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Quiz 8: Partial derivatives

Last unanswered question  Question  Next unanswered question
 

Question 1

 
 
Find the first order partial derivative with respect to x of f(x,y) = e3xcosy .
a) 3x 3x fx(x,y) = 3e cosy + e siny .   b) 3x 3x fx(x,y) = 3e cosy+ e cosy .
c) 3x fx(x,y) = - e siny .   d) fx(x,y) = 3e3xcosy .

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
Note that since we are differentiating with respect to x, we treat cosy as a constant.
 

Question 2

 
 
Find the first order partial derivative with respect to y of 3 2 y f(x,y) = x y + 3xe .
a) 2 2 y fy(x,y) = 3x y + 3e   b) fy(x,y) = 3x2y2 + 2x3y+ 3ey +3xey
c) fy(x,y) = 2x3y+ 3xey   d) 2 y fy(x,y) = 6x y+ 3e

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
We treat x3 and 3x as constants and differentiate term by term with respect to y.
Not correct. Choice (d) is false.
 

Question 3

 
 
Find the second order partial derivatives of f(x,y) = (3x+ 2y)4 .
a)
fxx(x,y) = 12(3x + 2y)3 f (x,y) = 24(3x + 2y)2 yy
  b)
fxx(x,y) = 36(3x + 2y)2 fyy(x,y) = 8(3x + 2y)3
c)
fxx(x,y) = 24(3x + 2y)2 fxy(x,y) = 32(3x + 2y)
  d)
2 fxy(x,y) = 108(3x+ 2y)2 fyy(x,y) = 48(3x+ 2y)

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
3 3 fx(x,y) = 12(3x+ 2y) , fy(x,y) = 8(3x + 2y),
2 2 fxx(x,y) = 108(3x+ 2y), fyy(x,y) = 48(3x+ 2y) .
 

Question 4

 
 
Find the second order partial derivatives of f(x,y) = 3x2y+ 4 cosxy .
a)
2 fxx(x,y) = 6y- 4y cosxy fyy(x,y) = - 4x2cosxy
  b)
2 fxx(x,y) = - 4x cos2xy fyy(x,y) = 6y- 4y cosxy
c)
f (x,y) = 6x- 4x2cosxy xx 2 fyy(x,y) = - 4y cosxy
  d)
fxx(x,y) = 6x - 4xycosxy fyy(x,y) = - 4xycosxy

 

Your answer is correct.
fx(x,y) = 6xy - 4y sinxy, fy(x,y) = 3x2 - 4xsin xy ,
fxx(x,y) = 6y- 4y2cosxy, fyy(x,y) = - 4x2cosxy.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
 

Question 5

 
 
What is the mixed, second order partial derivative of ∘ -------- f(x,y) = 2x2 + y2 ?
a)
2 2-1∕2 2 2 2 -3∕2 fxx(x,y) = 2(2x + y) - 4x (2x + y ) fyy(x,y) = (2x2 + y2)-1∕2 - y2(2x2 + y2)-3∕2
b) fxy(x,y) = - 2xy(2x2 +y2)-3∕2
c) 2 2 -1∕2 fxy(x,y) = - 4xy(2x +y )
d) 2 2 2 -3∕2 fxy(x,y) = - 4x (2x + y )

 

Not correct. Choice (a) is false.
Your answer is correct.
f (x,y) = 2x (2x2 + y2)-1∕2 x 2 2 -3∕2 fxy(x,y) = - 2xy(2x + y )
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
 

Question 6

 
 
If f(x,y) = x3 - 2xy + xy3 + 3y2 which of the following is true ?
a)
2 3 fx(x,y) = 3x - 2y+ y fy(x,y) = - 2x + 6y+ 3xy2 fxy(x,y) = 6x
b)
fx(x,y) = - 2x + 6y+ 3xy2 fy(x,y) = 3x2 - 2y+ y3 f (x,y) = - 2 + 3y2 xy
c)
fxx(x,y) = 6x f (x,y) = 6+ 6xy yy 2 fxy(x,y) = - 2+ 3y
d)
fxx(x,y) = 6+ 6xy f (x,y) = 6x yy 2 fxy(x,y) = - 2+ 3y

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
2 3 2 fx(x,y) = 3x - 2y + y, fy(x,y) = - 2x+ 3xy + 6y ,
fxx(x,y) = 6x, fyy(x,y) = 6xy+ 6 ,
2 fxy(x,y) = - 2 + 3y .
Not correct. Choice (d) is false.
 

