Math 1304 Calculus I 3.10 –Linear Approximation and Differentials

The tangent line approximation If y = f(x) is a function that has a values and value for its derivative at a point x = a, then L(x) = f(a) + f’(a) (x – a) gives a best approximating line to the curve y = f(x) at that point. The “curve” y = L(x) is the tangent line to f at x = a and is called the linearization of f at x = a.

Taylor’s Series The tangent line formula is part of a larger concept called Taylor’s series of the function f, centered at x = a, of order n T[a,n][f](x) = f(a) + f’(a) (x-a) + 1/2! f’’(a) (x-a) 2 + 1/3! f’’’(a) (x-a) 3 + … + 1/n! f (n) (a) (x-a) n

Examples of Linearization Approximate e –Solution: Let f(x) = e x and use Taylor series centered at x = 0, with x = 1. Approximate √4.01 –Solution: Let f(x) = x ½ and use the linearization (tangent formula, or 1 st order Taylor series) centered at x = 4, with x =4.01 Approximate √3.99 –Solution: Let f(x) = x ½ and use the linearization (tangent formula, or 1 st order Taylor series) centered at x = 4, with x = 4 -.1

The Differential If y = f(x) is differentiable at a point x = a, then dy = f’(a) dx is called the differential of f at a. The differential is a function of two variables: the point a, and a change in x. The formula for this function is df(a, h) = f’(a) h Note that if L(x) = f(a) + f’(a) (x-a), is the linearization of f centered at a, then df(a, x-a) = L(x) – f(a) L(x) = f(a) + df(a, x-a)

Rules for differentials Identity function: If F(x) = x, then dF(x, h) = h Sum rule: If F(x) = f(x) + g(x) then dF = f’(x) dx + g’(x) dx Product rule: If F(x) = f(x) g(x) then dF = f’(x) g(x) dx + f(x) g’(x) dx Chain rule: If F(x) = f(g(x), then dF = f’(g(x)) dg Power rule: If F(x) = f(x) n, then dF = n f(x) n-1 f’(x) dx Exponential rule: If F(x) = e f(x), then dF = e f(x) f’(x) dx

The Chain Rule Chain rule: If F(x) = f(g(x)), then dF(x, h) = f’(g(x)) g’(x) h = f’(g(x)) dg(x, h) dF = f’(g(x)) dg Special Case: If F(x) = x, then dF(x, h) = h

Examples See examples 3 and 4