Use linear approximation to estimate f (3.85) given that f(4) = 3 and f'(4) = 2.

Verified step by step guidance

1. Linear approximation is a method used to estimate the value of a function near a given point using the tangent line at that point.
2. The formula for linear approximation is: \( L(x) = f(a) + f'(a)(x - a) \), where \( a \) is the point at which the function is known, \( f(a) \) is the function value at \( a \), and \( f'(a) \) is the derivative at \( a \).
3. In this problem, we are given \( f(4) = 3 \) and \( f'(4) = 2 \). We want to estimate \( f(3.85) \).
4. Substitute \( a = 4 \), \( f(a) = 3 \), \( f'(a) = 2 \), and \( x = 3.85 \) into the linear approximation formula: \( L(3.85) = 3 + 2(3.85 - 4) \).
5. Simplify the expression \( L(3.85) = 3 + 2(-0.15) \) to find the estimated value of \( f(3.85) \).

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 a specific input value.

Tangent Line

The tangent line to a function at a given point is a straight line that touches the curve at that point and has the same slope as the function at that point. The equation of the tangent line can be expressed as y = f(a) + f'(a)(x - a), where 'a' is the point of tangency, f(a) is the function value, and f'(a) is the derivative at that point.

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 function's slope and is essential for finding the equation of the tangent line used in linear approximation.