Multivariable Functions
Functions of two variables: z(x, y), f(x,
y), g(x, y), h(x, y)
Arguments: x, y, t
Small changes in x, y, z, respectively: Δx, Δy, Δz.
- First Order Partial Derivatives
The partial derivative with respect to x
The partial derivative with respect to y
- Seconds Order Partial Derivativ
- Chain Rules
If f (x, y) = g (h, (x, y)) (g is a function of one
variable h), then
If h (t) = f (x(t),
y (t)), then
If z = f (x(u, v), y (u, v)), then
- Small Changes
- Local Maxima and Minima
f (x, y) has a local maximum at (x0, y0) if
f (x, y) ≤ f (x0, y0) for all (x, y) sufficiently close
to (x0, y0).
f (x, y) has a local minimum at (x0, y0) if
f (x, y) ≥ f (x0, y0) for all (x, y) sufficiently close
to (x0, y0).
- Stationary Points
Local maximum and local minimum occur at station
point.
- Saddle Point
A stationary point which is neither a local maximum
nor a local minimum.
- Second Derivative Test for Stationary Points
Let (x0, y0) be a stationary point
If D > 0, fxx (x0, y0) > 0, (x0, y0) is a point of local
minimum.
If D < 0, fxx (x0, y0) < 0, (x0, y0) is a point of local
maximum.
If D = 0, the test fails.
- Tangent Plane
The equation of the tangent plane to the surface
z = f (x, y) at (x0, y0, z0) is
z – z0 = fx (x0, y0)
(x – x0) + fy (x0, y0)
(y – y0).
- Normal to Surface
The equation of the normal to the surface z = f (x, y)
at (x0, y0, z0) is
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