Local Linearity and Linearization

Part 1 · Hook

Why this matters

Zoom in far enough on a smooth curve and it looks straight. That is why the ground looks flat even though Earth is round. Near a point, a differentiable function is almost its tangent line, and that line gives quick, good estimates. Knowing whether such an estimate is too high or too low is a classic free-response point.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. The line tangent to $y = x^2$ at $x = 3$ is

$y = 9 + 6(x - 3)$
$y = 6 + 9(x - 3)$
$y = 9 + 3(x - 3)$
$y = 6x$

Show the answer

Point $(3, 9)$ , slope $f'(3) = 6$ .

Correct: $y = 9 + 6(x - 3)$ :
$y = 6 + 9(x - 3)$ :
$y = 9 + 3(x - 3)$ :
$y = 6x$ :

2. If $f(x) = \sqrt{x}$ , then $f''(x)$ is

$-\frac{1}{4}x^{-3/2}$
$\frac{1}{2}x^{-1/2}$
$\frac{1}{4}x^{-3/2}$
$-\frac{1}{2}x^{-3/2}$

Show the answer

$f'(x) = \frac{1}{2}x^{-1/2}$ , then $f''(x) = -\frac{1}{4}x^{-3/2}$ .

Correct: $-\frac{1}{4}x^{-3/2}$ :
$\frac{1}{2}x^{-1/2}$ :
$\frac{1}{4}x^{-3/2}$ :
$-\frac{1}{2}x^{-3/2}$ :

Part 4 · See it

See it first

The graph of y = √x (crimson) and its tangent line at (4, 2), L(x) = 2 + (x − 4)/4 (blue). At x = 7 the line gives 2.75 while √7 ≈ 2.646; a zoomed panel shows the line above the concave down curve, so the estimate is an overestimate. — The tangent line to $\sqrt{x}$ at $x = 4$ gives $L(7) = 2.75$ , slightly above $\sqrt{7}$ : the graph is concave down, so the line lies above it. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

Near $x = a$ , a differentiable graph is nearly straightit is close to its tangent line $L(x)$ $= f(a) + f'(a)(x - a)$
So for $x$ near $a$ $f(x)$ $\approx f(a) + f'(a)(x - a)$ , the tangent line approximation
If $f'' > 0$ between $a$ and $x$ , the graph is concave up and lies above the linethe approximation is an underestimate
If $f'' < 0$ there, the graph is concave down and lies below the linethe approximation is an overestimate

Part 6 · Key ideas

Key ideas

The linearization at $a$ : $L(x)$ $= f(a) + f'(a)(x - a)$ . Use the point closest to where you estimate.
Concave up ( $f'' > 0$ ): tangent line below the graph, underestimate. Concave down ( $f'' < 0$ ): overestimate.
Justify with the concavity on the interval between $a$ and $x$ , not with the sign of $f'$ .
The closer $x$ is to $a$ , the better the approximation.

Part 7 · Misconception

A common mistake

The wrong idea: The tangent line approximation overestimates because $f$ is increasing.

What actually happens: Whether $f$ increases decides which way the line slopes, not which side of the graph it is on. Concavity decides: concave down means overestimate, concave up means underestimate.

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Table

Values of $f$ , $f'$ and $f''$

Selected values of a twice-differentiable function f. Also, f''(x) > 0 for 1.5 ≤ x ≤ 2.5 and f''(x) < 0 for 4.5 ≤ x ≤ 5.5.
$x$	$f(x)$	$f'(x)$	$f''(x)$
$2$	$5$	$-3$	$4$
$5$	$1$	$2$	$-1$

1. Using the table, write the linearization $L(x)$ of $f$ at $x = 2$ , in the form $y = \ldots$

Type an expression for $y =$ .

Show the answer

The point is $(2, f(2)) = (2, 5)$ and the slope is $f'(2) = -3$ : $L(x) = 5 - 3(x - 2)$ .

Answer: $y =$ $5 - 3(x - 2)$

2. Use the tangent line at $x = 2$ to approximate $f(2.1)$ .

No calculator. Type the exact value (a fraction, π, e or a root is fine).

