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Wize University Calculus 2 Textbook > Multivariable Functions

Functions of Several Variables

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Functions of Several Variables

A function of 2 variables
f(x,y)=zf(x,y)=z takes in a point (x,y)(x,y) in R2R^2 and outputs a real number.

A function of 3 variables
f(x,y,z)=wf(x,y,z)=w take in a point (x,y,z)(x,y,z) in R3R^3 and outputs a real number.

Example
Given the function z=f(x,y)=4−y2−x2z=f(x,y)=\sqrt{4-y^2-x^2}
a.) Find the domain and range
b.) Evaluate f(1,−1)f(1,-1)
a.) We can't have a negative number under the square root:
4−y2−x2≥04-y^2-x^2\ge0
4≥x2+y24\ge x^2+y^2
So, the domain includes all points in R2R^2 that are within the circle with radius 2 that is centered at the origin.

Since the value of −y2−x2-y^2-x^2 is always less than or equal to 0 for an x and y values, the range is z≤4z\le\sqrt{4}

Therefore, the domain is D={(x,y)∣x2+y2≤4}D=\left\{\left(x,y\right)|x^2+y^2\le 4\right\} and the range isR=[0,4]R=[0,4]

b.) f(1,−1)=4−(1)2−(−1)2=2f(1,-1)=\sqrt{4-(1)^2-(-1)^2}=\sqrt 2

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Level Curves/Contour Curves

To visualize the graph of a function z=f(x,y)z=f(x,y)
  1. we treat zz as the "heigth" of the graph
  2. we set z=kz=k for various k values
  3. graph the function on the xy-plane at different "heights"
Example
Sketch the graph of the function f(x,y)=1−y+xf(x,y)=1-y+x

Practice Question

State the domain of the function f(x,y)=ln⁡(10y−x2)tan⁡x\displaystyle f\left(x,y\right)=\frac{\ln \left(10y-x^2\right)}{\tan x}