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Few things in baseball are as exciting as a home run. In the summer of 1998, the entire baseball world got caught up in the excitement, as sluggers Mark McGwire of the St. Louis Cardinals and Sammy Sosa of the Chicago Cubs both approached and then passed Roger Maris' record for the most home runs in a single season. With each crack of the bat, spectators and players alike watched the flight of the ball, wondering if it would carry far enough to clear the fence for another home run or fall short and be caught by a waiting fielder. It's usually difficult to tell whether a ball will stay in the park and get caught or fly over the fence for a home run. It's reasonable then, to ask what factors determine which outcome will occur.
In sections 5.5 and 11.3, we developed the equations for the flight of such a projectile, although always under the (unrealistic) assumption of no air resistance. There, we discovered that a ball hit at angle  above the horizontal with initial velocity v will have a horizontal range of
R =  v2sin 2. |
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Using the properties of the sine function, we have been able to draw some interesting conclusions from this formula. However, notice that the equation for R differs from most of the functions we have studied as yet, in that R depends on two independent (unrelated) variables, v and . So far, we have only developed the calculus for functions of one variable.
It may have occurred to you that the situation is even more complicated than we've described. The range certainly depends on the initial velocity and launch angle, but it also depends on air effects in the form of air drag, wind velocity and the Magnus force. Air drag, in turn, varies with temperature, humidity and altitude, among other factors. It should now be clear that the range of a fly ball is not particularly easy to predict.
In this chapter, we introduce functions of several variables and extend the ideas of calculus to those functions. Often, we will gain valuable insight by considering the effect of one variable at a time. For instance, from the formula R = v2sin 2, we can conclude that for a given initial velocity, the maximum range is obtained with =  /4 (so that sin 2 = 1 ). For other functions, the interplay among variables may be more subtle, so we will learn to combine information from the individual variables into information about the entire function. In section 12.3, we introduce partial derivatives to analyze one aspect of the relationship among variables. The chain rule (introduced in section 12.5) and directional derivatives (discussed in section 12.6) extend our ability to analyze functions of two or more variables.
After studying the basic calculus for functions of several variables, you should be able to find extrema of relatively simple functions. Perhaps more importantly, you should understand enough about such functions to be able to approximate extrema of more complicated functions. Of course, in real applications problems, you are rarely given a convenient formula. Even so, the understanding of multivariable calculus that you develop here will help you to make sense of a broad range of complex phenomena. |