14-4 Tangent Plane & Linear Approximation - y = dy =...

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14.4 Tangent Plane & Linear approximation ex: z = f(x . y) = ln(x-3y) @ P(7.2) = P(7,2,0) Evaluate f(6.9, 2.06) - you need to approximate Approximate f(6.9,2.06) = ~ - 0.1 EQ of the tangent plane to z = ln (x - 3y) @ (7,2) z - z 0 = f x (x 0 ,y 0 ) (x-x 0 ) + f y (y-y 0 ) f x = (partial deriv) f / (partial deriv) x = 1/(x-3y) (1) | (7,2) = 1/(7-3(2) = 1 f y = (partial deriv)f/(partial deriv)y = 1/x-3y (-3) | (7,2) = 1/7-3(2) (-3) = -3 Differentials: Recall Mat114 /_\ y = f(y 0 +/_\y)-f(y 0 ) - the amount of change of the value of the function as we go from p 0 to p 1 in the curve y=f(x) . dy = amount of change of the tangent line as we go from p o to p 1 /_\ y != dy
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Unformatted text preview: /_\ y = dy = f'(x) dx /_\ z = dz = fx(x,y)dx + fy(x,y) dy- Use the differentials to estimate the max error in calculating the surface area of the box. Approximate surface area: /_\s ~ ds = Sx . dx + S y . dy + S z . dz s = 2xy + 2yz + 2xz w = f(x,y,z) /_\w ~ dw = fx . dx + fy . dy + fz . dz Sx = 2y+2z | (80,60,50) = 220 Sy = 2x + 2z | (80,60,50) = 260 S z = 2y+2x | (80,60,50) = 280 /_\ ~ ds = 220 (0.2) + 260(.2) + 280 (0.2) = 152 cm 2 S = 2xy + 2yz + 2xz Note: /_\x = dx /_\y = dy /_\z = dz-To measure change in surfaces, we use differentials....
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