Use differentials (or a linear approximation) to estimate (2.001)^5
To estimate (2.001)^5 using differentials or linear approximation, we can start by finding the derivative of the function f(x) = x^5. The derivative of f(x) with respect to x is denoted as f'(x) or df/dx and can be calculated as:
f'(x) = 5x^(5-1) = 5x^4.
Next, we need to choose a value close to x = 2.001 to use as our approximation point. Let's select x = 2 as it is close and easy to work with.
Now, we can use the linear approximation formula:
f(x + Δx) ≈ f(x) + f'(x) * Δx.
Substituting our values into the formula, we have:
f(2.001) ≈ f(2) + f'(2) * (2.001 - 2).
Calculating these values, we find:
f(2) = 2^5 = 32,
f'(2) = 5 * 2^4 = 80,
Δx = 2.001 - 2 = 0.001.
Substituting these values, we have:
f(2.001) ≈ 32 + 80 * 0.001.
Calculating the right-hand side of the equation, we get:
f(2.001) ≈ 32 + 0.08 = 32.08.
Therefore, (2.001)^5 is approximately equal to 32.08 using differentials or linear approximation.