Labware - MA35 Multivariable Calculus - Two Variable Calculus
 MA35 Labs 2 » Two Variable Calculus Contents2.1 Functions of Two Variables 2.1.1 Introduction 2.1.2 Linear Functions 2.1.3 Domain, Range & Function Graphs 2.1.4 Slice Curves 2.1.6 Continuity 2.4 Integration Search

Domain, Range & Function Graphs (Page: 1 | 2 )

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The domain of a function of two variables is a subset of the coordinate plane { (x,y) | x,y ∈ R }.

The most common domains are products of intervals of the form a ≤ x ≤ b, c ≤ y ≤ d.

Another common domain in the plane is a closed disc with center (x0,y0) and radius r given by or the open disc where we use less than instead of less than or equal to.

The range of a real-valued function f is the collection of all real numbers f(x,y) where (x,y) is in the domain of f.

The range of the constant function f(x,y) = k is the single number { k }. The range of a linear function L(x,y) = px + qy with p and q not both zero is all real numbers z. The next simplest functions are linear functions f(x,y) = px + qy + k, where p and q are the partial slopes and k is the z-intercept. The range of this function is all real numbers if p and q are not both zero and just the value { k } if p = 0 = q.

The graph of a function of two variables is the collection of points (x,y,f(x,y)) in 3-space where (x,y) is in the domain of f.

Demos

 Domain and Range This demonstration graphs a function f(x,y) over a rectangular domain. By default, the domain is set to -1 ≤ x ≤ 1, -1 ≤ y ≤ 1 and the function is set to f(x,y) = x^2 - y^2. In the window labeled Domain and Range, you can choose both the center and radius of a red disc domain in the xy-plane using the white and red hotspots respectively. The magenta line segment along the z-axis in this window shows the range of f(x,y) over the red disc domain.

Exercises

• 1. What is the range of the function f(x) = ax2 + cy2? (The answer will depend on the constants a and c.)
• 2. What is the range of the function f(x,y) = -x4 + 2x2 -y2 where the domain is all (x,y)? (Give reasons for your answer, without using the words "partial derivative".)
• 3. What is the range of the function f(x,y) = -x4 + 2x2 - y4 + 2y2?
• 4. What is the range of the function f(x,y) = x2 + 2bxy + y2? (The answer will depend on the constant b.)
• 5. What is the range of the function f(x,y) = ax2 + 2bxy + cy2? (The answer will depend on the constants a, b, and c.)