Other Aspects of Calculus

So far, we have studied functions of one variable only. Such functions can be represented visually using graphs in two dimensions; however, there is no good reason to restrict our investigation to two dimensions. Suppose, for example, that instead of determining the velocity of an object moving along a coordinate axis, we want to determine the velocity of a rock fired from a catapult at a given time, or of an airplane moving in three dimensions. We might want to graph real-value functions of two variables or determine volumes of solids of the type shown in Figure 2.11. These are only a few of the types of questions that can be asked and answered using multivariable calculus. Informally, multivariable calculus can be characterized as the study of the calculus of functions of two or more variables. However, before exploring these and other ideas, we must first lay a foundation for the study of calculus in one variable by exploring the concept of a limit.

A diagram in three dimensional space, over the x, y, and z axis where z = f(x,y). The base is the x,y axis, and the height is the z axis. The base is a rectangle contained in the x,y axis plane. The top is a surface of changing height with corners located directly above those of the rectangle in the x,y plane.. The highest point is above the corner at x=0, y=0. The lowest point is at the corner somewhere in the first quadrant of the x, y plane. The other two points are roughly the same height and located above the corners on the x axis and y axis. Lines are drawn connecting the corners of the rectangle to those of the surface.

Figure 2.11 We can use multivariable calculus to find the volume between a surface defined by a function of two variables and a plane.

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Limits

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