Question

1.

a) Let: z=rcis(t). Enter an argument of -z.

b) Let:z1=4cis(3π/4) and z2=2cis(π/3).

Calculate z1*(/z2) in polar form. Find its modulus and
principle argument. Calculate (z1/z2) in polar form. Find its
modulus and principle argument.

where /z2: the conjugate of z2

Answer #1

Find all values z1 and z2 such that (2, −1, 3), (1, 2, 2), and
(−4, z1, z2) do not span R3

1. Sketch the polar function r = (θ − π/4)(θ − 3π/4) on the
interval 0 ≤ θ ≤ 2π. Then find all lines tangent to this polar
function at the point (0, 0).
2. Find the area of the region enclosed by one loop of the curve
r = 5 sin(4θ).
3. Use the Monotone Sequence Theorem to determine that the
following sequence converges: an = 1/ 2n+3 .

Let x = [1, 1]T , y = [1, 1]T ∈ R 2 and let f : R 2 =⇒ R 2 with
f(z) =z1.x + z2.y for any z = [z1, z2] T ∈ R 2 . Further, z = g(r)
= [r 2 , r3 ] where r ∈ R . Show how chain rule is applied here
giving major steps of the calculation, write down the expression
for ∂f ∂r , and also evaluate ∂f/ ∂r at...

If we have a competitive industrial form that has the production
function q=z1^(1/4)*z2^(a)z3^(1/4)
q is the output, z1 z2 z3 is the production inputs and a is
parameter.
Assume that production input 2 (z2) is fixed in the short
run
1) Find the short run conditional input demand functions for the
firm
2) Find the short run cost function for the firm
3) Find the short run supply function for the firm
4) what happens to the conditional input demand,...

a.
r=3 - 3cos(Θ), enter value for r on a table
when;
Θ=0, (π/3),(π/2),(2π/3),π,(4π/3),(3π/2),(5π/3) & 2π
b. plot points from a, sketch graph
c. use calculus to find slope at (π/2),(2π/3),(5π/3)
& 2π
d. find EXACT area inside the curve in 1st
quadrant

Consider the sum of double integrals Z1 1/√2Zx √1−x2 xy dy dx
+Z√2 1 Zx 0 xy dy dx +Z2 √2Z√4−x2 0 xy dy dx .
a. [4] Combine into one integral and describe the domain of
integration in terms of polar coordinates. Give the range for the
radius r.
b. [4] Compute the integral.

Consider the sum of double integrals Z1 1/√2Zx √1−x2 xy dy dx
+Z√2 1 Zx 0
xy dy dx +Z2 √2Z√4−x2 0
xy dy dx .
a. [4] Combine into one integral and describe the domain of
integration in terms of polar coordinates. Give the range for the
radius r. b. [4] Compute the integral.

Let X,Y be independent [0,1]-uniform. Calculate expected values
of Z1=XY/(X+1), Z2= X/(Y+1),
Z3=(x+y)(x-2y). Calculate r[(X+Y), (X-Y)].

1.) Let f ( x , y , z ) = x ^3 + y + z + sin ( x + z ) + e^( x
− y). Determine the line integral of f ( x , y , z ) with respect
to arc length over the line segment from (1, 0, 1) to (2, -1,
0)
2.) Letf ( x , y , z ) = x ^3 * y ^2 + y ^3 * z^...

The temperature at a point (x,y,z) is given by
T(x,y,z)=200e−x2−y2/4−z2/9, where T is measured in degrees celsius
and x,y, and z in meters. There are lots of places to make silly
errors in this problem; just try to keep track of what needs to be
a unit vector. A. Find the rate of change of the temperature at the
point (0, -1, 2) in the direction toward the point (-1, 4, 2). b)In
which direction (unit vector) does the temperature...

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