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Linear Approximation
AP Calculus AB · Unit 4 Context of Derivatives
Learning Objectives
Use tangent line to approximate f(x) near a
Apply L(x) = f(a) + f′(a)(x − a)
Estimate √4.1, sin(0.1), etc. without calculator
Understand linearization error increases with distance
Formula
L(x) = f(a) + f′(a)(x − a)
Δy ≈ dy = f′(a)·dx
Error = |f(x) − L(x)|
Better approximation when x is closer to a
Tags
linear approximation
tangent line
differentials
linearization
FUNCTION
f(x) = √x, a = 4
f(x) = sin(x), a = 0
f(x) = eˣ, a = 0
f(x) = ln(x), a = 1
f(x) = 1/(1+x), a = 0
x value
4.1
f(x) exact
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L(x) approx
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Error
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% Error
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