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(a) Use forward difference approximation to estimate the first derivative of f(x)=4 x (x−3) at x=1.3 with h=0.005. Keep all the computations in six decimal places. (b) Apply two iterations of false position method to approximate the zero of the equation x 2 =sinx in the interval (0.7,1).

User Jamesh
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Explanation:

(a) To estimate the first derivative of the function f(x) = 4x(x-3) at x = 1.3 using forward difference approximation with h = 0.005, we can use the formula:

f'(x) ≈ (f(x + h) - f(x)) / h

Given x = 1.3 and h = 0.005, let's calculate the approximation:

f'(1.3) ≈ (f(1.3 + 0.005) - f(1.3)) / 0.005

First, let's calculate f(1.3):

f(1.3) = 4 * 1.3 * (1.3 - 3) = -8.32

Next, let's calculate f(1.3 + 0.005):

f(1.305) = 4 * 1.305 * (1.305 - 3) = -8.2896

Now, we can compute the forward difference approximation for the first derivative:

f'(1.3) ≈ (-8.2896 - (-8.32)) / 0.005

f'(1.3) ≈ (0.0304) / 0.005

f'(1.3) ≈ 6.08

Therefore, using forward difference approximation with h = 0.005, the estimated value of the first derivative of f(x) = 4x(x-3) at x = 1.3 is approximately 6.08.

(b) To apply two iterations of the false position method to approximate the zero of the equation x^2 = sin(x) in the interval (0.7, 1), let's follow the steps of the method:

Iteration 1:

Let a = 0.7 and b = 1.

Calculate the values of f(a) and f(b):

f(a) = a^2 - sin(a) = 0.49 - sin(0.7) ≈ 0.41612

f(b) = b^2 - sin(b) = 1 - sin(1) ≈ 0.15853

Calculate the value of c using the false position formula:

c = b - ((f(b) * (b - a)) / (f(b) - f(a)))

c ≈ 1 - ((0.15853 * (1 - 0.7)) / (0.15853 - 0.41612))

c ≈ 0.76089

Check the sign of f(c):

f(c) = c^2 - sin(c) = 0.76089^2 - sin(0.76089) ≈ -0.05746

Since f(c) is negative, the zero lies between c and b. Therefore, we update our interval as (c, b).

Iteration 2:

Let a = 0.76089 and b = 1.

Calculate the values of f(a) and f(b):

f(a) = a^2 - sin(a) = 0.5786 - sin(0.76089) ≈ 0.22401

f(b) = b^2 - sin(b) = 1 - sin(1) ≈ 0.15853

Calculate the value of c using the false position formula:

c = b - ((f(b) * (b - a)) / (f(b) - f(a)))

c ≈ 1 - ((0.15853 * (1 - 0.76089)) / (0.15853 - 0.22401))

c ≈ 0.82833

Check

the sign of f(c):

f(c) = c^2 - sin(c) = 0.82833^2 - sin(0.82833) ≈ -0.04243

Since f(c) is negative, the zero still lies between c and b. Therefore, we update our interval as (c, b).

After two iterations of the false position method, the approximated zero of the equation x^2 = sin(x) in the interval (0.7, 1) is approximately 0.82833.

User Goodson
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