Linear Approximation
The tangent line is a good approximation to a function near the point of tangency. This is called linear approximation or linearization.
Linear Approximation
f(x) ≈ f(a) + f'(a)(x - a)
Near x = a, the function approximately equals its tangent line
Approximate sqrt(4.1)
Medium
Use linear approximation to estimate sqrt(4.1)
1
Choose nearby known value
Use a = 4 (we know sqrt(4) = 2)
2
Set up
f(x) = sqrt(x), f'(x) = 1/(2sqrt(x))
f(4) = 2, f'(4) = 1/4
3
Apply formula
sqrt(4.1) ≈ 2 + (1/4)(4.1 - 4) = 2 + 0.025 = 2.025
Answer:
sqrt(4.1) ≈ 2.025
💡 Actual value: 2.0248... Very close!
Practice
1. Approximate e^0.1 using linear approximation at a = 0
e^x ≈ e^0 + e^0(x-0) = 1 + x. At x=0.1: 1 + 0.1 = 1.1
Key Takeaways
Linear approximation uses tangent line to estimate function values
Choose a nearby point where you know exact values
Formula: f(x) ≈ f(a) + f'(a)(x - a)
More accurate closer to a