Linear Approximation

Linear approximation is a method of estimating the value of a function near a given point using the tangent line at that point.

The formula is $$L(x) = f(c) + f'(c)(x - c)$$ .

The slope of the tangent line at point a is given by the derivative, f'(a).

The first step is to find the value of f(a).

The derivative at a point gives the slope of the tangent line, which is used to create the linear approximation.

Using linear approximation, f(4.5) ≈ f(4) + f'(4)(4.5 - 4) = 1 + 3(0.5) = 2.5.

Linear approximation is used to estimate the values of functions that are difficult to compute directly.

Linear approximation uses the tangent line at a point to estimate the value of the function near that point.

You can use the linear approximation of the cube root function near a known value.

The second derivative test is not needed for linear approximation; it is used to determine concavity and inflection points.