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multivariable_05_Curve_Sketching

multivariable_05_Curve_Sketching - 5 Curve Sketching...

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5 Curve Sketching Whether we are interested in a function as a purely mathematical object or in connection with some application to the real world, it is often useful to know what the graph of the function looks like. We can obtain a good picture of the graph using certain crucial information provided by derivatives of the function and certain limits. A local maximum point on a function is a point ( x, y ) on the graph of the function whose y coordinate is larger than all other y coordinates on the graph at points “close to” ( x, y ). More precisely, ( x, f ( x )) is a local maximum if there is an interval ( a, b ) with a < x < b and f ( x ) f ( z ) for every z in ( a, b ). Similarly, ( x, y ) is a local minimum point if it has locally the smallest y coordinate. Again being more precise: ( x, f ( x )) is a local minimum if there is an interval ( a, b ) with a < x < b and f ( x ) f ( z ) for every z in ( a, b ). A local extremum is either a local minimum or a local maximum. Local maximum and minimum points are quite distinctive on the graph of a function, and are therefore useful in understanding the shape of the graph. In many applied problems we want to find the largest or smallest value that a function achieves (for example, we might want to find the minimum cost at which some task can be performed) and so identifying maximum and minimum points will be useful for applied problems as well. Some examples of local maximum and minimum points are shown in figure 5.1. If ( x, f ( x )) is a point where f ( x ) reaches a local maximum or minimum, and if the derivative of f exists at x , then the graph has a tangent line and the tangent line must be horizontal. This is important enough to state as a theorem, though we will not prove it. 91
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92 Chapter 5 Curve Sketching . A B . A A B Figure 5.1 Some local maximum points ( A ) and minimum points ( B ). THEOREM 5.1 Fermat’s Theorem If f ( x ) has a local extremum at x = a and f is differentiable at a , then f ( a ) = 0. Thus, the only points at which a function can have a local maximum or minimum are points at which the derivative is zero, as in the left hand graph in figure 5.1, or the derivative is undefined, as in the right hand graph. Any value of x for which f ( x ) is zero or undefined is called a critical value for f . When looking for local maximum and minimum points, you are likely to make two sorts of mistakes: You may forget that a maximum or minimum can occur where the derivative does not exist, and so forget to check whether the derivative exists everywhere. You might also assume that any place that the derivative is zero is a local maximum or minimum point, but this is not true. A portion of the graph of f ( x ) = x 3 is shown in figure 5.2. The derivative of f is f ( x ) = 3 x 2 , and f (0) = 0, but there is neither a maximum nor minimum at (0 , 0).
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