Tutorial: Functions of several variables from the numerical, algebraic, and graphical viewpoints
Adaptive game version
This tutorial: Part B: Graphical viewpoint
(This topic is also in Section 15.1 in Finite Mathematics and Applied Calculus) #[I don't like this new tutorial. Take me back to the older tutorial!][No me gusta este nueve tutorial. ¡Regresame al tutorial más viejo!]#
#[Graph of a function of two variables][Gráfica de una función de dos variables.]#
The graph of a function $f(x,y)$ of two variables is the set of all points of the form $(x, y, z)$ where $(x,y)$ is in the domain of $f$ and $z = f(x,y).$ More simply, the graph is the set of all points of the form $(x,y,f(x,y))$ for $(x,y)$ in the domain of $f.$
#[Example][Ejemplo]#
#[Here is the graph of][Aquí está la gráfica de]# $f(x,y) = x^2 + y^2$ (#[Equation][Ecuación]# $z = x^2 + y^2$)
#[Noteworthy features][Características destacables]#
- #[Horizontal cross-sections][Secciones transversales horizontales]#: #[You get a horizontal cross-section by setting $z ={}$ constant in its equation ($z$ measures the height in the graph as drawn). For instance, $z = 1$ gives $1 = x^2+y^2$, which is the equation of the circle of radius 1 center $(0,0)$. Similarly, $z = 4$ gives a circle of radius 2.][Obtienes una sección transversal horizontal estableciendo $z ={}$ constante en su ecuación ($z$ mide la altura en la gráfica dibujada). Por ejemplo, $z = 1$ da $1 = x^2+y^2$, que es la ecuación del círculo de radio 1 galope $(0,0)$. De manera similar, $z = 4$ da un círculo de radio 2.]#
#[Here is the same graph again, showing the slices at $z = 1, 2, 3, 4$, and 5$, which are circle of radius $1, \sqrt{2}, \sqrt{3}, \sqrt{4} = 2$, and $\sqrt{5}.$][Aquí está nuevamente el mismo gráfico, que muestra los cortes en $z = 1, 2, 3, 4$ y 5$, que son círculos de radio $1, \sqrt{2}, \sqrt{3}, \sqrt{4 } = 2$ y $\sqrt{5}.$]# - #[Vertical cross-sections][Secciones transversales verticales]#: #[Its vertical slices through the $z$-axis are all the same shape becuase of the circular symmetry. For instance, the slice through the $yz$-plane is the parabola $z = y^2$ (put $x = 0$ to see why), explaining why why the surface looks like a parabola when seen end-on. Slices through planes parallel to that, through $x = \pm 1, \pm 2, ... $ are of the form $z = y^2 + {}$ constant, and so are also parabolas, as we see in the following ficure.][Sus cortes verticales a través del eje $z$ tienen la misma forma debido a la simetría circular. Por ejemplo, el corte a través del plano $xyz$ es la parábola $z = y^2$ (pongamos $x = 0$ para ver por qué), que explica por la cual la superficie parece una parábola cuando se ve de frente. Los cortes a través de planos paralelos a ese, a través de $x = \pm 1, \pm 2, ... $ tienen la forma $z = y^2 + {}$ constante, por lo que también son parábolas, como vemos en la siguiente figura.]#
Now try the exercises in Section 15.1 in Finite Mathematics and Applied Calculus.
or check o see if new tutorials have been added for this chapter.
Copyright © 2019 Stefan Waner and Steven R. Costenoble