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MATH1001 Quizzes

Quiz 6: Partial derivatives, limits and continuity
Question 1 Questions
The function f of two real variables x and y has the property that fy(x,y) = 3x2y y3. Check all options which are possible values of f(x,y). (Zero or more options can be correct)
a)
3x2 3y2
b)
3 2x2y2 1 4y4 + lnx
c)
3 2x2y2 1 4y4 + yex
d)
3 2x2y2 1 4y4 + sinx

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.
There is at least one mistake.
For example, choice (d) should be True.
Correct!
  1. False
  2. True
  3. False
  4. True
For which functions do the second partial derivatives 2f x2 and 2f y2 add together to give zero? Check each option with this property. (Zero or more options can be correct)
a)
f(x,y) = x2y 4xy3
b)
f(x,y) = x2 y2
c)
f(x,y) = xlny
d)
f(x,y) = xy + ex siny
e)
f(x,y) = x + y

There is at least one mistake.
For example, choice (a) should be False.
In this case we have fxx = 2y and fyy = 24xy.
There is at least one mistake.
For example, choice (b) should be True.
In this case we have fxx = 2 and fyy = 2, which add to zero.
There is at least one mistake.
For example, choice (c) should be False.
In this case we have fxx = 0 and fyy = x y2 .
There is at least one mistake.
For example, choice (d) should be True.
In this case we have fxx = ex siny and fyy = ex siny, which add to zero.
There is at least one mistake.
For example, choice (e) should be False.
In this case we have fxx = 1 4(x + y)32 = f yy.
Correct!
  1. False In this case we have fxx = 2y and fyy = 24xy.
  2. True In this case we have fxx = 2 and fyy = 2, which add to zero.
  3. False In this case we have fxx = 0 and fyy = x y2 .
  4. True In this case we have fxx = ex siny and fyy = ex siny, which add to zero.
  5. False In this case we have fxx = 1 4(x + y)32 = f yy.
What is the value of limx1x2 x 2 x2 2x ? (Hint: Decide whether the function is a continuous function in an interval about the point x = 1.) Exactly one option must be correct)
a)
2
b)
1
c)
1
d)
2
e)
This limit does not exist.

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
The function is defined and continuous at x = 1. Therefore the limit is simply the function value at this point. limx1x2 x 2 x2 2x = 1 1 2 1 2 = 2.
Choice (e) is incorrect
What is limx2x2 x 2 x2 2x ? (Hint: Factorise and simplify the expression.) Exactly one option must be correct)
a)
0
b)
1
c)
3 2
d)
The limit does not exist.
e)

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
limx2x2 x 2 x2 2x = limx2(x 2)(x + 1) x(x 2) = limx2x + 1 x = limx21 + 1 x = 1 + 1 2.
Choice (d) is incorrect
Choice (e) is incorrect
What is limx11 + 3x 1 ? Type your answer into the box.

Correct!
limx11 + 3x 1 = 1 + 3 1 = 1.
Incorrect. Please try again.
Try substituting x = 1 into 1 + 3x 1. (This is valid because the function is continuous at x = 1.)
What is lim(x,y)(0,0)x2 2y2 3x2 + y4 as (x,y) (0,0) along the xaxis? Exactly one option must be correct)
a)
0
b)
1 3
c)
1
d)
Function values increase without bound as (x,y) (0,0) and so there is no limit.

Choice (a) is incorrect
Choice (b) is correct!
Along the x–axis we have y = 0 so this limit becomes
limx0 x2 3x2 = limx01 3 = 1 3.
Choice (c) is incorrect
Choice (d) is incorrect
What is the value of the limit lim(x,y)(0,0)x2 2y2 3x2 + y4 if (x,y) approaches (0,0) along the line y = x? Exactly one option must be correct)
a)
2 3
b)
1 3
c)
0
d)
Function values increase without bound as (x,y) (0,0) and so there is no limit.

Choice (a) is incorrect
Choice (b) is correct!
Along the line y = x this limit becomes
limx0x2 2x2 3x2 + x4 = limx0 x2 3x2 + x4 = limx0 1 3 + x2 = 1 3.
Notice that this question combined with the previous one shows that the limit lim(x,y)(0,0)x2 2y2 3x2 + y4 does not exist.
Choice (c) is incorrect
Choice (d) is incorrect
At what value or values of x is the function
f(x) = x + 4if x 1 x2 if  1 < x < 1 2 x if x 1
NOT continuous? (You may check more than one box if you wish.) (Zero or more options can be correct)
a)
1
b)
0
c)
1
d)
2
e)
No values (f(x) is continuous everywhere)

There is at least one mistake.
For example, choice (a) should be True.
There is at least one mistake.
For example, choice (b) should be False.
There is at least one mistake.
For example, choice (c) should be False.
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 False.
Correct!
  1. True
  2. False
  3. False
  4. False
  5. False
At what value or values of x is the function
f(x) = |x + 1| 1if x < 0 x2 + x if 0 x < 1 3 x if 1 x
NOT continuous? Exactly one option must be correct)
a)
1, 0 and 1
b)
1
c)
0
d)
0 and 1
e)
No values (f(x) is continuous everywhere)

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is correct!
For what values of a will the function
f(x) = x3if x a x2 if x > a
be continuous for all x? Exactly one option must be correct)
a)
1 and 2
b)
2 and 1
c)
1 and 1
d)
0 and 1
e)
1 and 0

Choice (a) is incorrect
As 2322 the function is not continuous if a = 2.
Choice (b) is incorrect
As (2)3(2)2 the function is not continuous if a = 2.
Choice (c) is incorrect
As (1)3(1)2 the function is not continuous if a = 1.
Choice (d) is correct!
For the function to be continuous at x = a, we must have or a3 = a2. Hence a = 0 or 1.
Choice (e) is incorrect
As (1)3(1)2 the function is not continuous if a = 1.