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**Unformatted text preview: **deﬁned
implicitly by the equation above. Then using implicit diﬀerentiation on the equation
∂z
∂z
∂z
∂z
above we can calculate that ∂ x (2, 1) = 4 and ∂ y (2, 1) = −5 so dz = ∂ x dx + ∂ y dy =
4(.1) − 5(−.1) = .9 and z = −.1 as before. Page 2 of 8 Tuesday, October 19, 2010 1st Midterm Exam MTH 164 (Multivariable Calculus) (c) Find the maximum rate of change of the height of the surface at P (2, 1, −1) above the
xy plane and the direction in the xy -plane at which it occurs.
Answer:
dz = 4dz − 5dy = 4, −5 · dx, dy = 4, −5 dx, dy cos(θ) So the maximum
rate of a change (for a unit vector) will be when the vector is parallel to 4, −5. The
√
optimal direction is = √4,−5 and the rate of change is 16 + 25.
u...

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