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Quiz 5: Partial derivatives and tangent planes

Last unanswered question  Question  Next unanswered question
 

Question 1

 
 
Which option correctly gives the two first order partial derivatives of the following function?
f(x,y) = ex + x-+ (2x+ y)4 y
a) x 1- 3 3 fx = e + y + 8(2x + y), fy = xlny + 4(2x + y)
b) f = ex + x2 + 8(2x + y)3, f = - x-+ 8(2x+ y)3 x y y y2
c) 2 fx = ex + x + 8(2x + y)3, fy = x lny + 8(2x + y)3 y
d) 1 x fx = ex +--+ 8(2x + y)3, fy = --2 + 4(2x+ y)3 y y
e) x2 x fx = ex +-- + 8(2x + y)3, fy = ex --2 + 8(2x+ y)3 y y

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
Not correct. Choice (e) is false.
 

Question 2

 
 
Find the two first order partial derivatives, with respect to x and y, of
z = cos(x2y)+ sin y.
a) ∂z-= - 2xsin (x2y) ∂x and ∂z-= - x2ysin(x2y)- cosy ∂y
b) -∂z= - 2xysin(x2y) ∂x and ∂z= x2 sin(x2y)- cosy ∂y
c) ∂z ∂x-= 2x sin(x2y)- cosy and ∂z ∂y-= x2sin(x2y)
d) -∂z 2 ∂x = - 2xysin(x y) and ∂z- 2 2 ∂y = - x sin (x y)+ cosy
e) -∂z= - x2sin (x2y) ∂x and ∂z-= - 2xsin(x2y)+ cosy ∂y

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
Not correct. Choice (e) is false.
 

Question 3

 
 
Find the first order partial derivative with respect to y of
2 f(x,y) = xexy + ln(xy).
a) 2 2 1 1 exy + 2yx2exy + y-+ x-   b) 2 1 2yx2exy + y-
c) 2 1 xexy + xy-   d) 2 2 1 exy + y2xexy + x-
e) 2 xy2 1 2yx e + xy-

 

Not correct. Choice (a) is false.
Your answer is correct.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
Not correct. Choice (e) is false.
 

Question 4

 
 
Find the first order partial derivative with respect to x of
( --1---) f(x,y) = sin x2 + y2 .
a) ( ) 2x cos ---1--- x2 + y2   b) ( ) ---2y----cos ---1--- (x2 + y2)2 x2 + y2
c) ( ) -----2x---cos --1---- (x2 + y2)2 x2 + y2   d) ( ) 2y ln(x2 + y2)cos --1---- x2 + y2
e) ( ) - sin ----1---- (x2 + y2)2

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
Not correct. Choice (d) is false.
Not correct. Choice (e) is false.
 

Question 5

 
 
Find the equation of the tangent line to y = √------ e5x + 3 at x = 0.
a) x - 4y + 8 = 0   b) x - 4y - 8 = 0
c) 5x - 4y + 8 = 0   d) 5x - 4y + 4√ - 3 = 0

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
Not correct. Choice (d) is false.
 

Question 6

 
 
Find the equation of the tangent plane to the surface
2 2 z = x - y
at the point (5,-4,9).
a) z - 9 = 10(x - 5) + 8(y + 4)   b) z - 9 = 10(x - 5) - 8(y - 4)
c) z - 9 = 8(x + 4) + 10(y - 5)   d) z - 9 = 8(x- 5) - 10(y - 4)
e) z - 9 = 5(x- 8) + 4(y - 8)

 

Your answer is correct.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
Not correct. Choice (e) is false.
 

