Calculus I - Week 7
Linear Approximation and Differentials
The tangent line to a function at a point is the best linear approximation to the function near that point. This week formalizes that idea with linearization and differentials, then connects it to Newton's method and error estimation in physics and engineering.
Learning Objectives
- Find the linearization L(x) of a function at a given point. - Use linear approximation to estimate function values near a known point. - Define differentials dx and dy and interpret them geometrically. - Use differentials to estimate errors in measurements. - Understand when linear approximation is accurate and when it breaks down. - Apply linearization to physics formulas involving small changes.
1. The Tangent Line as an Approximation
Near x = a, a smooth function looks like its tangent line. The tangent line at x = a has equation: y = f(a) + f'(a)(x - a) For x close to a, f(x) is approximately equal to this line. This is the linear approximation or linearization of f at x = a.
2. The Linearization Formula
The linearization of f at x = a is: L(x) = f(a) + f'(a)(x - a) When x is near a, f(x) ≈ L(x). Example: Linearize f(x) = sqrt(x) at a = 4. f(4) = 2, f'(x) = 1/(2*sqrt(x)), f'(4) = 1/4. L(x) = 2 + (1/4)(x - 4) Estimate sqrt(4.1) ≈ L(4.1) = 2 + (1/4)(0.1) = 2.025. Actual: sqrt(4.1) ≈ 2.0248. Very close!
3. More Linearization Examples
Example: Estimate (1.02)^10 using linearization of f(x) = x^10 at a = 1. f(1) = 1, f'(x) = 10x^9, f'(1) = 10. L(x) = 1 + 10(x - 1). L(1.02) = 1 + 10(0.02) = 1.2. Actual: (1.02)^10 ≈ 1.2190. Close for a 2% change. Example: Estimate cube root of 27.3. f(x) = x^(1/3) at a = 27. f(27) = 3, f'(x) = (1/3)x^(-2/3), f'(27) = 1/27. L(x) = 3 + (1/27)(x - 27). L(27.3) = 3 + (0.3/27) = 3 + 1/90 ≈ 3.011.
4. Differentials
If y = f(x), the differential dy is defined as: dy = f'(x) dx Here dx is an arbitrary small change in x, and dy is the corresponding change in y predicted by the tangent line (not the actual change in f). The actual change in y is Delta_y = f(x + dx) - f(x). For small dx: Delta_y ≈ dy = f'(x) dx.
5. Using Differentials for Error Estimation
Example: A sphere has radius measured at r = 10 cm with possible error dr = 0.1 cm. Estimate the error in the computed volume. V = (4/3)*pi*r^3 dV = 4*pi*r^2 dr = 4*pi*(100)(0.1) = 40*pi cm^3. Relative error: dV/V = 3*(dr/r) = 3*(0.1/10) = 0.03 = 3%. This is a useful formula in physics: relative error in volume is three times the relative error in radius.
6. Differentials for Area
Example: A square has side s = 5 cm ± 0.05 cm. Estimate the error in area. A = s^2 dA = 2s ds = 2(5)(0.05) = 0.5 cm^2. Relative error: dA/A = 2*(ds/s) = 2*(0.05/5) = 2%. Note: doubling the exponent doubles the relative error — area error is twice radius error.
7. Accuracy of Linear Approximation
Linear approximation works well when: - x is very close to a - f does not curve sharply (small second derivative) The error in linear approximation is approximately (1/2)*f''(a)*(x - a)^2. If f'' is large (strong curvature), linear approximation becomes inaccurate more quickly.
8. Physics Application: Small Angle Approximation
For small angles theta (in radians): sin(theta) ≈ theta cos(theta) ≈ 1 tan(theta) ≈ theta These come from linearizing sin(x) at x = 0: L(x) = sin(0) + cos(0)(x - 0) = x. The small angle approximation is used everywhere in physics: pendulums, optics, and mechanics.
9. Newton's Method Preview
Linear approximation underlies Newton's method for solving f(x) = 0. Starting with a guess x_0, the next guess is: x_1 = x_0 - f(x_0)/f'(x_0) This is the x-intercept of the tangent line at x_0. If f is well-behaved, the guesses converge rapidly to the root.
10. Common Mistakes
- Confusing dy with Delta_y — dy is the tangent line approximation, Delta_y is the actual change. - Forgetting to evaluate f'(a) at the specific point a, not at a general x. - Using a point a that is too far from the target x. - Sign errors when computing f'(a).