Final answer:
The linear approximation is found by calculating the derivative of the function at the point of interest and using the point-slope form to create the tangent line equation. This approximation can then be used to estimate function values near that point without computing the full function.
Step-by-step explanation:
The linear approximation of a function at a given point is a process by which we can estimate the function's values near that point using a linear function. This linear function is the tangent line to the original function at the point of approximation. To find the linear approximation, we first need to calculate the derivative of the function and evaluate it at the point of interest (this gives us the slope of the tangent line). Then, we use the point-slope form of a line to write the equation of this tangent line, which serves as our linear approximation.
The linear approximation, also known as the tangent line approximation or the first-order Taylor approximation, is given by the formula L(x) = f(a) + f'(a)(x-a), where f(a) is the function value and f'(a) is the derivative at the point a. Using this approximation, we can estimate values of the function near a without having to compute the full function, which can be especially handy for complex functions or when a quick, rough estimate is needed.
To estimate the given function value using the approximation, simply plug the value you want to estimate into the linear approximation equation. The result will be a close approximation if the value is near the point a.