Question

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) = 20

VII. fx(7, 8, 9) = 30

VIII. xu(1, 2, 3) = −5

IX. xu(7, 8, 9) = −7

Answer #1

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.

Let u, v, and w be vectors in Rn. Determine which of the
following statements are always true. (i) If ||u|| = 4, ||v|| = 5,
and ?||u + v|| = 8, then u?·?v = 4. (ii) If ||u|| = 2 and ||v|| =
3, ?then |u?·?v| ? 5. (iii) The expression (v?·?w)u is both
meaningful and defined. (A) (ii) and (iii) only (B) (ii) only (C)
none of them (D) all of them (E) (i) only (F) (i) and...

5. Le tv=〈1,1,1〉 and w=2i+j−3k.
(a) Calculate the following.
(i) 4v + 3w
(ii) ∥w∥
(iii) ew (i.e., the unit vector in the direction of w)
(iv) w·v
(v) cos(θ), where θ is the angle between w and v
(vi) projv(w) 3 (Forreference: v=〈1,1,1〉andw=2i+j−3k.)
(vii) The decomposition w = w|| + w⊥ with respect to v
(viii) w × v
(ix) sin(θ), where θ is the angle between v and w
(b) Are v and w orthogonal? Justify your answer!...

Solve the following problem:
max 3x+y
s.t. 4x+y≤ 8
x+y≥3
x+4y≤8
x,y for all integers
Part i) Is this problem convex ? why?
Part ii)Is (x,y)=(1,4) a feasible solution?
Part iii)Is (x,y)=(1,1) a feasible solution?
Part iv)s (x,y)=(1.5,0.5) a feasible solution?
Part v)What is the value of the objective function that
corresponds to each of the previous three solutions?
Part vi)Do each of these three values correspond to a lower
bound, upper bound, non or both?
Part vii)If we eliminate...

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i. f(2) = 3 and f(1) = −1
ii. lim x→−4 f(x) = −∞
iii. limx→∞ f(x) = 1
iv. lim x→−∞ f(x) = −2
v. lim x→−1+ f(x) = ∞
vi. lim x→−1− f(x) = −∞
vii. f 0 (x) > 0 on (−4, −3.5) ∪ (−2.5, −1.5) ∪ (1, 2) ∪ (2,
∞)
viii. f 0 (x) < 0 on (−∞, −4)...

Evaluate the following.
f(x, y) = x + y
S: r(u, v) = 5
cos(u) i + 5 sin(u)
j + v k, 0 ≤ u
≤ π/2, 0 ≤ v ≤ 3

Let the linear transformation T: V--->W be such that T (u) =
u2 If a, b are Real. Find T (au + bv) ,
if u = (x, y) v = (z, w) and uv = (xz-yw, xw + yz)
Let the linear transformation T: V---> W be such that T (u)
= T (x, y) = (xy, 0) where u = (x, y), with 2, -3. Then, if u = (
1.0) and v = (0.1). Find the value...

1. Let u(x) and v(x) be functions such that
u(1)=2,u′(1)=3,v(1)=6,v′(1)=−1
If f(x)=u(x)v(x), what is f′(1). Explain how you arrive at your
answer.
2. If f(x) is a function such that f(5)=9 and f′(5)=−4, what is the
equation of the tangent line to the graph of y=f(x) at the point
x=5? Explain how you arrive at your answer.
3. Find the equation of the tangent line to the function
g(x)=xx−2 at the point (3,3). Explain how you arrive at your
answer....

Suppose f is a differentiable function of x
and y, and
g(u, v) =
f(eu
+ sin(v),
eu +
cos(v)).
Use the table of values to calculate
gu(0, 0)
and
gv(0, 0).
f
g
fx
fy
(0, 0)
0
5
1
4
(1, 2)
5
0
6
3
gu(0, 0)
=
gv(0, 0)
=

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.

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