Linear Approximation and Differentials — Mathematics Knowledge System

Overview

Linear approximation (also called linearization) uses the tangent line to approximate f(x) near a point a: L(x) = f(a) + f'(a)*(x-a). Near a, f(x) ~= L(x). This approximation is valid when x is close to a and f is smooth. The related concept of differentials: if y = f(x), then dy = f'(x)*dx formalizes how a small change dx in x causes a change dy in y. Linear approximation underlies Newton's method, numerical root-finding, and sensitivity analysis in applied mathematics.

Intuition

Zoom in on any smooth function near a point: the graph looks more and more like a straight line (the tangent line). The tangent line IS the linear approximation. For x close to a, f(x) ~= f(a) + f'(a)*(x-a) because the tangent line and the curve are nearly indistinguishable at small scale. Example: sqrt(4.1) ~= sqrt(4) + (1/(2*sqrt(4)))*(4.1-4) = 2 + (1/4)*0.1 = 2.025. The true value is 2.0248... -- excellent approximation!

Formal Definition

Definition

If f is differentiable at a, the linearization (linear approximation) of f at a is: L(x) = f(a) + f'(a)*(x-a). This is the first-order Taylor polynomial. The error: f(x) - L(x) = o(|x-a|) as x->a, meaning the error grows slower than linearly in |x-a|. If f'' is continuous, the error is O(|x-a|^2): |f(x)-L(x)| <= (M/2)*(x-a)^2 where M = max|f''| on [a,x]. Differentials: if y=f(x), the differential is dy = f'(x) dx, interpreting dx as an increment and dy as the corresponding tangent-line increment.

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

Linearization at a

f(x) \approx L(x) \text{ for } x \approx a

Approximation

dy = f'(x)\,dx

Differential

|f(x) - L(x)| \le \frac{M}{2}(x-a)^2,\quad M = \max|f''|

Error bound (Taylor)

Notation

Notation	Meaning
$L(x) = f(a) + f'(a)(x-a)$	Linearization (tangent line)
$dy = f'(x)\,dx$	Differentials
$\Delta y = f(x+\Delta x) - f(x)$	Actual change in y

Theorems

Theorem 1: Linear Approximation Error

If f is differentiable at a, then f(x) = L(x) + o(|x-a|) as x->a (the error is little-o of |x-a|). More precisely, if f'' exists and is continuous on [a,b], then for x in [a,b]: |f(x) - L(x)| <= (1/2)*max_{[a,b]}|f''|*(x-a)^2. This is a consequence of the Lagrange form of the remainder in Taylor's theorem.

Theorem 2: Standard Linear Approximations near 0

For x near 0: (1+x)^n ~= 1 + nx; sin(x) ~= x; cos(x) ~= 1; e^x ~= 1+x; ln(1+x) ~= x; 1/(1-x) ~= 1+x. These all follow from f(0) + f'(0)*x with: f'(x)=n(1+x)^{n-1}, f'(x)=cos(x), f'(x)=-sin(x), f'(x)=e^x, f'(x)=1/(1+x), f'(x)=1/(1-x)^2 respectively.

Theorem 3: Propagation of Error via Differentials

If a measured quantity x has uncertainty dx, and y=f(x) is derived from it, then the uncertainty in y is approximately |dy| = |f'(x)|*|dx|. For products y=u*v: dy/y = du/u + dv/v (relative errors add). For y=x^n: dy/y = n*dx/x.

Worked Examples

1
f(x) = (1+x)^5 near a=0. f(0) = 1, f'(x) = 5(1+x)^4, f'(0) = 5.
2
L(x) = 1 + 5x. With x = 0.02: L(0.02) = 1 + 0.1 = 1.1.
$L(0.02) = 1 + 5(0.02) = 1.10$
3
True value: (1.02)^5 = 1.1040808... The approximation 1.10 has error ~0.004.
$(1.02)^5 \approx 1.10$

✓ Answer

(1.02)^5 ~= 1.10 (true value: 1.1041).

Practice Problems

Easyapplication

Use linear approximation to estimate sin(0.1) (radians). What is the error bound if |f''| <= 1 on [0, 0.1]?

Common Mistakes

Common Mistake

Confusing dy (the differential) with Delta y (the actual change).

dy = f'(x)*dx is the change in the tangent line (linear approximation), not the actual change in f. The actual change is Delta y = f(x + Delta x) - f(x). They are equal only to first order: Delta y = dy + O((Delta x)^2). For small dx, dy ~= Delta y, which is why differentials are useful for approximations. But they are not exactly equal.

Quiz

The differential dy for y = x^3 at x=2 with dx=0.01 is:

0.120.0012120.012

Historical Background

The tangent line as a local approximation to a curve was implicit in Leibniz's and Newton's development of calculus. Leibniz introduced the notation dy, dx for differentials (1684). The idea that every smooth curve 'looks linear' under sufficient magnification is captured precisely by differentiability. Linear approximation is formalized in the first-order Taylor expansion, which generalizes to higher dimensions (Jacobians) and higher orders (Taylor series).

1684
Leibniz introduces differential notation dy, dx
Gottfried Wilhelm Leibniz
1715
Taylor series developed, generalizing linear approximation to all orders
Brook Taylor
1860s
Rigorous analysis: linearization defined via derivative (Weierstrass)
Karl Weierstrass

Summary

Linearization: L(x) = f(a) + f'(a)*(x-a) is the best linear approximation to f near a.
Error: |f(x)-L(x)| is O((x-a)^2) when f'' exists and is bounded.
Differentials: dy = f'(x)*dx relates small input changes to output changes.
Standard approximations near 0: sin(x)~=x, cos(x)~=1, e^x~=1+x, (1+x)^n~=1+nx.

References

BookStewart, J. Calculus: Early Transcendentals. 8th ed. Cengage, 2015.
BookSpivak, M. Calculus. 4th ed. Publish or Perish, 2008.
WebsiteWikipedia -- Linear approximation