Question

Let z = cos (- (5y^{2} + 2x) ). Then

a) The rate of change in z at (-1,-1) as we change x but hold y fixed is ___

and

b) The rate of change in z at (-1, -1) as we change y but hold x fixed is ____.

Answer #1

Let F~ (x, y, z) = x cos(x 2 + y 2 − z 2 )~i + y cos(x 2 + y 2 −
z 2 )~j − z cos(x 2 + y 2 − z 2 ) ~k be the force acting on a
particle at location (x, y, z). Under this force field, the
particle is moved from the point P = (1, 1, 1) to Q = (0, 0, √ π).
What is the work done by...

True or false? and explain please.
(i) Let Z = 2X + 3. Then, Cov(Z,Y ) = 2Cov(X,Y ).
(j) If X,Y are uncorrelated, and Z = −5−Y + X , then V ar(Z) = V
ar(X) + V ar(Y ).
(k) Let X,Y be uncorrelated random variables (Cov(X,Y ) = 0) with
variance 1 each. Let A = 3X + Y . Then, V ar(A) = 4.
(l) If X and Y are uncorrelated (meaning that Cov[X,Y ] =...

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...

Verify the identity: z=x+iy
|sin z|^2=1/2*(cosh(2y)-cos(2x))

Verify that the functions y1 = cos x − cos 2x and y2 = sin x −
cos 2x both satisfy the differential equation y′′ + y = 3 cos
2x.

Let w = (x 2 -z)/ y4 ,
x = t3+7,
y = cos(2t),
z = 4t.
Use the Chain Rule to express dw/ dt in terms of t. Then
evaluate dw/ dt at t = π/ 2

Let F ( x , y , z ) =< e^z sin( y ) + 3x , e^x cos( z ) + 4y
, cos( x y ) + 5z >, and let S1 be the sphere x^2 + y^2 + z^2 =
4 oriented outwards Find the flux integral ∬ S1 (F) * dS. You may
with to use the Divergence Theorem.

Let x,y,zx,y,z be (non-zero) vectors and suppose
w=10x+10y−4zw=10x+10y−4z.
If z=2x+2yz=2x+2y, then w=w= x+x+ yy.
Using the calculation above, mark the statements below that must
be true.
A. Span(x, z) = Span(w, z)
B. Span(w, y, z) = Span(x, y)
C. Span(w, z) = Span(w, y)
D. Span(w, x) = Span(x, y, z)
E. Span(w, y) = Span(w, x, y)

The temperature at a point (x, y, z) is given by T(x, y, z) =
400e−x2 − 5y2 − 9z2 where T is measured in °C and x, y, z in
meters.
(a) Find the rate of change of temperature at the point P(2, −1,
2) in the direction towards the point (3, −5, 6). °C/m
(b) In which direction does the temperature increase fastest at
P?
(c) Find the maximum rate of increase at P.

solve the given DE equation y''-4y'+20y= (x+1)e^2x cos x + 2x^2
e^2x sinx

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