Use linear approximation to estimate f (5.1) given that f(5) = 10 and f'(5) = -2.

Verified step by step guidance

1. Identify the function value and derivative at the point of interest: f(5) = 10 and f'(5) = -2.
2. Recall the formula for linear approximation: L(x) = f(a) + f'(a)(x - a), where a is the point of tangency.
3. Substitute the given values into the linear approximation formula: L(x) = 10 + (-2)(x - 5).
4. Set x = 5.1 in the linear approximation formula to estimate f(5.1): L(5.1) = 10 + (-2)(5.1 - 5).
5. Simplify the expression to find the estimated value of f(5.1) using the linear approximation.

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 if a function is differentiable, its behavior can be closely approximated by a linear function in the vicinity of that point. The formula for linear approximation is f(x) ≈ f(a) + f'(a)(x - a), where 'a' is the point of tangency.

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 a fundamental concept in calculus that provides information about the slope of the tangent line to the function at that point. In the context of linear approximation, the derivative at a specific point is used to determine the slope of the tangent line, which is essential for estimating function values nearby.

Function Evaluation

Function evaluation involves calculating the output of a function for a given input. In this context, we are interested in estimating f(5.1) using known values of f(5) and f'(5). Understanding how to evaluate functions and apply the linear approximation formula is crucial for making accurate estimates based on the information provided.