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Local Linearity &
Linearization

Learn how to approximate function values using tangent lines: the powerful idea that every smooth curve looks like a straight line when you zoom in close enough.

AP Calculus BC

What is Local Linearity?

If you zoom in far enough on any differentiable function at a point, the curve becomes virtually indistinguishable from a straight line: its tangent line at that point.

zoom in here tangent line f(x) y x

Why Does Local Linearity Work?

The definition of the derivative tells us that near \( x = a \), the difference between the function and a linear approximation shrinks faster than the distance from \( a \):

This means the error of the linear approximation vanishes so quickly that the tangent line and the curve become indistinguishable at the point of tangency.

When Does It Fail?

Local linearity requires the function to be differentiable at the point. It breaks down at:

Corners
Like \( f(x)=|x| \) at \( x=0 \): left and right slopes disagree.
Cusps
Like \( f(x)=x^{2/3} \) at \( x=0 \): the tangent line becomes vertical.
Discontinuities
If \( f \) is not even continuous at \( a \), no tangent line can exist.

The Linearization Formula

This is simply the equation of the tangent line to \( f \) at the point \( (a, f(a)) \), written in point-slope form. To approximate \( f(x) \) near \( a \):

Visualizing \( L(x) \)

(a, f(a)) a f(a) L(x) f(x) y x

Linearization vs. Differentials

Linearization Differentials
Formula \( L(x) = f(a) + f'(a)(x-a) \) \( dy = f'(x)\,dx \)
Gives you Approximate value of \( f(x) \) Approximate change in \( f \)
Key variable \( x \) (the input) \( dx \) (the change in input)
Relationship \( L(x) = f(a) + dy \) where \( dx = x - a \)

How to Apply Linearization

1
Choose the base point \( a \): Pick a value near the target where you can compute \( f(a) \) and \( f'(a) \) exactly.
2
Compute \( f(a) \): Evaluate the function at the base point.
3
Compute \( f'(a) \): Find the derivative and evaluate it at \( a \).
4
Build \( L(x) \): Write \( L(x) = f(a) + f'(a)(x - a) \).
5
Substitute: Plug in the target value for \( x \) to get your 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 known exactly.

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

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

Let \( f(x) = \sin x \). Choose \( a = 0 \) since \( \sin(0) \) and \( \cos(0) \) are known.

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

Example 3: \( e^{0.05} \) Approximation

Let \( f(x) = e^x \) with base point \( a = 0 \).

\( f(0) \)
\( e^0 = 1 \)
\( f'(0) \)
\( e^0 = 1 \)
\( L(x) \)
\( 1 + 1 \cdot (x - 0) = 1 + x \)

Over or Under Estimate?

The concavity of \( f \) at \( a \) tells you whether the tangent line sits above or below the curve:

Concave Up (\( f''(x) > 0 \))
Tangent line lies below the curve: linearization is an underestimate.
Concave Down (\( f''(x) < 0 \))
Tangent line lies above the curve: linearization is an overestimate.
Concave Up underestimate Concave Down overestimate

Bounding the Error

If \( f''(x) \) is continuous near \( a \), the error in linearization satisfies:

This is the Taylor remainder for the first-order approximation. The error grows like the square of the distance from \( a \), so staying close matters.

Must-Know Linearizations at \( a = 0 \)

\( f(x) \) \( L(x) \approx \) Estimate Type
\( \sin x \) \( x \) Depends on sign of \(x\)
\( \cos x \) \( 1 \) Overestimate
\( e^x \) \( 1 + x \) Underestimate
\( \ln(1+x) \) \( x \) Overestimate
\( (1+x)^n \) \( 1 + nx \) Depends on \( n \)
\( \tan x \) \( x \) Depends on sign of \(x\)

Common Mistakes

✕ Wrong Base Point
Choosing an \( a \) where you can't compute \( f(a) \) exactly, or one too far from the target value.
✕ Forgetting the Derivative
Writing \( L(x) = f(a)(x - a) \) instead of \( f(a) + f'(a)(x - a) \). Both parts are needed.
✕ Confusing \( x \) and \( dx \)
Using \( x = 0.1 \) when you mean \( dx = x - a = 0.1 \). Mixing up absolute vs. relative input.

Question 1 of 5

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

Question 2 of 5

Using linearization, approximate \( e^{-0.02} \).

Question 3 of 5

If \( f \) is concave up near \( x = a \), the linearization \( L(x) \) is:

Question 4 of 5

Let \( g(x) = \ln x \). Use the linearization at \( a = 1 \) to approximate \( \ln(1.05) \).

Question 5 of 5

The linearization of \( f(x) = (1+x)^4 \) at \( a = 0 \) is: