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MATH1001 Quizzes

Quiz 5: Partial derivatives and tangent planes
Question 1 Questions
Which option correctly gives the two first order partial derivatives of the following function?
f(x,y) = ex + x y + (2x + y)4
Exactly one option must be correct)
a)
fx = ex + 1 y + 8(2x + y)3,fy = xlny + 4(2x + y)3
b)
fx = ex + x2 y + 8(2x + y)3,fy = x y2 + 8(2x + y)3
c)
fx = ex + x2 y + 8(2x + y)3,fy = xlny + 8(2x + y)3
d)
fx = ex + 1 y + 8(2x + y)3,fy = x y2 + 4(2x + y)3
e)
fx = ex + x2 y + 8(2x + y)3,fy = ex x y2 + 8(2x + y)3

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
Choice (e) is incorrect
Find the two first order partial derivatives, with respect to x and y, of
z = cos(x2y) + siny.
Exactly one option must be correct)
a)
z x = 2xsin(x2y) and z y = x2ysin(x2y) cosy
b)
z x = 2xysin(x2y) and z y = x2 sin(x2y) cosy
c)
z x = 2xsin(x2y) cosy and z y = x2 sin(x2y)
d)
z x = 2xysin(x2y) and z y = x2 sin(x2y) + cosy
e)
z x = x2 sin(x2y) and z y = 2xsin(x2y) + cosy

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
Choice (e) is incorrect
Find the first order partial derivative with respect to y of
f(x,y) = xexy2 + ln(xy).
Exactly one option must be correct)
a)
exy2 + 2yx2exy2 + 1 y + 1 x
b)
2yx2exy2 + 1 y
c)
xexy2 + 1 xy
d)
exy2 + y2xexy2 + 1 x
e)
2yx2exy2 + 1 xy

Choice (a) is incorrect
Choice (b) is correct!
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is incorrect
Find the first order partial derivative with respect to x of
f(x,y) = sin 1 x2 + y2 .
Exactly one option must be correct)
a)
2xcos 1 x2 + y2
b)
2y (x2 + y2)2 cos 1 x2 + y2
c)
2x (x2 + y2)2 cos 1 x2 + y2
d)
2yln(x2 + y2)cos 1 x2 + y2
e)
sin 1 (x2 + y2)2

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
Choice (d) is incorrect
Choice (e) is incorrect
Find the equation of the tangent line to y = e5x + 3 at x = 0. Exactly one option must be correct)
a)
x 4y + 8 = 0
b)
x 4y 8 = 0
c)
5x 4y + 8 = 0
d)
5x 4y + 43 = 0

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
Choice (d) is incorrect
Find the equation of the tangent plane to the surface
z = x2 y2
at the point (5,4,9). Exactly one option must be correct)
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)

Choice (a) is correct!
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is incorrect
Find the equation of the tangent plane to the surface
z = exy
at the point where x = 2 and y = 0. Exactly one option must be correct)
a)
z = 2x + 2y + 5
b)
z = y + 2
c)
z = 2y + 1
d)
z = x + 2y 1
e)
z = x y 1

Choice (a) is incorrect
We have z x = yexy and z x = xexy. Now evaluate these at x = 2 and y = 0.
Choice (b) is incorrect
We have z x = yexy and z x = xexy. Now evaluate these at x = 2 and y = 0.
Choice (c) 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.
Choice (d) is incorrect
We have z x = yexy and z x = xexy. Now evaluate these at x = 2 and y = 0.
Choice (e) is incorrect
We have z x = yexy and z x = xexy. Now evaluate these at x = 2 and y = 0.
Find the equation of the tangent plane to the surface
z = 3x2y + 5y2 cosx
at the point where x = 0 and y = 2. Exactly one option must be correct)
a)
z = 6x 10y + 40
b)
z = 10x + 20y + 22
c)
z = 20y + 12
d)
z = 20y 20
e)
z = 5x y 18

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
Choice (e) is incorrect
Which of the options satisfies the following equation? xz x + yz y = z Exactly one option must be correct)
a)
z = xey
b)
z = x+y xy
c)
z = x2 + y
d)
z = x2 + y2
e)
z = ln(x2 y)

Choice (a) is incorrect
Since z x = ey and z y = xey, we see that in this case, xz x + yz y = xey + xyeyz
Choice (b) is incorrect
Since z x = 2y (xy)2 and z y = 2x (xy)2 , we see that in this case, xz x + yz y = 0z
Choice (c) is incorrect
Since z x = 2x and z y = 1, we see that in this case, xz x + yz y = 2x2 + yz
Choice (d) is correct!
Here we have z x = x x2 +y2 and z y = y x2 +y2 . In this case, xz x + yz y = x2+y2 x2 +y2 = x2 + y2 = z
Choice (e) is incorrect
Since z x = 2x x2y and z y = 1 x2y, we see that in this case, xz x + yz y = 2x2y x2y z
Find the equation of the tangent plane to the surface
z = ln(x2 + y2)
at the point where x = 1 and y = 2. Exactly one option must be correct)
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

Choice (a) is incorrect
The first partial derivatives are z x = 2x x2+y2 and z x = 2y x2+y2 . Now evaluate these at x = 1 and y = 2.
Choice (b) is incorrect
The first partial derivatives are z x = 2x x2+y2 and z x = 2y x2+y2 . Now evaluate these at x = 1 and y = 2.
Choice (c) is incorrect
The first partial derivatives are z x = 2x x2+y2 and z x = 2y x2+y2 . Now evaluate these at x = 1 and y = 2.
Choice (d) is incorrect
The first partial derivatives are z x = 2x x2+y2 and z x = 2y x2+y2 . Now evaluate these at x = 1 and y = 2.
Choice (e) is correct!