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Let f : R3 → R be de?ned by: f(x, y, z) = xy2+ x3z4+ y5z6
a) Compute ~ ∇f(x, y, z) , and evaluate ~ ∇f(2, 1, 1) .
b) Brie?y explain why f must be a di?erentiable function (you just need to "look" at the equations for the partial derivatives).
c) Find D~uf(2, 1, 1) where ~u is a unit vector in the direction of ~v = h4, 3,-1i .
d) Find an equation for the plane which is tangent to the surface de?ned by
xy2+ x3z4+ y5z6= 11
at the point (2, 1, 1) .
e) Use di?erentials to approximate the value of f(2.01, 1.02, 0.97) . Use a calculator to?nd a more accurate value, and compare.
Marks obtained by 70 students are given below: M arks 20 70 50 60 75 90 40 No.
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Two tangents PA and PB are drawn to the circle with center O, such that ∠APB=120 o . Prove that OP=2AP. Ans: Given : - ∠APB = 120o Construction : -Join OP To prove : -
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