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MATH1011 Quizzes

Quiz 8: Partial derivatives
Question 1 Questions
Find the first order partial derivative with respect to x of f(x,y) = e3x cosy.
a)
fx(x,y) = 3e3x cosy + e3x siny.
 b)
fx(x,y) = 3e3x cosy + e3x cosy.
c)
fx(x,y) = e3x siny.
 d)
fx(x,y) = 3e3x cosy.

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
Note that since we are differentiating with respect to x, we treat cosy as a constant.
Find the first order partial derivative with respect to y of f(x,y) = x3y2 + 3xey.
a)
fy(x,y) = 3x2y2 + 3ey
 b)
fy(x,y) = 3x2y2 + 2x3y + 3ey + 3xey
c)
fy(x,y) = 2x3y + 3xey
 d)
fy(x,y) = 6x2y + 3ey

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
We treat x3 and 3x as constants and differentiate term by term with respect to y.
Choice (d) is incorrect
Find the second order partial derivatives of f(x,y) = (3x + 2y)4.
a)
fxx(x,y) = 12(3x + 2y)3 fyy(x,y) = 24(3x + 2y)2
 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)
fxy(x,y) = 108(3x + 2y)2 fyy(x,y) = 48(3x + 2y)2

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
fx(x,y) = 12(3x + 2y)3, f y(x,y) = 8(3x + 2y)3,
fxx(x,y) = 108(3x + 2y)2, f yy(x,y) = 48(3x + 2y)2.
Find the second order partial derivatives of f(x,y) = 3x2y + 4cosxy.
a)
fxx(x,y) = 6y 4y2 cosxy fyy(x,y) = 4x2 cosxy
 b)
fxx(x,y) = 4x2 cosxy fyy(x,y) = 6y 4y2 cosxy
c)
fxx(x,y) = 6x 4x2 cosxy fyy(x,y) = 4y2 cosxy
 d)
fxx(x,y) = 6x 4xycosxy fyy(x,y) = 4xycosxy

Choice (a) is correct!
fx(x,y) = 6xy 4ysinxy, fy(x,y) = 3x2 4xsinxy,
fxx(x,y) = 6y 4y2 cosxy, f yy(x,y) = 4x2 cosxy.
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
What is the mixed, second order partial derivative of f(x,y) = 2x2 + y2 ?
a)
fxx(x,y) = 2(2x2 + y2)12 4x2(2x2 + y2)32 fyy(x,y) = (2x2 + y2)12 y2(2x2 + y2)32
b)
fxy(x,y) = 2xy(2x2 + y2)32
c)
fxy(x,y) = 4xy(2x2 + y2)12
d)
fxy(x,y) = 4x2(2x2 + y2)32

Choice (a) is incorrect
Choice (b) is correct!
fx(x,y) = 2x(2x2 + y2)12 fxy(x,y) = 2xy(2x2 + y2)32
Choice (c) is incorrect
Choice (d) is incorrect
If f(x,y) = x3 2xy + xy3 + 3y2 which of the following is true ?
a)
fx(x,y) = 3x2 2y + y3 fy(x,y) = 2x + 6y + 3xy2 fxy(x,y) = 6x
b)
fx(x,y) = 2x + 6y + 3xy2 fy(x,y) = 3x2 2y + y3 fxy(x,y) = 2 + 3y2
c)
fxx(x,y) = 6x fyy(x,y) = 6 + 6xy fxy(x,y) = 2 + 3y2
d)
fxx(x,y) = 6 + 6xy fyy(x,y) = 6x fxy(x,y) = 2 + 3y2

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
fx(x,y) = 3x2 2y + y3, f y(x,y) = 2x + 3xy2 + 6y,
fxx(x,y) = 6x, fyy(x,y) = 6xy + 6,
fxy(x,y) = 2 + 3y2.
Choice (d) is incorrect
Find the second order partial derivative with respect to x of f(x,y) = cosx + xyexy + xsiny.
a)
fxx(x,y) = cosx + 2y2exy + xy3exy + siny
b)
fxx(x,y) = 2y2exy + xy3exy
c)
fxx(x,y) = exy + 2y2exy + xy3exy cosy
d)
fxx(x,y) = cosx + 2y2exy + xy3exy
e)
fxx(x,y) = 2x2exy + x3yexy

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
Choice (e) is incorrect
Find the second order partial derivative with respect to y of f(x,y) = cosx + xyexy + xsiny.
a)
fyy(x,y) = 2x2exy + x3yexy xsiny
b)
fyy(x,y) = cosx + 2y2exy + x2y2exy + cosy
c)
fyy(x,y) = exy + 3xyexy + x2y2exy + cosy
d)
fyy(x,y) = x2exy + x3yexy xsiny

Choice (a) is correct!
fy(x,y) = xexy + x2yexy + xcosy fyy(x,y) = x3yexy + x2exy + x2exy xsiny = 2x2exy + x3yexy xsiny.
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
What is the mixed, second order partial derivative of
f(x,y) = 3x x + y?
a)
fxx(x,y) = 6y (x + y)3
b)
fyy(x,y) = 6x (x+y)3
c)
fxy(x,y) = 3(y x) (x + y)3
d)
fxy(x,y) = 6(x y) (x + y)3
e)
fxy(x,y) = 3(x y) (x + y)3

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is correct!
fx = 3y (x+y)2 , fxy = 3(x+y)26y(x+y) (x+y)4 = 3(xy) (x+y)3 .
If f(x,y) = sin(y x) which of the following are true ? (More than one answer may be correct.)
a)
2f x2 = 2y x3 cos(y x)
b)
2f y2 = 1 x2 sin(y x)
c)
2f xy is defined for all x and y
d)
2f x2 = 2y x3 cos(y x) y2 x4 sin(y x)
e)
2f xy = 2f yx provided x0
f)
2f x2 = 2y x3 cos(y x) y2 x4 sin(y 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 xy 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 xy = 2f yx if both mixed derivatives exist and are continuous.
There is at least one mistake.
For example, choice (f) should be True.
Correct!
  1. False
  2. True
  3. False f(x,y) itself is not defined when x = 0. So 2f xy is certainly not defined when x = 0.
  4. False
  5. True By lectures, 2f xy = 2f yx if both mixed derivatives exist and are continuous.
  6. True