Question

g (u, v) is a differentiable function and g (1,2) = 100, gu (1,2) = 3, gv (1,2) = 7 are given. The function f is defined as f (x, y, z) = g (xyz, x ^ 2, y^2z). Find the equation of the tangent plane at the point (1,1,1) of the f (x, y, z) = 100 surface.

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)
=

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 =

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(b) Graph the function. Can you spot a point “a” such that the
tangent line through (a, f(a)) does not exist? If yes, show using
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1. Let u(x) and v(x) be functions such that
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If f(x)=u(x)v(x), what is f′(1). Explain how you arrive at your
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2. If f(x) is a function such that f(5)=9 and f′(5)=−4, what is the
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An implicitly defined function of x, y and z is given along with
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Use the gradient ∇F to:
(a) find the equation of the normal line to the surface at
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(b) find the equation of the plane tangent to the surface at
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r(u,v)=〈3u,−5v-5u^2,−5v^2〉 at the point (3,0,−5)

8).
a) Find an equation of the tangent plane to the surface z = x at
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