Answer:

tangent line to f(x)=1/x at x=2 provides a good approximation to f for x NEAR 2 . (b) At x=2.1 , the VALUE of y on the tangent line to f(x)=1/x is 0.475 . The actual value of f(2.1) is 1/2.1 , which is approximately 0.47619 .

A tangent line to the graph of a function at a point (\(a,f(a)\)) is the line that secant lines through (\(a,f(a)\)) approach as they are TAKEN through points on the function with \(x\)-values that approach \(a\); the slope of the tangent line to a graph at \(a\) measures the rate of change of the function at \(a\)

In GENERAL, for a differentiable function f , the equation of the tangent line to f at x=a can be used to approximate f(x) for x near a . Therefore, we can write

f(x)≈f(a)+f′(a)(x−a) for x near a .

We call the linear function

L(x)=f(a)+f′(a)(x−a)(4.2.1)

the linear approximation, or tangent line approximation, of f at x=a . This function L is also known as the linearization of f at x=a.

To show how useful the linear approximation can be, we look at how to find the linear approximation for f(x)=x−−√ at x=9.