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Math
Posted 21 days ago

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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 nम 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 2x2x+e2x3=02 x^{2}-x+e^{2 x}-3=0.
(a) Rearrange the above equation as f(x)=g(x)\mathrm{f}(x)=\mathrm{g}(x), where f\mathrm{f} and g\mathrm{g} are functions of xx. 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 2x2x+e2x3=02 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 21 days ago
Solution by Steps
step 1
Given the equation 2x2x+e2x3=02x^2 - x + e^{2x} - 3 = 0, we need to rearrange it as f(x)=g(x)f(x) = g(x). One possible rearrangement is f(x)=2x2xf(x) = 2x^2 - x and g(x)=3e2xg(x) = 3 - e^{2x}
step 2
To sketch the graphs of f(x)=2x2xf(x) = 2x^2 - x and g(x)=3e2xg(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)=xf(x) = x and g(x)=3e2x2xg(x) = \frac{3 - e^{2x}}{2x}
step 5
Sketch the graphs of f(x)=xf(x) = x and g(x)=3e2x2xg(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)=e2xf(x) = e^{2x} and g(x)=32x2+xg(x) = 3 - 2x^2 + x
step 8
Sketch the graphs of f(x)=e2xf(x) = e^{2x} and g(x)=32x2+xg(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 x0.97x \approx -0.97 and x0.54x \approx 0.54, the intervals are (1,0)(-1, 0) and (0,1)(0, 1)
step 11
Verify the existence of the real roots in the intervals by checking the sign changes of the function 2x2x+e2x32x^2 - x + e^{2x} - 3 within the intervals
Answer
The equation 2x2x+e2x3=02x^2 - x + e^{2x} - 3 = 0 has two real roots, approximately at x0.97x \approx -0.97 and x0.54x \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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