Question

1)

a) Let z=x4 +x2y, x=s+2t−u, y=stu2: Find:

( I ) ∂z ∂s

( ii ) ∂z ∂t

( iii ) ∂z ∂u

when s = 4, t = 2 and u = 1

1) b> Let ⃗v = 〈3, 4〉 and w⃗ = 〈5, −12〉. Find a vector (there’s more than one!) that bisects the angle between ⃗v and w⃗.

Answer #1

Let s = f(x; y; z) and x = x(u; v; w); y = y(u; v; w); z = z(u;
v; w). To calculate ∂s ∂u (u = 1, v = 2, w = 3), which of the
following pieces of information do you not need?
I. f(1, 2, 3) = 5
II. f(7, 8, 9) = 6
III. x(1, 2, 3) = 7
IV. y(1, 2, 3) = 8
V. z(1, 2, 3) = 9
VI. fx(1, 2, 3)...

For the following equation x=2t^2,y=3t^2,z=4t^2;1 less
or equal to t less or equal to 3
i) Write the positive vector tangent of the curve with parametric
equations above.
ii) Find the length function s(t) for the curve.
iii) Write the position vector of s and verify by differentiation
that this position vector in terms of s is a unit tangent to the
curve.

5. The utility function of a consumer is u(x, y)
=x2y
i. Find the demand functions x(p1,p2,m)
and y(p1,p2,m).
ii. What is the consumers indirect utility?

1. f(x, y, z) = 2x-1 − 3xyz2 + 2z/
x4
2. f(s, t) = e-bst − a ln(s/t) {NOTE: it is
-bst2 }
Find the first and second order partial derivatives for question
1 and 2.
3. Let z = 4exy − 4/y and x =
2t3 , y = 8/t
Find dz/dt using the chain rule for question 3.

. Find the flux of the vector field F~ (x, y, z) =
<y,-x,z> over a surface with downward orientation, whose
parametric equation is given by r(s, t) = <2s, 2t, 5 − s 2 − t 2
> with s^2 + t^2 ≤ 1

1. Let T(x, y, z) = (x + z, y − 2x, −z + 2y) and S(x, y, z) =
(2y − z, x − z, y + 3x). Use matrices to find the composition S ◦
T.
2. Find an equation of the tangent plane to the graph of x 2 − y
2 − 3z 2 = 5 at (6, 2, 3).
3. Find the critical points of f(x, y) = (x 2 + y 2 )e −y...

Let X, Y ∼ U[0, 1], be independent and let Z = max{X, Y }. (a)
(10 points) Calculate Pr[Z ≤ a]. (b) (10 points) Calculate the
density function of Z. (c) (5 points) Calculate V ar(Z).

Let f(x, y) =sqrt(1−xy) and consider the surface S defined by
z=f(x, y).
find a vector normal to S at (1,-3)

If z=(x+4y)ex+y,x=ln(u),y=v,z=(x+4y)ex+y,x=ln(u),y=v, find
∂z∂u∂z∂u and ∂z∂v∂z∂v. The variables are restricted to domains on
which the functions are defined.

1.Let y=6x^2. Find a parametrization of the
osculating circle at the point x=4.
2. Find the vector OQ−→− to the center of the
osculating circle, and its radius R at the point
indicated. r⃗
(t)=<2t−sin(t),
1−cos(t)>,t=π
3. Find the unit normal vector N⃗ (t)
of r⃗ (t)=<10t^2, 2t^3>
at t=1.
4. Find the normal vector to r⃗
(t)=<3⋅t,3⋅cos(t)> at
t=π4.
5. Evaluate the curvature of r⃗
(t)=<3−12t, e^(2t−24),
24t−t2> at the point t=12.
6. Calculate the curvature function for r⃗...

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