Quiz 5

Question 1

what do partial derivative represent?

Answer 1

f_x and f_y measure rate of change in a direction in a the x or y direction respectively

Question 2

Tangent plane the graph of a function equation

Answer 2

z=f_x (a,b) * (x-a) + _y(a,b) * (y-b) + c. c = f(a,b)

Question 3

Multivariable Chain Rule Forumla

Answer 3

dz/dt = ∂z/∂x * dx/dt + ∂z/∂y * dy/dt

Can also be written as: <∂z/∂x, ∂z/∂y> * <dx/dt, dy/dt> since it looks like a dot product

Question 4

what’s the typical setting for the chain rule?

Answer 4

we have a function f(x,y) describing a quantity based on location and a position function. (How fast?)
→
r = R → V_2

→
r(t) = <x(t),y(t)>

Question 5

what is the gradient?

Answer 5

<∂z/∂x, ∂z/∂y> or <f_x, f_y>

Question 6

what’s the definition of a directional derivative?

Answer 6

Let f: R^2 -> R and u be a unit vector <a,b>
(D_u)f = lim f(x+ah,y+bh) - f(x,y)/h
h->0

Question 7

what’s the point of directional derivatives?

Answer 7

measure rate of change in the direction of the unit vector.

Question 8

how to calc the directional derivative

Answer 8

D_u f = u * ∇f(x,y)

Question 9

uses of gradient

Answer 9

• directional derivative - D_u f = u * ∇f(x,y)
• chain rule: df/dt = r’(t) or <x(t),y(t)> * ∇f(x,y)
• direction of max increase: literrally just the gradient ∇f(f_x (x,y),f_y (x,y))

Question 10

differential formula

Answer 10

dz = ∂z/∂x * dx + ∂z/∂y * dy

Question 11

how do you find the maximum change? (not the direction)

Answer 11

take the length of the gradient ∇f(f_x (x,y),f_y (x,y))

Question 12

Tangent plane a level set of a function equation

Answer 12

f_x (a,b) * (x-a) + f_y(a,b) * (y-b) + f_z(a,b) * (z-c) = 0