Final answer:
The linear approximation of the expression (1+x)ᵗ is 1 + kx, using the derivative at x=0. None of the provided options match this result, suggesting a possible misprint in the question.
Step-by-step explanation:
When using a linear approximation to estimate the expression for (1+x)ᵗ, we want to find the linear function that best approximates the function at x=0. The linear approximation of a function f(x) at x=a can be found by using the formula f(a) + f'(a)(x-a), where f'(a) represents the derivative of f(x) at x=a. In this case, the function f(x) is (1+x)ᵗ, and we are looking to approximate it at x=0.
The first derivative of the function f(x) = (1+x)ᵗ is f'(x) = k(1+x)ᵗ⁻¹. Applying the linear approximation formula at x=0, we get f(0) + f'(0)(x-0) = 1 + kx, which simplifies to 1 + kx. Therefore, the linear approximation of (1+x)ᵗ is 1 + kx, which is not one of the provided options (a), (b), (c), or (d). This means the question may contain a misprint or need clarification.