Lesson 2: Linear Approximation and Differentials
Estimated time: 30-40 minutes
Learning Objectives
By the end of this lesson, you will be able to:
- Write the linearization of a function at a point
- Use linearization to approximate function values
- Understand and compute differentials dy and dx
- Estimate errors using differentials
Linearization
Near a point x = a, a differentiable function is well approximated by its tangent line. This tangent-line approximation is called the linearization.
Linearization
The linearization of f at x = a is: L(x) = f(a) + f'(a)(x − a)
For x close to a: f(x) ≈ L(x).
Example 1: Approximating a Square Root
Approximate √4.1 using linearization of f(x) = √x at a = 4.
Step 1: f(4) = 2, f'(x) = 1/(2√x), so f'(4) = 1/4.
Step 2: L(x) = 2 + (1/4)(x − 4).
Step 3: L(4.1) = 2 + (1/4)(0.1) = 2 + 0.025 = 2.025.
Actual value: √4.1 ≈ 2.02485. The error is about 0.00015.
Example 2: Approximating sin(0.1)
Use linearization of f(x) = sin x at a = 0.
Step 1: f(0) = 0, f'(0) = cos 0 = 1.
Step 2: L(x) = 0 + 1(x − 0) = x.
Step 3: sin(0.1) ≈ 0.1. (Actual: 0.09983...)
Differentials
Differentials
If y = f(x), the differential is: dy = f'(x) dx
Here dx is an independent variable (a small change in x), and dy approximates the resulting change in y.
The actual change is Δy = f(x + dx) − f(x). The differential dy approximates Δy when dx is small.
Example 3: Using Differentials
A cube's side is measured as 5 cm with an error of ±0.02 cm. Estimate the error in the computed volume.
Step 1: V = s³, so dV = 3s² ds.
Step 2: dV = 3(25)(0.02) = 1.5 cm³.
The volume error is approximately ±1.5 cm³ out of V = 125 cm³, a relative error of 1.2%.
Example 4: Relative and Percentage Error
If a sphere's radius is measured as 10 cm with error dr = 0.1 cm, estimate the percentage error in volume.
Step 1: V = (4/3)πr³, dV = 4πr² dr.
Step 2: Relative error: dV/V = (4πr² dr)/((4/3)πr³) = 3 dr/r = 3(0.1/10) = 0.03.
Result: 3% error in volume from a 1% error in radius.
Key Takeaways
- Linearization L(x) = f(a) + f'(a)(x − a) gives a tangent-line approximation.
- The approximation is best when x is close to a.
- Differentials: dy = f'(x) dx approximates the actual change Δy.
- Relative error = dy/y; percentage error = (dy/y) × 100%.
Check Your Understanding
1. Use linearization of f(x) = x1/3 at a = 8 to approximate 8.11/3.
2. If y = x4 and x changes from 2 to 2.01, estimate dy.
3. A circle's radius is 20 cm with error ±0.5 cm. Estimate the error in the area.
4. For f(x) = cos x, find the linearization at a = π/2.