Use the linear approximation (1+x)^k=1+kx to find an approximation for the function (3+3x)^(1/3) for values of x near zero.
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To use the linear approximation `(1+x)^k ≈ 1 + kx` to find an approximation for the function `(3+3x)^(1/3)` for values of x near zero, we need to follow these steps:
Step 1: Identify the values needed for the approximation
We are considering values of x near zero, which means we want to find an approximation for `(3+3x)^(1/3)` when x is close to zero.
Step 2: Define the function and its variables
Let's define the function we want to approximate as `f(x) = (3+3x)^(1/3)`.
Step 3: Choose a base point
Select a value of x for which the calculation can be easily done. In this case, we choose `x = 0`.
Step 4: Calculate the first derivative
Calculate the first derivative of the function `f(x)` to find the slope at the base point x=0. In this case, we take the derivative of `f(x)` with respect to x using the power rule:
f'(x) = d/dx [(3+3x)^(1/3)]
= (1/3)(3+3x)^(-2/3)(3)
= (1/3)(3+3x)^(-2/3)
Step 5: Evaluate the first derivative at the base point
Substitute the base point `x = 0` into the first derivative expression obtained in the previous step:
f'(0) = (1/3)(3+3(0))^(-2/3)
= (1/3)(3)^(-2/3)
= (1/3)(1/3)
= 1/9
Step 6: Construct the linear approximation equation
Using the linear approximation formula `(1+x)^k ≈ 1 + kx`, we can write the linear approximation equation as:
f(x) ≈ f(0) + f'(0)(x - 0)
≈ (3+3(0))^(1/3) + (1/9)(x - 0)
≈ 3^(1/3) + (1/9)x
So, the linear approximation for the function `(3+3x)^(1/3)` near x=0 is approximately given by `f(x) ≈ 3^(1/3) + (1/9)x`.