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Unformatted text preview: Section 15.6 Directional Derivatives and the Gradient Vector Finding rates of change in different directions Recall that when we first started considering derivatives of functions of more than one variable, we had to introduce the concept of a partial derivative since there is more than one variable we can differentiate with respect to. For a function of two variables, the partial derivatives f x and f y had a geometric interpretation - specifically, they represented the rate of change of f in the vector i direction and vector j directions respectively. In this section we consider the more general case of determining the rate of change of a function f in the direction of any arbitrary vector vectorv by using partial derivaitives. 1. Directional Derivatives As with single variable calculus, by the rate of change of f in the direction of the vector vectorv , we mean the slope of a tangent line to f at the point ( a, b ) in the direction of vectorv . In order to calculate this, we simply generalize the ideas from single variable calculus. Suppose that we want to find the rate of change of f ( x, y ) in the direction of some unit vector vectoru = u 1 vector i + u 2 vector j at the point P ( a, b ) (it is important that vectoru is a unit vector or this will not work). Let Q = ( a + u 1 h, b + u 2 h ) (so the point in the xy-plane a distance h from P along the vector vectoru ). h P Q The average rate of change from P to Q can be calculated using a difference quotient. Specifically, the average rate of change will equal f ( Q ) f ( P ) | vector PQ | = f ( a + u 1 h, b + u 2 h ) f ( a, b ) h 1 2 (observe that since vectoru is a unit vector, the length of vector PQ will be h (WHY?)). As with single variable, we can take the limit h 0, and as h gets smaller, the difference quotient approximates the slope of f at P in the direction of vectoru more accurately. This motivates the following definition. Definition 1.1. The directional derivative of f ( x, y ) at ( a, b ) in the direction of a unit vector vectoru is D vectoru f ( a, b ) = lim h f ( a + u 1 h, b + u 2 h ) f ( a, b ) h if this limit exists. Example 1.2. For an arbitrary function f ( x, y ), determine the direc- tional derivatives D vector i f ( x, y ) and D vector j f ( x, y ). By definition, D vector i f ( x, y ) = lim h f ( x + h, y ) f ( x, y ) h = f x and D vector j f ( x, y ) = lim h f ( x + u 1 h, y ) f ( x, y ) h = f y . Example 1.3. If the following is a contour diagram for f ( x, y ) with the z = 0 contour at the origin, going up by 1 for each concentric circle, approximate the rate of change of f ( x, y ) at (1 , 1) in the direction of vectoru = vector i + vector j ....
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