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tangent

# tangent - TANGENT PLANES O Knill Math21a REMINDER TANGENT...

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Unformatted text preview: TANGENT PLANES O. Knill, Math21a REMINDER: TANGENT LINE. Because vectorn = ∇ f ( x , y ) = ( a, b ) is perpendicular to the level curve f ( x, y ) = c through ( x , y ), the equation for the tangent line is ax + by = d , a = f x ( x , y ), b = f y ( x , y ), d = ax + by Example: Find the tangent to the graph of the function g ( x ) = x 2 at the point (2 , 4). Solution: the level curve f ( x, y ) = y − x 2 = 0 is the graph of a function g ( x ) = x 2 and the tangent at a point (2 , g (2)) = (2 , 4) is obtained by computing the gradient ( a, b ) = ∇ f (2 , 4) = (− g ′ (2) , 1 ) = (− 4 , 1 ) and forming − 4 x + y = d , where d = − 4 · 2+1 · 4 = − 4. The answer is − 4 x + y = − 4 which is the line y = 4 x − 4 of slope 4. Graphs of 1D functions are curves in the plane, you have computed tangents in single variable calculus.-3-2-1 1 2 3-6-4-2 2 4 6 8 GRADIENT IN 3D. If f ( x, y, z ) is a function of three variables, then ∇ f ( x, y, z ) = ( f x ( x, y, z ) , f y (...
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