Graphing and Analyzing Functions

This video introduces functions of two variables, focusing on determining function values and domains. It explains the concept of a function with two variables, where z is defined in terms of x and y, and discusses the domain and range. The video showcases graphs of functions like z = cos(x)sin(y) and z = ln(1-x-y), highlighting their surfaces. It demonstrates evaluating function values using substitution and determining domains through inequalities. The video concludes with a summary of the key points covered.

What is the domain of a function of two variables?

• All possible values of z

• All possible values of x, y, and z

• All possible values of x and y

• All possible values of x

What is the range of a function of two variables?

• All possible values of x

• All possible values of y

• All possible values of z

• All possible values of x and y

Which function produces a surface with hills and valleys?

• Z = ln(1 - x - y)

• Z = e^(1 - x^2 - y^2)

• Z = cos(x) * sin(y)

• Z = sqrt(4 - x^2 - y^2)

Which graphing software was mentioned as having difficulty plotting certain surfaces?

• Maple

• MATLAB

• Python

• R

What is the z-coordinate when x = -3 and y = 3π/2 for the function Z = -3 * cos(3π/2)?

• 1

• -3

• 0

• 3

What is the ordered triple for x = 2 and y = π/3 in the function Z = 2 * cos(π/3)?

• (2, π/3, 0)

• (2, π/3, 3)

• (2, π/3, 1)

• (2, π/3, 2)

For the function Z = ln(1 - x - y), what must be true for the domain?

• x + y >= 1

• x + y = 1

• x + y < 1

• x + y > 1

What is the domain of the function Z = sqrt(x) in terms of one variable?

• x < 0

• x > 0

• x >= 0

• x <= 0

How is the domain of the function Z = sqrt(4 - x^2 - y^2) expressed?

• x^2 + y^2 <= 4

• x^2 + y^2 >= 4

• x^2 + y^2 > 4

• x^2 + y^2 < 4

What shape does the graph of x^2 + y^2 = 4 represent?

• An ellipse

• A circle

• A triangle

• A square