Question 7

 
 
Find the equation of the tangent plane to the surface
z = exy
at the point where x = 2 and y = 0.
a) z = -2x + 2y + 5   b) z = -y + 2
c) z = 2y + 1   d) z = x + 2y - 1
e) z = x - y - 1

 

Not correct. Choice (a) is false.
We have ∂z ∂x = yexy and -∂z ∂x = xexy. Now evaluate these at x = 2 and y = 0.
Not correct. Choice (b) is false.
We have -∂z ∂x = yexy and ∂z ∂x = xexy. Now evaluate these at x = 2 and y = 0.
Your answer is correct.
At (2,0) we have ∂z ∂x = 0, ∂z ∂y- = 2 and z = 1. So the tangent plane has equation z - 1 = 0(x - 2) + 2(y - 0), that is, z = 2y + 1.
Not correct. Choice (d) is false.
We have ∂∂xz = yexy and ∂∂zx = xexy. Now evaluate these at x = 2 and y = 0.
Not correct. Choice (e) is false.
We have ∂∂zx = yexy and ∂∂zx = xexy. Now evaluate these at x = 2 and y = 0.
 

Question 8

 
 
Find the equation of the tangent plane to the surface
2 2 z = 3x y+ 5y cosx
at the point where x = 0 and y = 2.
a) z = 6x - 10y + 40   b) z = 10x + 20y + 22
c) z = 20y + 12   d) z = 20y - 20
e) z = 5x - y - 18

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
Not correct. Choice (e) is false.
 

Question 9

 
 
Which of the options satisfies the following equation?
∂z- ∂z- x∂x + y ∂y = z
a) z = xey   b) z = x+y x-y
c) z = x2 + y   d) z = ∘ -2---2- x + y
e) z = ln(x2 - y)

 

Not correct. Choice (a) is false.
Since ∂z y ∂x = e and ∂z y ∂y = xe , we see that in this case, x ∂∂zx + y ∂∂zy = xey +xyey ⁄= z
Not correct. Choice (b) is false.
Since -2y ∂∂zx = (x-y)2 and ∂∂zy = (x2-xy)2 , we see that in this case, x ∂∂zx + y ∂∂zy = 0 ⁄= z
Not correct. Choice (c) is false.
Since ∂∂zx = 2x and ∂z = 1 ∂y , we see that in this case, x-∂z + y ∂z = 2x2 + y ⁄= z ∂x ∂y
Your answer is correct.
Here we have ∂z √--x--- ∂x = x2+y2 and ∂z √--y--- ∂y = x2+y2 . In this case, x ∂z+ y ∂z= √x2+y2-= ∘x2-+-y2-= z ∂x ∂y x2+y2
Not correct. Choice (e) is false.
Since ∂z= 22x-- ∂x x-y and ∂∂zy = x-21-y , we see that in this case, 2 x ∂∂zx + y ∂∂zy = 2xx2--yy-⁄= z
 

Question 10

 
 
Find the equation of the tangent plane to the surface
z = ln(x2 + y2)
at the point where x = 1 and y = -2.
a) 5z = 2(x - 1) - 4(y + 2)   b) 5z = ln5 + 2x - 4y + 10
c) 5z = 5ln5 + 4y - 8   d) 5z = 5ln5 - 2x + 4y + 4
e) 5z = 5ln5 + 2x - 4y - 10

 

Not correct. Choice (a) is false.
The first partial derivatives are ∂∂zx = x22+xy2- and ∂∂zx = x22+yy2- . Now evaluate these at x = 1 and y = -2.
Not correct. Choice (b) is false.
The first partial derivatives are ∂z= -22x2 ∂x x +y and -∂z --2y-- ∂x = x2+y2 . Now evaluate these at x = 1 and y = -2.
Not correct. Choice (c) is false.
The first partial derivatives are -∂z --2x-- ∂x = x2+y2 and ∂z -2y-- ∂x = x2+y2 . Now evaluate these at x = 1 and y = -2.
Not correct. Choice (d) is false.
The first partial derivatives are ∂z= --2x-- ∂x x2+y2 and -∂z --2y-- ∂x = x2+y2 . Now evaluate these at x = 1 and y = -2.
Your answer is correct.