Questions in PDF

Assignment Set 2 (Due May 21, 2pm)
Ruipeng Shen
May 20, 2014
2.6.12 Let z = f (r), where r =
yzx − xzy = 0 for all (x, y) 6= (0, 0).
2.6.16
p
x2 + y 2 and f is a differentiable function. Prove that
Let w = f (x, y, z), where x = r cos θ and y = r sin θ. Find ∂w/∂r, ∂w/∂θ and ∂w/∂z.
2.7.20 Determine the maximum rate of change of the function f (x, y, z) =
p = (3, 3, 2), and the direction in which it occurs.
4.1.8
√
xyz at the point
Find the second partial derivatives zxx , zxy , zyx and zyy of the function z = x arctan(y/x).
4.1.16 Explain why there is no c2 function f (x, y) such that fx (x, y) = ex + xy and fy (x, y) =
ex + xy.
4.2.8 Find the second-order Taylor formula for the function f (x) = cos x at the point x0 = π/3.
Give the remainder in integral form.
4.2.24 Find the second-order Taylor formula for the function f (x, y) = ln(x2 + y 2 + 1) at the
point (0, 1).
4.3.10 Find all critical points (if any) of the given function f (x, y) and determine whether they
are local extreme points or saddle points.
f (x, y) = x3 + y 3 + 3x2 y − 3y.
4.3.14 Find all critical points (if any) of the given function f (x, y) and determine whether they
are local extreme points or saddle points.
f (x, y) = ln(x2 + y 2 + 2).
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