# MA 227 Standard S05

Multivariable Functions

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)