Calculus I review: Suppose f : R → R. Recall the tangent line to y = f(x) at x = a is

y = f(a) + f′(a)(x − a).

Thus y = f(a) + f′(a)(x − a) is the best linear approximation of f near x = a. Ex: Find the best linear approximation for f(x) = 2x + 5. Answer: Note slope = f′(x) = 2.

Ex: Find the best linear approximation for h(x) = x^{2} at x = 3. h′(x) = 2x. Thus slope of tangent line at x = 3 is h′(3) = 2(3) = 6. Hence \frac{y−9}{x−3} = 6 Equation of tangent line: y = 9 + 6(x − 3)

Estimate h(3.1):

The gradient of f : R^{n} → R^{1} is denoted by

∇f(a) = (\frac{∂f}{∂x_{1}} (a), ..., \frac{∂f}{∂x_{n}} (a))

Defn: The Jacobian matrix of f : R^{n} → R^{m} at a is

Df(a) = [ \frac{∂f_{i}}{∂x_{j}} (a) ]_{m×n} = [ \begin{matrix}\frac{∂f_{1}}{∂x_{1}} (a) & & ... & & \frac{∂f_{1}}{∂x_{n}} (a) \\ . & . & & & . \\ . & & . & & . \\ . & & & . & . \\ \frac{∂f_{m}}{∂x_{1}} (a) & & ... & & \frac{∂f_{m}}{∂x_{n}} (a) \end{matrix} ]

Thm: If f is differentiable at a, then 1.) f is continuous at a. 2.) \frac{∂f_{i}}{∂x_{j}} exists at a for all i, j. 3.) The derivative of f at a = Df(a)

= the Jacobian matrix of f at a.

Thm: Let f : R^{n} → R^{m}, f = (f_{1}, ..., f_{m}). If \frac{∂f_{i}}{∂x_{j}} exists and are continuous in a neighborhood of a for all i, j, then f is differentiable at a