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Engineering Calculus Notes 253

# Engineering Calculus Notes 253 - maximum value which is 1...

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3.3. DERIVATIVES 241 the point. From Equation ( 3.6 ), we see that the directional derivative gives the rate at which f ( −→ x ) changes as we move in the direction −→ u at speed one . In the plane, a unit vector has the form −→ u α = (cos α ) −→ ı + (sin α ) −→ where α is the angle our direction makes with the x -axis. In this case, the directional derivative in the direction given by α is d −→ x 0 f ( −→ u α ) = ∂f ∂x ( −→ x 0 ) cos α + ∂f ∂y ( −→ x 0 ) sin α. Equation ( 3.10 ) tells us that the directional derivative in the direction of the unit vector −→ u is the dot product d −→ x 0 f ( −→ u ) = −→ f · −→ u which is related to the angle θ between the two vectors, so also −→ f · −→ u = vextenddouble vextenddouble vextenddouble −→ f ( −→ x 0 ) vextenddouble vextenddouble vextenddouble bardbl −→ u bardbl cos θ = vextenddouble vextenddouble vextenddouble −→ f ( −→ x 0 ) vextenddouble
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Unformatted text preview: maximum value, which is 1, when θ = 0, which is to say when −→ u points in the direction of −→ ∇ −→ x , and its minimum value of − 1 when −→ u points in the opposite direction. This gives us a geometric interpretation of the gradient, which will prove very useful. Remark 3.3.5. The gradient vector −→ ∇ f ( −→ x ) points in the direction in which the directional derivative has its highest value, known as the direction of steepest ascent , and its length is the value of the directional derivative in that direction. As an example, consider the function f ( x,y ) = 49 − x 2 − y 2 at the point −→ x = (4 , 1) ....
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