Multivariate Functions

Question 1

What is the level curve of a multivariate function?

Answer 1

Curves with equations f(x, y) =k where k is constant in the range of f
Therefore, set of all points in Domain of f where f takes on the value k

Question 2

What do level curves represent?

Answer 2

Trace of the surface in the horizontal plane z = k projected down to the xy-plane

Question 3

What is the logic behind the derivative of a multivariate function?

Answer 3

If we let one variable vary while keeping the other constant, we are considering a function of a single variable, e.g g(x) = f(x, b)

Question 4

How do we calculate the partial derivatives of a function?

Answer 4

fx(a, b) = g’(a) where g(x) = f(x, b)
Therefore, derive fx by keeping y constant and defirentiatng with respect to x

Question 5

What are the other notations for the partial derivatives?

Answer 5

Partial symbols on fractions
D1f or Dxf or f1
all symbolise fx

Question 6

What are the two interpretations for partial derivatives?

Answer 6

Letting y =b, we look at Curve C1 in which the plane y=b intersects S
This means C1 is the trace of S in y =b

1. fx and fy are geometrically interpreted as the slopes of the tangent lines at P(a,b,c) to the traces C1, C2 of S in the planes x=a, y=b
2. Rates of change. z/x represents how z changes with respect to x when y is fixed.

Question 7

What is Clairaut’s Theorem?

Answer 7

Suppose f is defined on disk D that contains (a, b). If f_xy and f_yx both continuous on D, then they are equal.

Question 8

What does f_xy and f_yx mean?

Answer 8

Differentiate with respect to x then y

Question 9

What is the tangent plane of a surface?

Answer 9

If function z has continuous partial derivatices and we have a point P on S,
then C1 and C2 both contain point P
The tangent plane is the plane that contains the tangent lines to the trace curves.

z - z0 = fx(x0, y0)(x-x0) + fy(x0, y0)(y-y0)

Question 10

What is the linearization of f?

Answer 10

Linear function whose graph is the tangent plane:
L(x,y) = f(a, b) + fx(a,b)(x-a)+fy(a,b)(y-b)

Question 11

What is the linear approximation of f?

Answer 11

Same equation as linearization
f is approximated by L, or L is linear approximation of f close to the point (a,b)

Question 12

How do we know when the linearization of a function is a good approximation?

Answer 12

If f is differentiable at (a,b) then the linearization close to a, b is a good approximation

Question 13

When is a function differentiable at a, b?

Answer 13

• fx, fy exists on S
• both are continuous at (a,b)