Question

does there exist a surface x=x(u,v) with E=1, F=0, G=(cos^2)u and e=(cos^2)u, f=0, g=1 ?

Answer #1

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)
=

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

Differential Geometry:
Let a be a positive constant, and define x(u,v) := (v cos u, v
sin u, a u), where 0 < u < 2π and v real. Compute the
principal curvatures and the Gaussian curvature at each point of
the surface defined by x.

Prove the identity
1) sin(u+v)/cos(u)cos(v)=tan(u)+tan(v)
2) sin(u+v)+sin(u-v)=2sin(u)cos(v)
3) (sin(theta)+cos(theta))^2=1+sin(2theta)

Consider the surface σ(u,v) = (f(u)cosv,f(u)sinv,g(u)).
Calculate the normal vector. Also identify the surface.

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....

If f (2) = 1, \ f '(2) = - 3 and g (x) = log_2 (x ^ 2f (x) / cos (f (x)) then g' (2) is equal to:
a)-9.625983
B)-3.327777
c)-4.800967
d)-12.01444
E)The answer doesn´t exist

Let g(u, v) = f(u 3 − v 3 , v3 − u 3 ). Prove that v^2 ∂g/∂u −
u^2 ∂g/∂v = 0, using the Chain Rule

For the function w=f(x,y) , x=g(u,v) , and
y=h(u,v). Use the Chain Rule to
Find ∂w/∂u and
∂w/∂v when u=2 and v=3 if
g(2,3)=4, h(2,3)=-2,
gu(2,3)=-5,
gv(2,3)=-1 ,
hu(2,3)=3,
hv(2,3)=-5,
fx(4,-2)=-4, and
fy(4,-2)=7
∂w/∂u=
∂w/∂v =

Find f.
f ''(x) = 3 + cos(x), f(0) =
−1, f(5π/2) = 0

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