Math · Calculus I · Worked example
Estimate a square root with a tangent line
Use a linear approximation to estimate √4.1.
Choose f and a
Let f(x) = √x and a = 4, a nearby point where the square root is exact: f(4) = 2.
Find the slope at a
f′(x) = 1/(2√x), so f′(4) = 1/4.
Write the linearization
L(x) = f(4) + f′(4)(x − 4) = 2 + (x − 4)/4.
Evaluate near a
L(4.1) = 2 + (1/4)(0.1) = 2.025.
Compare with the true value
√4.1 = 2.024846…, so the error is about 0.00015. The estimate is slightly high because √x is concave down: its graph bends below the tangent line.
Result
√4.1 ≈ 2.025. The true value is 2.02485 to five decimal places.
Your turn
Use the linearization of f(x) = ∛x at a = 27 to estimate ∛27.5.
Show the answer and explanation
∛27.5 ≈ 3.0185.
Write f(x) = x^(1/3). Then f(27) = 3 and f′(x) = (1/3)x^(−2/3), so f′(27) = 1/(3 · 9) = 1/27. L(27.5) = 3 + 0.5/27 = 3.0185. The true value is 3.01841, so the estimate is slightly high, as it is for √x.
Keep exploring
Open the rows in Math and add L(3.9) = 1.975: the same tangent line estimates √3.9, and the checker confirms the value.
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