Determine whether the statement is true or false. Explain your answer. Suppose that z = f x , y has continuous first partial derivatives in the interior of a region R in the x y -plane , and set q = 1 , 0 , ∂ z / ∂ x and r = 0 , 1 , ∂ z / ∂ y . Then the surface area of the surface z = f x , y over R is ∬ R q × r d A
Determine whether the statement is true or false. Explain your answer. Suppose that z = f x , y has continuous first partial derivatives in the interior of a region R in the x y -plane , and set q = 1 , 0 , ∂ z / ∂ x and r = 0 , 1 , ∂ z / ∂ y . Then the surface area of the surface z = f x , y over R is ∬ R q × r d A
Determine whether the statement is true or false. Explain your answer.
Suppose that
z
=
f
x
,
y
has continuous first partial derivatives in the interior of a region R in the
x
y
-plane
,
and set
q
=
1
,
0
,
∂
z
/
∂
x
and
r
=
0
,
1
,
∂
z
/
∂
y
.
Then the surface area of the surface
z
=
f
x
,
y
over R is
Suppose that D is the ellipse
D={tr, ») € R°: + v* < 10}
{(x, y) € R²:
+ y? < 10}
D =
4
and that f is a differentiable function defined on all of R². Suppose that (xo, Yo) is in
ƏD, the boundary of D. Denote by (xo, Yo) the derivative of ƒ in the direction of the
outward pointing unit normal at the point (xo, Yo). Given that
fe
(4, 3) = 2 and
df
(4, 3) = 2,
dy
calculate (4, 3).
H. Use the gradient to find the equation of the tangent plane to each of the surfaces at the given point.
a) x² + 3x²y-z = 0 at (1,1,4) (Answ: 9x+3y-z = 8)
b) z = f(x, y, z) = r²y³z at (2,1,3) (Answ: 4x - 3y -z = 2)
I. In electrostatics the force (F) of attraction between two particles of opposite charge is given by
(Coulomb's law) where k is a constant and r = (x, y, z). Show that F is the gradient
T
(Hint: ||||||(x, y, z)||). Important problem!
F(r) = k₁
of P(7)
||7-1³
-k
||1|
=
Sketch the contour map of the function f(x, y) = 4 – Va2 + y?
Label the level curves at c=4,3, 2, 1, 0.
Chapter 14 Solutions
Calculus Early Transcendentals, Binder Ready Version
Calculus, Single Variable: Early Transcendentals (3rd Edition)
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Area Between The Curve Problem No 1 - Applications Of Definite Integration - Diploma Maths II; Author: Ekeeda;https://www.youtube.com/watch?v=q3ZU0GnGaxA;License: Standard YouTube License, CC-BY