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Local Linearity &
Linearization

Learn how to zoom into a curve until it looks like a straight line, then use that tangent line to approximate function values near a point.

AP Calculus AB

What Is Local Linearity?

If a function \( f \) is differentiable at a point \(x = a\), then when you zoom in closely enough around that point, the graph of \( f \) becomes virtually indistinguishable from a straight line: the tangent line at \( a \).

Zooming In on a Curve

x y y = f(x) tangent line
\((a, f(a))\)
 ZOOMED IN

Why Does This Matter?

Approximation
Compute hard-to-evaluate expressions like \(\sqrt{4.02}\) without a calculator.
Error Analysis
Understand how small changes in input produce small changes in output via \(\Delta y \approx f'(a)\,\Delta x\).
Foundation
Linearization is the first-degree Taylor polynomial: the gateway to higher-order approximations.

The Linearization Formula

The linearization of \( f \) at \( x = a \) is the tangent-line function:

This is simply the point-slope form of the tangent line at \( (a,\, f(a)) \) with slope \( f'(a) \). For values of \( x \) near \( a \):

Linearization Visualized

x y 1 2 3 4 5 error
\(f(x)\)
\(L(x)\)
\((a, f(a))\)

Tangent Line vs. Linearization

Tangent Line Linearization
Form \( y - f(a) = f'(a)(x - a) \) \( L(x) = f(a) + f'(a)(x - a) \)
What it is An equation of a line A function that approximates \( f \)
Purpose Describe the line itself Approximate \( f(x) \) near \( a \)
Same math? Yes: identical formula, different perspective

Differentials: \( dx \) and \( dy \)

Differentials give us a compact notation for the linear change along the tangent line:

\( dx \)
A small change in \( x \): we set \( dx = \Delta x = x - a \).
\( dy \)
The corresponding change along the tangent line: \( dy = f'(a) \, dx \).

How to Apply Linearization

1
Identify a nearby "nice" point \( a \) where \( f(a) \) and \( f'(a) \) are easy to compute.
2
Compute \( f(a) \): the function value at the nice point.
3
Compute \( f'(a) \): the derivative at that same point.
4
Build the linearization: \( L(x) = f(a) + f'(a)(x - a) \).
5
Plug in your target \( x \): evaluate \( L(x) \) for the approximation.

Example 1: Approximate \( \sqrt{4.1} \)

Let \( f(x) = \sqrt{x} \). We want \( f(4.1) \). Choose \( a = 4 \) because \( \sqrt{4} = 2 \) is exact.

1
\( f(a) = f(4) = \sqrt{4} = \) \( 2 \)
2
\( f'(x) = \frac{1}{2\sqrt{x}} \), so \( f'(4) = \frac{1}{2 \cdot 2} = \) \( 0.25 \)
3
\( L(x) = 2 + 0.25(x - 4) \)
4
\( L(4.1) = 2 + 0.25(0.1) = 2 + 0.025 = \) \( 2.025 \)

Example 2: Approximate \( \sin(0.1) \)

Let \( f(x) = \sin x \). Choose \( a = 0 \) because \( \sin(0) = 0 \) and \( \cos(0) = 1 \) are easy.

1
\( f(0) = \sin(0) = \) \( 0 \)
2
\( f'(x) = \cos x \), so \( f'(0) = \cos(0) = \) \( 1 \)
3
\( L(x) = 0 + 1 \cdot (x - 0) = x \)
4
\( L(0.1) = \) \( 0.1 \)

Example 3: Approximate \( (1.02)^5 \)

Let \( f(x) = x^5 \). We want \( f(1.02) \). Choose \( a = 1 \).

1
\( f(1) = 1^5 = \) \( 1 \)
2
\( f'(x) = 5x^4 \), so \( f'(1) = \) \( 5 \)
3
\( L(x) = 1 + 5(x - 1) \)
4
\( L(1.02) = 1 + 5(0.02) = 1 + 0.10 = \) \( 1.10 \)

Over- or Under-Estimate?

Concave Up Underestimate Concave Down Overestimate

The Concavity Rule

\( f''(a) > 0 \)
Concave up near \( a \). Tangent lies below the curve. Linearization is an underestimate: \( L(x) \le f(x) \).
\( f''(a) < 0 \)
Concave down near \( a \). Tangent lies above the curve. Linearization is an overestimate: \( L(x) \ge f(x) \).

Common Mistakes

✕ Wrong Base Point
Choosing an \( a \) that is far from the target \( x \) or where \( f(a) \) is hard to compute. Always pick the nearest "nice" value.
✕ Forgetting \( f'(a) \)
Using \( f'(x) \) instead of \( f'(a) \). The slope must be evaluated at the base point, not at the target.
✕ Sign of \( (x - a) \)
Getting the sign wrong when \(x < a\). Remember: if your target is below the base point, \( (x - a) \) is negative.

Summary Reference

Concept Formula / Rule
Linearization \( L(x) = f(a) + f'(a)(x - a) \)
Approximation \( f(x) \approx L(x) \) when \( x \approx a \)
Differential \( dy = f'(a)\,dx \)
Underestimate \( f''(a) > 0 \) (concave up)
Overestimate \( f''(a) < 0 \) (concave down)
Best near \( a \) Error grows as \(|x - a|\) increases

Question 1 of 5

What is the linearization of \( f(x) = \sqrt{x} \) at \( a = 9 \)?

Question 2 of 5

Using the linearization of \( f(x) = \cos x \) at \( a = 0 \), what is the approximate value of \( \cos(0.05) \)?

Question 3 of 5

If \( f(x) = e^x \) and we use the linearization at \( a = 0 \), is the estimate for \( f(0.5) \) an overestimate or underestimate?

Question 4 of 5

Let \( g(x) = \ln x \). Using linearization at \( a = 1 \), approximate \( \ln(1.1) \).

Question 5 of 5

If \( f(3) = 7 \) and \( f'(3) = -2 \), use linearization to estimate \( f(3.1) \).