Question 7

 
 
Find the second order partial derivative with respect to x of f(x,y) = cosx + xyexy + xsin y .
a) fxx(x,y) = - cosx+ 2y2exy + xy3exy + sin y
b) fxx(x,y) = 2y2exy + xy3exy
c) xy 2 xy 3 xy fxx(x,y) = e + 2y e + xy e - cosy
d) 2 xy 3 xy fxx(x,y) = - cosx+ 2y e + xy e
e) f (x,y) = 2x2exy + x3yexy xx

 

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 8

 
 
Find the second order partial derivative with respect to y of f(x,y) = cosx + xyexy + xsin y .
a) 2 xy 3 xy fyy(x,y) = 2x e + x ye - xsiny
b) 2 xy 2 2xy fyy(x,y) = - cos x+ 2y e + x y e + cosy
c) xy xy 2 2 xy fyy(x,y) = e + 3xye + x y e + cosy
d) f (x,y) = x2exy + x3yexy - xsin y yy

 

Your answer is correct.
fy(x,y) = xexy +x2yexy +x cosy 3 xy 2 xy 2 xy fyy(x,y) = x y2exy+ x3e x+y x e - x siny = 2x e + x ye - x siny.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
 

Question 9

 
 
What is the mixed, second order partial derivative of
f(x,y) =-3x--? x +y
a) fxx(x,y) =----6y- fyy(x,y) = --6x-3 (x + y)3 (x+y)
b) 3(y--x) fxy(x,y) = (x+ y)3
c) 6(x--y) fxy(x,y) = (x+ y)3
d) f (x,y) = 3(x--y) xy (x+ y)3

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
fx = (x3y+y)2, fxy = 2 3(x+y)(x+-y6y)(4x+y) = 3((xx+-yy)3).
 

Question 10

 
 
If y f(x,y) = sin(x) which of the following are true ? (More than one answer may be correct.)
a) ∂2f-= 2y cos(y-) ∂x2 x3 x
b) 2 ∂-f2 = - 12-sin(y) ∂y x x
c) ∂2f ----- ∂x ∂y is defined for all x and y
d) ∂2f 2y y y2 y -∂x2 = x3 cos(x)- x4 sin(x)
e) -∂2f- = ∂2f-- ∂x ∂y ∂y∂x provided x≠0
f) ∂2f 2y y y2 y ---2 = -3 cos(-)--4 sin(-) ∂x x x x x

 

There is at least one mistake.
For example, choice (a) should be false.
There is at least one mistake.
For example, choice (b) should be true.
There is at least one mistake.
For example, choice (c) should be false.
f(x,y) itself is not defined when x = 0. So ∂2f ----- ∂x ∂y is certainly not defined when x = 0.
There is at least one mistake.
For example, choice (d) should be false.
There is at least one mistake.
For example, choice (e) should be true.
By lectures, -∂2f-= ∂2f-- ∂x ∂y ∂y∂x if both mixed derivatives exist and are continuous.
There is at least one mistake.
For example, choice (f) should be true.
Your answers are correct
  1. False.
  2. True.
  3. False. f(x,y) itself is not defined when x = 0. So ∂2f ----- ∂x ∂y is certainly not defined when x = 0.
  4. False.
  5. True. By lectures, -∂2f-= ∂2f-- ∂x ∂y ∂y∂x if both mixed derivatives exist and are continuous.
  6. True.