1. Why Approximations Are Useful
Many functions are difficult to compute exactly for values slightly away from a known point. Approximations using differentials provide a quick way to estimate values using the tangent line. In simple notes-style: near a point, the curve behaves almost like a straight line, so the tangent line gives a close estimate.
2. Idea of Linear Approximation
Linear approximation replaces the function with its tangent line near a chosen point. The tangent line is easy to compute and gives a good estimate for small changes in the input.
2.1. Formula
f(x + \Delta x) \approx f(x) + f'(x)\Delta x
Here, \( \Delta x \) is a small change in x.
2.2. Geometric View
The tangent line touches the curve at a point and almost overlaps it nearby. So we use the tangent line's value as an estimate of the function.
3. Differentials
Differentials help describe very small changes. If y = f(x), then:
3.1. Definition
dy = f'(x) dx
Here, dx is a small change in x, and dy is the corresponding change in y predicted by the tangent line.
3.2. Relation to Approximation
Since dy ≈ Δy for small dx, we get:
\Delta y \approx dy = f'(x) dx
4. Examples of Linear Approximation
These examples show how the tangent line provides quick approximations.
4.1. Example 1 — Approximating a Square Root
Estimate \( \sqrt{4.1} \).
Let f(x) = √x, approximate near x = 4.
f(4) = 2, \; f'(x) = \frac{1}{2\sqrt{x}} → f'(4) = \frac{1}{4}
Δx = 0.1
f(4.1) \approx 2 + \frac{1}{4}(0.1) = 2.025
4.2. Example 2 — Approximating a Cube
Estimate \( (2.02)^3 \).
Let f(x) = x³, approximate near x = 2.
f(2) = 8, \; f'(x) = 3x^2 → f'(2) = 12
f(2.02) \approx 8 + 12(0.02) = 8.24
4.3. Example 3 — Approximating Logarithm
Estimate ln(1.05).
Let f(x) = ln x, approximate near x = 1.
f(1) = 0, \; f'(1) = 1
\ln(1.05) \approx 0 + 1(0.05) = 0.05
5. When Linear Approximation Works Best
Linear approximation is most accurate when:
- Δx is small
- The function is smooth around the chosen point
- The derivative does not change too rapidly
6. Why Approximations Using Differentials Matter
Approximations simplify calculations, especially when exact values are difficult. They are widely used in physics, finance, engineering, and anywhere quick estimates are needed.