^{2x}$ and $g(x) = 3 - 2x^2 + x$

step 8

Sketch the graphs of $f(x) = e^{2x}$ and $g(x) = 3 - 2x^2 + x$ on the same coordinate axes

step 9

The graphs intersect at two points, further confirming the existence of two real roots

step 10

To state the intervals where the roots exist, observe the points of intersection from the graphs. For example, if the intersections occur at $x \approx -0.97$ and $x \approx 0.54$, the intervals are $(-1, 0)$ and $(0, 1)$

step 11

Verify the existence of the real roots in the intervals by checking the sign changes of the function $2x^2 - x + e^{2x} - 3$ within the intervals

Answer

The equation $2x^2 - x + e^{2x} - 3 = 0$ has two real roots, approximately at $x \approx -0.97$ and $x \approx 0.54$.

Key Concept

Rearranging and graphing functions to find intersections

Explanation

By rearranging the given equation into different forms and plotting the graphs, we can visually identify the points of intersection, which correspond to the real roots of the equation.

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