Linear approximation Find the linear approximation to the following functions at the given point a.


f(x) = 4x² + x; a = 1

Verified step by step guidance

1. Identify the function f(x) = 4x² + x and the point a = 1 where we need to find the linear approximation.
2. Recall the formula for the linear approximation of a function at a point a, which is L(x) = f(a) + f'(a)(x - a).
3. Calculate f(a) by substituting x = 1 into the function: f(1) = 4(1)² + 1.
4. Find the derivative of the function, f'(x). For f(x) = 4x² + x, use the power rule to get f'(x) = 8x + 1.
5. Evaluate the derivative at the point a: f'(1) = 8(1) + 1. Substitute f(a) and f'(a) into the linear approximation formula to get L(x) = f(1) + f'(1)(x - 1).

Key Concepts

Linear Approximation

Linear approximation is a method used to estimate the value of a function near a given point using the tangent line at that point. It is based on the idea that a function can be closely approximated by a linear function when evaluated near a specific value. The formula for linear approximation is f(a) + f'(a)(x - a), where f'(a) is the derivative of the function at point a.

Derivative

The derivative of a function at a point measures the rate at which the function's value changes as its input changes. It is defined as the limit of the average rate of change of the function as the interval approaches zero. In the context of linear approximation, the derivative at point a provides the slope of the tangent line, which is crucial for constructing the linear approximation.

Function Evaluation

Function evaluation involves calculating the output of a function for a specific input value. In the context of linear approximation, it is essential to evaluate both the function f(x) and its derivative f'(x) at the point a. This allows us to determine the function's value and the slope of the tangent line, which are necessary components for creating the linear approximation.