Show the answer

$f(2.1)$ $\approx L(2.1)$ $= 5 - 3(0.1)$ $= 4.7$ .

Answer: $\frac{47}{10}$ ≈ 4.700

3. Is the approximation of $f(2.1)$ from the tangent line an overestimate or an underestimate?

An underestimate, because $f'' > 0$ makes $f$ concave up there
An overestimate, because $f'(2) < 0$
An overestimate, because $f'' > 0$ there
Neither: the tangent line gives the exact value of $f(2.1)$

Show the answer

On $1.5 \le x \le 2.5$ , $f'' > 0$ , so the graph is concave up and lies above the tangent line. $L(2.1)$ is below $f(2.1)$ : an underestimate.

Correct: An underestimate, because $f'' > 0$ makes $f$ concave up there: A concave up graph lies above its tangent lines.
An overestimate, because $f'(2) < 0$ : The sign of $f'$ says which way the line slopes, not which side of the graph it lies on.
An overestimate, because $f'' > 0$ there: Concave up puts the tangent line below the graph, giving an underestimate.
Neither: the tangent line gives the exact value of $f(2.1)$ : The line matches $f$ only at $x = 2$ ; elsewhere the graph bends away from it.

4. Use the line tangent to the graph of $f$ at $x = 5$ to approximate $f(4.8)$ .

No calculator. Type the exact value (a fraction, π, e or a root is fine).

Show the answer

$L(x) = 1 + 2(x - 5)$ , so $f(4.8)$ $\approx 1 + 2(-0.2)$ $= 0.6$ .

Answer: $\frac{3}{5}$ ≈ 0.600

5. Is the approximation of $f(4.8)$ an overestimate or an underestimate?

An underestimate, because $f'' < 0$ there
An overestimate, because $f'' < 0$ makes $f$ concave down there
An underestimate, because 4.8 lies to the left of the point $x = 5$
An overestimate, because $f'(5) > 0$

Show the answer

On $4.5 \le x \le 5.5$ , $f'' < 0$ : concave down, so the tangent line lies above the graph and $L(4.8)$ overestimates $f(4.8)$ .

An underestimate, because $f'' < 0$ there: Concave down puts the tangent line above the graph.
Correct: An overestimate, because $f'' < 0$ makes $f$ concave down there: A concave down graph lies below its tangent lines.
An underestimate, because 4.8 lies to the left of the point $x = 5$ : The side of the point does not matter; the concavity does.
An overestimate, because $f'(5) > 0$ : The sign of $f'$ does not decide; the reason must be about $f''$ .

6. If the graph of $f$ is concave up on an interval containing $x = a$ , the line tangent to the graph at $x = a$

lies above the graph near $a$ , so it overestimates $f(x)$
crosses the graph at $x = a$
lies below the graph near $a$ , so it underestimates $f$
gives exact values near $a$

Show the answer

Concave up: the graph bends upward, away from the tangent line, so the line lies below it and tangent line approximations are underestimates.

lies above the graph near $a$ , so it overestimates $f(x)$ : That happens when the graph is concave down.
crosses the graph at $x = a$ : With constant concavity, the graph stays on one side of its tangent line near $a$ .
Correct: lies below the graph near $a$ , so it underestimates $f$ : Concave up graphs bend up away from their tangent lines.
gives exact values near $a$ : It matches $f$ exactly only at $x = a$ .

Part 9 · Summary

Summary

Near

x = a

,

f(x)

\approx L(x)

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

. The tangent line lies below a concave up graph (

f'' > 0

), giving an underestimate, and above a concave down graph (

f'' < 0

), giving an overestimate. The estimate is best close to

a

.

Part 10 · Up next

What comes next

Part 11 · Connections

Why this matters

What this builds on

Quick check before you start

See it first

How it works, step by step

Key ideas

A common mistake

Check yourself

Values of ff, f′f' and f″f''

Summary

What comes next

Connections

Values of $f$ , $f'$ and $f''$