Linear Approximation & Differentials

AP Calculus · Axiom Academy

LESSON Linear Approximation & Differentials Using the tangent line to estimate function values When you zoom in close enough on a smooth curve, it looks like a straight line. That straight line is the tangent line, and it's a perfect approximation for the function near that point. This idea is powerful: to estimate without a calculator, you can use the simpler tangent line instead of the original function. The tangent line at (a, f(a)) is an excellent approximation for f(x) near x = a. We introduce new notation to make this clearer: : The change in x (we can choose this) : The change in y estimated by the tangent line With this notation, linear approximation says: In other words: The differential dy is the change in y that the tangent line predicts. We want to approximate without a calculator. Step 1: Choose a nearby point where we know the value. Since = 8, use with a = 64. Step 2: Calculate the derivative: Step 4: Use the linear approximation with dx = 1: Result: ≈ 8.0625 (actual value ≈ 8.0623)

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