The Linear Approximation Formula

You are staring at a curve that refuses to give a clean answer. √4.1, sin(0.05), ln(1.1), the calculator will do it, but the exam expects you to produce a number by hand. The linear approximation formula, L(x) = f(a) + f'(a)(x−a), replaces the curve with its tangent line at a known point a. That line gives you a usable estimate for x near a. The formula is the tangent line written in point-slope form, and it is the entire basis for every linearization you will ever compute.

The formula works because at the point of tangency a the line matches the function in two ways: the same y-value, f(a), and the same slope, f'(a). No other straight line can do that at that exact point. For x values close to a, the curve and the line stay close. Move away from a, and the error grows quadratically, not linearly.

L(x) is the estimated function value at x. f(a) is the exact function value at the anchor point a. f'(a) is the derivative at a, the slope of the tangent line. (x−a) is the horizontal distance from the anchor point to the point you care about. Multiply slope by distance, add to the starting value, and you get L(x).

The formula is the point-slope form of a line, y − f(a) = f'(a)(x−a), solved for y. That line is called the linearization of f at a. Every course, from AP Calculus to Calculus III, uses this same expression. Stewart 'Calculus' §3.10 presents it as L(x) = f(a) + f'(a)(x−a).

The most common mistake: using the wrong anchor point. Pick a too far from x, and the estimate becomes useless. Pick a where f(a) is simple, 4 for √x, 0 for sin x, and the arithmetic stays clean.

The words 'linear approximation' and 'tangent line' refer to the same object. The distinction is only in how you use it. When you call it the tangent line, you are thinking geometrically: a line that touches the curve at one point. When you call it the linearization, you are thinking computationally: a formula that produces estimated y-values.

The tangent line passes through (a, f(a)) and has slope f'(a). That is exactly the information the linearization formula encodes. If you know the function value and the slope at a, you can write the line equation without memorizing anything beyond the point-slope pattern.

Concavity Determines Overestimate Or Underestimate

Local linearity is the property that a differentiable function, when magnified near a point, looks indistinguishable from its tangent line. Zoom in on any smooth curve at enough magnification, and the bend disappears. That is why the linear approximation works at all: the curve is practically straight within a small window.

This is not true for non-differentiable functions. At a corner, like the absolute value function at x = 0, no unique tangent line exists. Zooming in still shows a corner. The linear approximation formula cannot apply there.

Overestimate versus underestimate is determined by concavity. If f''(a) > 0 (concave up), the curve lies above its tangent line, so L(x) is an underestimate. If f''(a) < 0 (concave down), L(x) is an overestimate. Stewart 'Calculus' §3.10 uses this concavity test to decide the sign of the approximation error.

Memorize These Five Common Linearizations

Five linearizations appear so often that they are worth memorizing. Each one replaces a common function with its tangent line at x = 0. The small-angle approximation used in physics is a special case of these.

sin x ≈ x, with error about −x³/6. At x = 0.1 rad, the error is 0.00017. At x = 0.5 rad, the error jumps to 0.021. Radians are mandatory; degrees break the formula.

cos x ≈ 1, but the next term in the Taylor series is −x²/2. The linearization of cos x at 0 is just 1, which is useful only for very tiny angles.

e^x ≈ 1 + x. The exponential function grows fast; this approximation works for x up to about 0.1 before the error becomes noticeable.

ln(1+x) ≈ x. At x = 0.1, ln(1.1) ≈ 0.0953, versus an exact 0.0953, the error is only 0.0003.

(1+x)^k ≈ 1 + kx. This is the binomial approximation for any real exponent k.

Function	Linearization	Next Term (Error Direction)
sin x	x	−x³/6 (underestimate for x>0)
cos x	1	−x²/2 (underestimate)
e^x	1 + x	x²/2 (overestimate for x>0)
ln(1+x)	x	−x²/2 (underestimate for x>0)
(1+x)^k	1 + kx	k(k−1)x²/2 (sign depends on k)

Bound the Error With Taylor's Inequality

The linear approximation is the Taylor polynomial of degree 1. A Taylor polynomial of degree n matches the function's value and its first n derivatives at the point a. For n = 1, that means matching f(a) and f'(a), which is exactly what the tangent line does. Taylor's inequality for n = 1 bounds the error: |R_1(x)| ≤ (M/2)|x−a|², where M is an upper bound for |f''(z)| on the interval between a and x.

This is how you put a number on the error without knowing the true function value. If f'' is small near a, the bound is tight. If f'' is large, the bound warns you that the linearization is unreliable. Stewart 'Calculus' §11.11 gives the full inequality. For most calculus homework, the M you pick is the maximum of |f''| on the interval from a to x.

The 2-variable version, the tangent plane, extends the same idea. For a function of two variables, the linearization is L(x,y) = f(a,b) + f_x(a,b)(x−a) + f_y(a,b)(y−b). Calculus III students use this to approximate values on surfaces.