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Linear approximation uses the tangent line at a known point to estimate function values nearby: f(x) is approximately equal to L(x) = f(a) + f'(a)(x - a) for x near a. This is the simplest and most practical consequence of differentiability. It is the foundation for differentials, Newton's method, and Taylor polynomials. The quality of the approximation depends on how close x is to a and how curved the function is.
Approximate values like sqrt(4.1) by linearizing sqrt(x) at x = 4. Compare the approximation with the true value to see the error. Discuss when the approximation is good (f is nearly linear near a) vs. poor (high curvature).
You already know that the derivative f'(a) is the slope of the tangent line to y = f(x) at x = a. Linear approximation takes the next step: that tangent line is not just a line that touches the curve at one point — it is the best linear approximation to the function near that point. If you zoom in close enough to any differentiable function, the function and its tangent line become indistinguishable. Linear approximation exploits this fact to estimate function values that would otherwise require a calculator.
The formula is L(x) = f(a) + f'(a)(x − a). This is simply the point-slope equation of the tangent line at (a, f(a)), rearranged in a useful way. To approximate √4.1, choose a = 4, where you know the exact value: f(x) = √x, f(4) = 2, f'(x) = 1/(2√x), f'(4) = 1/4. The linear approximation is L(x) = 2 + (1/4)(x − 4). At x = 4.1: L(4.1) = 2 + (1/4)(0.1) = 2.025. The true value is approximately 2.02485 — the approximation is excellent because 4.1 is very close to 4, and √x is nearly linear in a small neighborhood of 4.
The quality of the approximation depends on two things: how far x is from a, and how curved the function is near a. The curvature is captured by the second derivative — a large |f''(a)| means the function bends sharply away from the tangent line, and the approximation degrades quickly. A small |f''(a)| means the function is nearly linear, and the tangent line stays close for a wider range of x. This is why L(x) works well for √x at a = 4 but would be less accurate at a = 0.01, where the function curves steeply.
Linear approximation is the foundation for several tools you'll encounter next. Differentials restate it using the notation df = f'(a)dx, emphasizing the infinitesimal change perspective. Newton's method for root-finding iteratively applies linear approximation: it replaces the curve with its tangent line, finds where the tangent crosses zero, and uses that as the next approximation. Taylor polynomials extend the idea by adding higher-degree correction terms, starting with the quadratic correction (1/2)f''(a)(x−a)². Linear approximation is the first-order Taylor polynomial — the foundation of a hierarchy of increasingly accurate polynomial approximations.