Find the first four iterates of the function using the given value.
f(x)= x(2.5 - x); x sub zero = 3
To find the iterates of the function, we need to repeatedly apply the function to the initial value. The first iterate is obtained by substituting the initial value into the function:
f(x_0) = x_0(2.5 - x_0)
By substituting x_0 = 3 into the function, we get:
f(3) = 3(2.5 - 3)
= 3(-0.5)
= -1.5
So, the first iterate is -1.5.
To find the second iterate, we substitute the first iterate (-1.5) back into the function:
f(f(x_0)) = f(-1.5) = -1.5(2.5 - (-1.5))
= -1.5(2.5 + 1.5)
= -1.5(4)
= -6
So, the second iterate is -6.
To find the third iterate, we substitute the second iterate (-6) back into the function:
f(f(f(x_0))) = f(-6) = -6(2.5 - (-6))
= -6(2.5 + 6)
= -6(8.5)
= -51
So, the third iterate is -51.
To find the fourth iterate, we substitute the third iterate (-51) back into the function:
f(f(f(f(x_0)))) = f(-51) = -51(2.5 - (-51))
= -51(2.5 + 51)
= -51(53.5)
= -2728.5
So, the fourth iterate is -2728.5.
Therefore, the first four iterates of the function f(x) = x(2.5 - x) with x_0 = 3 are -1.5, -6, -51, and -2728.5.
To find the iterates of the function, we will start with the initial value x₀ = 3 and repeatedly apply the function f(x) = x(2.5 - x). Let's calculate the first four iterates step by step:
Step 1:
Plug in x₀ = 3 into the function:
f(3) = 3(2.5 - 3) = 3(-0.5) = -1.5
Step 2:
Plug in the result from step 1 into the function:
f(-1.5) = (-1.5)(2.5 - (-1.5)) = (-1.5)(2.5 + 1.5) = (-1.5)(4) = -6
Step 3:
Plug in the result from step 2 into the function:
f(-6) = (-6)(2.5 - (-6)) = (-6)(2.5 + 6) = (-6)(8.5) = -51
Step 4:
Plug in the result from step 3 into the function:
f(-51) = (-51)(2.5 - (-51)) = (-51)(2.5 + 51) = (-51)(53.5) = -2728.5
Therefore, the first four iterates of the function f(x) using the initial value x₀ = 3 are:
x₁ = -1.5
x₂ = -6
x₃ = -51
x₄ = -2728.5