Homework 10
1. Suppose f is continuous on (a, b] and lim f (x) exists. Show that
x→a+
R ∞ sin2 x
result to show that 0 x2 dx exists.
Rb
a
f (x) exists. Use this
2. Determine whether each integral is convergent or not. Evaluate those that are convergent.
R∞ x
(a) 0 e2xe +3 dx.
R∞
(b) −∞ cos πtdt.
R∞
1
dv.
(c) 2 v2 +2v−3
3. Find all the first partial derivatives of the following functions.
(a) g(x, y, z) = exz sin(y/z).
(b) s(u, v) =
(c) f (x, y) =
4. Find
∂z
∂x
u+2v
u2 +v2 .
Ry
x
and
2
et dt.
∂2z
∂x∂y
at (0, 0) if z is defined implicitly as a function of x and y by
xy 2 z 3 + x3 y 2 z = x + y + z.
5. Find the tangent plane of the given surfaces at the specified points.
(a) z = ex cos y at (0, 0, 1).
(b) z = 3x2 − y 2 + 2x at (1, −2, 1).
(c) xy + yz + zx = 3 at (1, 1, 1).
(d) sin(xyz) = x + 2y + 3z at (2, −1, 0).
6. If z = f (u, v), where u = xy, v = y/x, and f has continuous second partial derivatives, show
that
x2
2
∂2z
∂2z
∂z
2∂ z
−
y
=
−4uv
+ 2v .
2
2
∂x
∂y
∂u∂v
∂v
7. Find the gradient and the Hessian matrix for the following functions at the specified points.
(a) f (x, y, z) = x3 y 2 z at (1, 1, 1).
(b) v = r cos(s + 2t) at (1, 0, π2 ).
(c) z = 4x3 − xy 2 at (−1, 1).
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