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.