Local linearization

Warm-up

Three quick questions from earlier lessons.

Discover

The violet curve is $y = \sqrt{x}$ ; the blue line is its tangent at $x = a$ . Move $a$ : near the point of contact, curve and line almost agree.

The curve y = √x and its tangent line at x = a. Near x = a the two are almost the same; farther away the line rises above the curve.

At $a = 4$ the tangent is $L(x) = 2 + \tfrac{1}{4}(x - 4)$ . Use it to estimate $\sqrt{4.4}$ .

2.1
2.4

Key idea

Near $x = a$ , a differentiable function is almost its tangent line.

The local linearization of

f

at

x = a

is

L(x) = f(a) + f'(a)(x - a)

, and

f(x) \approx L(x)

for

x

near

a

. If

f

is concave up near

a

,

L

underestimates; if concave down, it overestimates.

Worked example

$f(x) = \sqrt{x}$ , $f'(x) = \dfrac{1}{2\sqrt{x}}$ . Estimate $\sqrt{9.6}$ .

f(9) = 3

,

f'(9) = \tfrac{1}{6}

A nearby point with an exact root: a = 9.

L(9.6) = 3 + \tfrac{1}{6}(0.6) = 3.1

L(x) = f(9) + f'(9)(x − 9).

\sqrt{9.6} \approx 3.1

(true value

3.098…

)

√x is concave down: the estimate is a little too big.

Your turn, guided

Your turn: $f(2) = 7$ and $f'(2) = 3$ . Estimate $f(2.1)$ .

f(2.1) \approx 7 + 3(0.1) = 7.3

L(x) = 7 + 3(x − 2).

Practice ★

$f(2) = 3$ and $f'(2) = 2$ . Write the local linearization $L(x)$ of $f$ at $x = 2$ .
$f(4) = -5$ and $f'(4) = -4$ . Write the local linearization $L(x)$ of $f$ at $x = 4$ .
$f(4) = 4$ and $f'(4) = -1$ . Write the local linearization $L(x)$ of $f$ at $x = 4$ .

Watch out!

Practice ★★

$f(x) = \sqrt{x}$ has $f'(x) = \dfrac{1}{2\sqrt{x}}$ . Use the local linearization at $x = 4$ to estimate $\sqrt{ 6 }$ . Give an exact fraction.
$f(x) = \sqrt{x}$ has $f'(x) = \dfrac{1}{2\sqrt{x}}$ . Use the local linearization at $x = 4$ to estimate $\sqrt{ 8 }$ . Give an exact fraction.

Near $x = 4$ , the graph of $f$ is concave up. Does the local linearization at $x = 4$ over- or underestimate $f(x)$ for $x$ near $4$ ?
1. Underestimate: the tangent line lies below the curve.
2. Overestimate: the tangent line lies above the curve.
3. Neither: near $x = 4$ the linearization is exact.

Exit ticket

Three questions to check you have got it.

$f(2) = 3$ and $f'(2) = 2$ . Write the local linearization $L(x)$ of $f$ at $x = 2$ .
$f(4) = -5$ and $f'(4) = -4$ . Write the local linearization $L(x)$ of $f$ at $x = 4$ .
$f(4) = 4$ and $f'(4) = -1$ . Write the local linearization $L(x)$ of $f$ at $x = 4$ .

Summary card

Local linearization

$L(x) = f(a) + f'(a)(x - a)$ , the tangent line used as an estimate.
Good near $a$ ; concave up underestimates, concave down overestimates.

Watch out: multiply the step $x - a$ by the slope $f'(a)$ .

About this lesson