The main topics covered in this course included:

  1. Graphing surfaces: We learned how to graph a surface given by an equation of the form z=f(x,y). We used the section method and level curves to visualize the surface and gain insights into its properties.
    • Section method: for $x (\text{or } y) = 0, 1, 2, 3, …$, draw $z=f(x,y)$ on a $yz$(or $yx$)-plain.
    • level curves: for $z = 0, 1, 2, 3, …$, draw on a $xy$-plain.
    • Important example: $z^2 = x^2+y^2$ and $z^2 = x^2-y^2$
    • test: $x^2$

  2. Partial derivatives: We introduced the concept of partial derivatives, which were derivatives with respect to one variable while holding all other variables constant. We learned how to compute partial derivatives of a function of two variables and interpret them geometrically as slopes of tangent lines.

  3. Tangent planes: We used partial derivatives to find the equation of the tangent plane to a surface at a given point. We also explored the geometric interpretation of the tangent plane as an approximation of the surface near the given point.

  4. Multivariable functions: We extended the concept of the tangent plane to functions of more than two variables. We learned how to compute partial derivatives and find tangent planes to these functions.