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MA 227 Standard S05

Multivariable Functions
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At the end of the course, each student should be able to…

  • S05: MulivarFunc. Sketch and analyze the domain, level curves, and graph of a two-variable real-valued function.

S05: Multivariable Functions

  • Functions of multiple variables are evaluated in the same way as single-variable functions.
    • Take \(f(x,y,z)=x^2yz-\frac{ze^x}{y^2+4}\) for example.
    • Then \(f(0,1,-3)=(0)^2(1)(-3)-\frac{(-3)e^{(0)}}{(1)^2+4}=-\frac{3}{5}\).
    • The domain of a function is the set of allowable inputs for that function.
  • Two-variable functions may be analyzed/sketched in 2D or 3D as follows.
    • The level curves of a two-variable function are the curves where \(f(x,y)=k\) for some value of \(k\). These curves lay in the domain and illustrate a topographical map of the function’s values.
    • The graph of a two-variable function is the surface \(z=f(x,y)\). It may be sketched from the equation \(z=f(x,y)\) or by lifting the level curves to appropriate heights in \(xyz\) space.

Textbook References

  • University Calculus: Early Transcendentals (3rd Ed)
    • 13.1 (exercises 1-16, 31-52)