Use Regula Falsi Method to find a real root of the equation xe^x - 2=0 correct to two decimal places within the interval [0,1].
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To use the Regula Falsi method to find a real root of the equation xe^x - 2 = 0 within the interval [0, 1], follow these steps:
Step 1: Define the function f(x) = xe^x - 2.
Step 2: Check the endpoints of the interval. Calculate f(0) and f(1) to determine if there is a root within the interval.
f(0) = (0)e^(0) - 2 = -2
f(1) = (1)e^(1) - 2 ≈ -0.37
Since f(0) is negative and f(1) is slightly negative, there is a root within the interval [0, 1].
Step 3: Compute the value of x1 using the formula:
x1 = x0 - (f(x0)(x1 - x0)) / (f(x1) - f(x0))
In this case, x0 = 0 and x1 = 1.
Step 4: Compute f(x1) and check its sign.
f(0) ≈ -2
f(1) ≈ -0.37
Since f(x1) is negative, the root lies between x0 = 0 and x1 = 1.
Step 5: Compute the next approximation, x2, using the formula:
x2 = x1 - (f(x1)(x2 - x1)) / (f(x2) - f(x1))
In this case, x1 = 1, and we substitute it into the formula to find x2.
Step 6: Repeat steps 4 and 5 until the desired accuracy is reached.
You can continue this process until you achieve the desired accuracy of two decimal places.
To solve the equation xe^x - 2 = 0 using the Regula Falsi method, also known as the False Position method, follow these steps:
Step 1: Set up the interval
Start by selecting an interval [a, b] within which the equation has a real root. In this case, we are given the interval [0, 1].
Step 2: Evaluate the function at the interval endpoints
Evaluate the function f(x) = xe^x - 2 at the endpoints of the interval: f(a) and f(b).
When x = 0, f(0) = 0e^0 - 2 = -2.
When x = 1, f(1) = 1e^1 - 2 = -1 + 2 - 1 = 0.
Since f(a) and f(b) have opposite signs, there must be a real root within the interval [0, 1].
Step 3: Calculate the x-value for the intersection of the secant line
Find the x-value, let's call it c, at which the secant line intersects the x-axis. The secant line is determined by the points (a, f(a)) and (b, f(b)).
c = a - f(a) * (b - a) / (f(b) - f(a))
Plugging in the values:
c = 0 - (-2) * (1 - 0) / (0 - (-2))
c = 0 + 2 * (1 - 0) / 2
c = 1
Step 4: Evaluate the function at the estimated root
Evaluate the function f(x) at the estimated root c.
When x = 1, f(1) = 0.
Step 5: Update the endpoint for the new interval
Now, we need to update either a or b based on the sign of f(c).
Since f(c) = 0, the root itself is found, and we can stop.
Therefore, a real root of the equation xe^x - 2 = 0 within the interval [0, 1] is x = 1, correct to two decimal places.