Prove: If x = x t and y = y t are differentiable at t, and if z = f x , y is differentiable at the point x t , y t , then d z d t = ∇ z ⋅ r' t where r t = x t i + y t j .
Prove: If x = x t and y = y t are differentiable at t, and if z = f x , y is differentiable at the point x t , y t , then d z d t = ∇ z ⋅ r' t where r t = x t i + y t j .
Prove: If
x
=
x
t
and
y
=
y
t
are differentiable at t, and if
z
=
f
x
,
y
is differentiable at the point
x
t
,
y
t
,
then
d
z
d
t
=
∇
z
⋅
r'
t
where
r
t
=
x
t
i
+
y
t
j
.
With integration, one of the major concepts of calculus. Differentiation is the derivative or rate of change of a function with respect to the independent variable.
Let v = (3,4, 12) . Find the directional derivative of f(r, y, z) = x – y? +32³ in the direction of v.
Prove that if f is a differentiable function such that ∇f(x0, y0) = 0, then the tangent plane at (x0, y0) is horizontal.
B. Let n > 3 be an odd number and f(a, y) = (x" + y")/".
(a) Determine if f(x, y) is differentiable at (0,0).
(b) Find all the points where f(a,y) is not differentiable. Justify your answer,
(c) Find the unit vector(s) i € R? such that the directional derivative Daf(0,0) is minimum.
(d) Compute
or show that it does not exist.
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