Tangent Lines & Linearization

Equation of tangent line at x = a:

y - f(a) = f'(a)(x - a)

This IS the linear approximation.

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

Near x = a, L(x) ≈ f(x).

Example: Approximate √(4.1) using linearization.

f(x) = √x, a = 4. f(4) = 2, f'(x) = 1/(2√x), f'(4) = 1/4.

L(4.1) = 2 + (1/4)(0.1) = 2.025. Actual: 2.0248...

Over/underestimate:

If f is concave up, tangent line is BELOW the curve → underestimate.

If f is concave down, tangent line is ABOVE → overestimate.

AP exam: "Use the tangent line at x = 3 to approximate f(3.1)." This is linearization. Write the tangent line equation first.

Reference:

Wikipedia: Linear Approximation

https://en.wikipedia.org/wiki/Linear_approximation