Chapter 14 - Module 1-5

Question 1

How to take a limit in the Multivariable case

Answer 1

Check from all directions; the limit must match in all directions in order to be valid

Question 2

Formal Partial Derivative Definitions

Answer 2

Question 3

Clairaut’s Theorem

Answer 3

f_xy=f_yx

Question 4

The tangent plane to a surface defined by z=f(x,y) at the point (x₀,y₀,z₀) is

Answer 4

z = z₀ + f_x(x₀,y₀)(x-x₀) + f_y(x₀,y₀)(y-y₀)

Question 5

Linear Approximation

Answer 5

L(x,y) = f(x₀,y₀) + f_x(x₀,y₀)(x-x₀) + f_y(x₀,y₀)(y-y₀)

Question 6

Total Differential f(x,y,z)

Answer 6

df = f_xdx + f_ydy + f_zdz

= (∂f/∂x)dx + (∂f/∂y)dy + (∂f/∂z)dz

Question 7

The General Chain Rule

Answer 7

(d/dt)f(x(t),y(t)) = (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt)

Question 8

Implicit Differentiation Formula

Answer 8

dy/dx = -F_x/Fy

Question 9

Directional Derivative Definition

Answer 9

D_uf(x,y) = f_x(x,y)a + f_y(x,y)b

= f_x(x,y)cosθ + f_y(x,y)sinθ

Question 10

The Gradient of a function f(x,y) is the vector function ∇f defined by

Answer 10

∇f(x,y) = < fx(x,y), f_y(x,y) >

= (∂f/∂x)i + (∂f/∂y)j

Question 11

Combined Directional Derivative Definition

Answer 11

D_uf(x,y) = ∇f(x,y) • u

Question 12

The maximum value of the directional derivative D_uf(x) is

Answer 12

|∇f(x)|, which occurs when u has the same direction as the gradient vector ∇f(x)

Question 13

Second Derivatives Test

Answer 13

If f_x(a,b) and f_y(a,b) = 0 [that is, (a,b) is a critical point of f]. Let

D = D(a,b) = f_xx(a,b)f_yy(a,b) - [f_xy(a,b)]²

(a) If D > 0 and f_xx(a,b) > 0, then f(a,b) is a local minimum.
(b) If D > 0 and f_xx(a,b) < 0, then f(a,b) is a local maximum.
(c) If D < 0, then f(a,b) is not a local maximum or minimum.

Question 14

To find the absolute maximum and minimum values of a continuous function f on a closed, bounded set D:

Answer 14

1. Find the values of f at the critical points of f in D
2. Find the extreme values of f on the boundary of D
3. The largest of the values from steps 1 and 2 is the absolute maximum value; the smallest of these is the absolute minumum value

Question 15

Method of Lagrange Multipliers

Answer 15

(a) Find all values of x, y, z, and λ such that

∇f(x,y,z) = λ∇g(x,y,z)

and g(x,y,z) = k

(b) Evaluate f at all the points 9x,y,z) that result from step (a). The larhest is the max value of f; the smallest is the min value of f.

Question 16

Lagrange Multipliers with Two Constraints

Answer 16

∇f(x₀,y₀,z₀) = λ∇g(x₀,y₀,z₀) + µ∇h(x₀,y₀,z₀)