In arithmetic, use a linear approximation to estimate the fee of a wellknown feature (f(x)) via the usage of linear expressions. that is also called tangent line approximation, which is the approach of figuring out the road equation that is nearer estimation for entered linear capabilities at any given fee of x. So, the linear approximation calculator approximates the value of the feature and unearths the spinoff of the function to evaluate the spinoff to locate slope with the help of the linearization method.
A linear approximation equation can simplify the conduct of complicated features. The point \( x = k \) is the correct linear approximation. As we get farther faraway from the factor \( x = k \), the estimation becomes less accurate. A easy curve linear approximation envies the course of the curve. however, it does no longer expect the concavity of any curve.
Example: Find the value of \( f(4.2) \) by using the linear approximation \( x_0 = 4 \), for a differentiable function where \( f(4) = 10 \) and \( f'(4) = 3 \). Solution: Using the linear approximation formula: \[ L(x) \approx f(x_0) + f'(x_0)(x - x_0) \] Substitute the given values into the formula: $$ L(x) = f(4) + f'(4)(x - 4) $$ $$ L(x) = 10 + 3(x - 4) $$ Simplify: $$ L(x) = 10 + 3x - 12 $$ $$ L(x) = 3x - 2 $$ Now, calculate \( f(4.2) \): $$ f(4.2) \approx L(4.2) $$ $$ f(4.2) = 3(4.2) - 2 $$ $$ f(4.2) = 12.6 - 2 $$ $$ f(4.2) = 10.6 $$ **Final Answer:** The approximated value is \( f(4.2) = 10.6 \). Moreover, an Online Integral Calculator helps you to evaluate the integrals of functions with respect to the variable involved.
The online linearization calculator will estimate the values of a given function by using linear approximation formula with the following steps:
