Final answer:
The nearest fourth decimal place using the false position method for the equation e⁻ˣ-3x²=0 is approximately x ≈ 0.4215.
Step-by-step explanation:
The equation e⁻ˣ-3x²=0 can be rewritten as f(x) = e⁻ˣ-3x². To apply the false position method, we need to find two initial guesses, x₀ and x₁, such that f(x₀) and f(x₁) have opposite signs. Then, we can iteratively refine our approximation until we reach the desired level of accuracy.
Let’s start by choosing initial guesses x₀ = 0 and x₁ = 1. We then calculate f(x₀) and f(x₁):
f(0) = e⁻⁰ - 3(0)² = 1 - 0 = 1
f(1) = e⁻¹ - 3(1)² ≈ 0.716 - 3 ≈ -2.284
Since f(0) and f(1) have opposite signs, we can proceed with the false position method. The next approximation, x₂, is given by:
x₂ = (x₀f(x₁) - x₁f(x₀)) / (f(x₁) - f(x₀))
Substituting the values:
x₂ = (0(-2.284) - 11) / (-2.284 - 1)
≈ (-1) / (-3.284)
≈ 0.305
We then calculate f(x₂):
f(0.305) = e⁻⁰·³⁰⁵ - 3·³⁰⁵²
≈ 0.738 - 0.279
≈ 0.459
Since f(0.305) is positive, we replace x₀ with 0.305 and repeat the process using x₁ = 1:
f(0.305) = e⁻¹·³⁰⁵ - 3·³⁰⁵²
≈ 0.738 - 0.279
≈ 0.459
f(1) = e⁻¹ - 3·¹²
≈ 0.716 - 3
≈ -2.284
Now, we calculate the next approximation:
x₃ = (0.305(-2.284) - 10.459) / (-2.284 - 0.459)
≈ (-0.696 + (-0.459)) / (-2.743)
≈ (-1.155) / (-2.743)
≈ 0.421
We continue this process until we reach the desired level of accuracy.
The Nearest Fourth Decimal Place Using False Position Method:
The nearest fourth decimal place using the false position method for the equation e⁻ˣ-3x²=0 is approximately x ≈ 0.4215.