Local linearity and error direction | Code Scholars Tutoring

Understand the idea

The tangent line provides a local approximation near a point with known function and derivative values. The differential estimates output change from input change. Concavity determines whether the tangent estimate is above or below the true value locally. An estimate is most useful near the base point; carry the input change with its sign.

L(x)=f(a)+f'(a)(x-a),\qquad dy=f'(a)\,dx

Worked example

Approximate $\displaystyle \sqrt{4.1}$ using the tangent at $\displaystyle x=4$ .

For

\displaystyle f(x)=\sqrt{x}

\displaystyle f(4)=2

and

\displaystyle f'(4)=1/4

. Thus

\displaystyle L(4.1)=2+(1/4)(0.1)=2.025

. Since

\displaystyle f''(x)<0

for

\displaystyle x>0

, the tangent line overestimates the square root.

Try it yourself

Use linearization of $\displaystyle \ln x$ at $\displaystyle 1$ to estimate $\displaystyle \ln(0.98)$ .

Show solution and reasoning

$\displaystyle L(x)=x-1$ , so the estimate is $\displaystyle -0.02$ . Because $\displaystyle \ln x$ is concave down, this is an overestimate.

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