Multivariable Calculus: Differentiation of Functions of Several Variables flashcards

Question 1

What is a function of two variables?

Accepted Answer

A function z = f(x, y) that assigns a unique real number z to each ordered pair (x, y) in a subset D of R^2.

Question 2

What is a level curve of f(x, y)?

Accepted Answer

The set of points with f(x, y) = c, where c is a real number in the range of f.

Question 3

What is a plot of several level curves of f(x, y) called?

Accepted Answer

A contour map.

Question 4

What is a level surface of f(x, y, z)?

Accepted Answer

The set of points with f(x, y, z) = c, where c is a real number in the range of f.

Question 5

When is P0 a boundary point of a set R?

Accepted Answer

When every δ disk centered at P0 contains points of R and points not in R.

Question 6

What is a closed set?

Accepted Answer

A set that contains all of its boundary points.

Question 7

When is a function of two variables continuous at a point?

Accepted Answer

When the function is defined there, its limit exists there, and the limit equals the function value.

Question 8

What is a partial derivative?

Accepted Answer

A derivative of a function of several independent variables taken with all variables but one held constant.

Question 9

What are mixed partial derivatives?

Accepted Answer

Partial derivatives of second or higher order in which at least two differentiations are with respect to different variables.

Question 10

What is a partial differential equation?

Accepted Answer

An equation involving an unknown function of two or more independent variables and at least one of its partial derivatives.

Question 11

What is the equation of the tangent plane to z = f(x, y) at (x0, y0), where f is differentiable?

Accepted Answer

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

Question 12

What is the total differential of z = f(x, y) at (x0, y0)?

Accepted Answer

dz = fx(x0, y0) dx + fy(x0, y0) dy

Question 13

What is a directional derivative?

Accepted Answer

The derivative of a function along the direction of a given unit vector.

Question 14

What is the gradient of f(x, y)?

Accepted Answer

∇f(x, y) = (∂f/∂x)i + (∂f/∂y)j

Question 15

What direction does the gradient of a function point in?

Accepted Answer

The direction of greatest change of the function.

Question 16

When is (x0, y0, f(x0, y0)) a saddle point of z = f(x, y)?

Accepted Answer

When fx(x0, y0) = 0 and fy(x0, y0) = 0 but f has no local extremum at (x0, y0).

Question 17

What is the discriminant D used in the second derivative test for f(x, y)?

Accepted Answer

D = fxx(x0, y0) fyy(x0, y0) - (fxy(x0, y0))^2

Question 18

In an optimization problem, what is the objective function?

Accepted Answer

The function to be maximized or minimized.

Question 19

What is a constraint in an optimization problem?

Accepted Answer

An equation or inequality in one or more variables that limits the possible solutions.

Question 20

What is the method of Lagrange multipliers used for?

Accepted Answer

Optimizing a function subject to one or more constraints.

Question 21

With one constraint, what variable usually represents the Lagrange multiplier?

Accepted Answer

λ (lambda).

Question 22

What is a δ ball around (x0, y0, z0)?

Accepted Answer

All points in R^3 at a distance less than δ from (x0, y0, z0).

Multivariable Calculus: Differentiation of Functions of Several Variables flashcards

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