Math 1304 Calculus I

1. Math 1304 Calculus I 3.10 –Linear Approximation and Differentials

2. 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.

3. 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

4. Examples of Linearization • Approximate e • Solution: Let f(x) = ex and use Taylor series centered at x = 0, with x = 1. • Approximate √4.01 • Solution: Let f(x) = x½ and use the linearization (tangent formula, or 1st 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 1st order Taylor series) centered at x = 4, with x = 4 -.1

5. 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)

6. 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) = ef(x), then dF = ef(x) f’(x) dx

7. 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

8. Examples • See examples 3 and 4