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sn8_1 - Math 2433 Week 8 Popper008 15.4 The Gradient as a...

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Math 2433 – Week 8 Popper008
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15.4 - The Gradient as a Normal; Tangent Lines and Tangent Planes Suppose that f ( x , y ) is a nonconstant function that is continuously differentiable. That means f is differentiable and its gradient f is continuous. We saw last week that at each point in the domain f (if 0 ) points in the direction of the most rapid increase of f . Also, at each point of the domain, the gradient vector f (if 0 ) is perpendicular to the level curve of f that passes through that point. The vector Is perpendicular to the gradient so it is the tangent vector . The equation of the tangent line is: And the equation of the normal line is:
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Example: Write an equation for the tangent line and an equation for the normal line at point P . For f(x,y,z) , the equation for the tangent plane is And the normal line is
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Example: Find an equation for the tangent plane and scalar parametric equations for the normal line at the point P .
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