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MATH1111 Quizzes

The Partial Derivative Quiz
Web resources available Questions

This quiz tests the work covered in the lecture on partial derivatives and corresponds to Section 14.1 of the textbook Calculus: Single and Multivariable (Hughes-Hallett, Gleason, McCallum et al.).
There is a discussion on partial derivatives at http://www.bio.brandeis.edu/biomath/populate/surface.html

There are more web quizzes at Wiley, select Section 1. This quiz has 15 questions.

Suppose f(x,y) = 2x2y. Which of the following gives the best estimate of the quotient fx(2,1) and fy(2,1) where h = 1? Exactly one option must be correct)
a)
fx(2,1) = 8 and fy(2,1) = 8.
b)
fx(2,1) = 2 and fy(2,1) = 1.
c)
fx(2,1) = 1 and fy(2,1) = 4.
d)
fx(2,1) = 10 and fy(2,1) = 8.

Choice (a) is incorrect
Try again, these are the exact values of the partial derivatives.
Choice (b) is incorrect
Try again, check your multiplication.
Choice (c) is incorrect
Try again, you may have forgotten to square your x values.
Choice (d) is correct!
limh02 × (2 + h)2 × 1 2 × 22 × 1 h = 18 8 1 = 10 when h = 1
and limh02 × 22 × (1 + h) 2 × 22 × 1 h = 16 8 1 = 8 when h = 1.
Refer to the table on page 686 of Calculus: Single and Multivariable (Hughes-Hallett, Gleason, McCallum et al.) giving the temperature, T, in oC of a plate, as a function of its distance from the bottom corner of the plate.

Which of the following statements are the most accurate? (Zero or more options can be correct)
a)
Tx(3,2) 45oCm when h = 1
b)
Tx(3,2) 25oCm when h = 1
c)
Ty(3,2) 15oCm when h = 1
d)
Ty(3,2) 10oCm when h = 1

There is at least one mistake.
For example, choice (a) should be True.
T(3,2) = 145 and T(4,2) = 190 so Tx(3,2) 190 145 1 = 45oCm when h = 1.
There is at least one mistake.
For example, choice (b) should be False.
You may have been looking at T(2,3). This is also a good approximation for Tx(2,2).
There is at least one mistake.
For example, choice (c) should be False.
This is a good approximation for Ty(3,1).
There is at least one mistake.
For example, choice (d) should be True.
T(3,2) = 145 and T(3,3) = 135 so Ty(3,2) 135 145 1 = 10oCm when h = 1.
Correct!
  1. True T(3,2) = 145 and T(4,2) = 190 so Tx(3,2) 190 145 1 = 45oCm when h = 1.
  2. False You may have been looking at T(2,3). This is also a good approximation for Tx(2,2).
  3. False This is a good approximation for Ty(3,1).
  4. True T(3,2) = 145 and T(3,3) = 135 so Ty(3,2) 135 145 1 = 10oCm when h = 1.
Use the difference quotient with x = 0.2 and y = 0.1 to estimate fx(1,2) and fy(1,2) where f(x,y) = e2x cosy.
Which of the following are the correct estimations? Exactly one option must be correct)
a)
fx(1,2) e2.4 cos2 e2 cos2 0.2 and fy(1,2) e2 cos2.2 e2 cos2 0.2
b)
fx(1,2) e2.4 cos2 e2 cos2 0.2 and fy(1,2) e2 cos2.1 e2 cos2 0.1
c)
fx(1,2) e2.2 cos2 e2 cos2 0.2 and fy(1,2) e2 cos2.2 e2 cos2 0.2
d)
fx(1,2) e2.2 cos2 e2 cos2 0.2 and fy(1,2) e2 cos2.1 e2 cos2 0.1

Choice (a) is incorrect
Try again, you have correctly estimated fx(2,1) but you have not used the correct value for y.
Choice (b) is correct!
You have carefully substitute into the formulae
fx(1,2) e2×(1+x) cos2 e2×1 cos2 x and
fy(1,2) e2×1 cos(2 + y) e2×1 cos2 y .
Choice (c) is incorrect
Try again, neither answer is correct. Carefully substitute into the formulae
e2×(1+x) cos2 e2×1 cos2 x and
e2×1 cos(2 + y) e2×1 cos2 y .
Choice (d) is incorrect
Try again, you have correctly estimated fy(2,1) but you have not correctly used the value for x.
Consider the contour diagram below for f(x,y).
PIC
Which of the following are the most accurate estimates for f(2,4),fx(2,4) and fy(2,4)? Exactly one option must be correct)
a)
f(2,4) = 6fx(2,4) = 3 and fy(2,4) = 3 2
b)
f(2,4) = 3fx(2,4) = 3 2 and fy(2,4) = 3
c)
f(2,4) = 6fx(2,4) = 3 2 and fy(2,4) = 3
d)
f(2,4) = 3fx(2,4) = 3 and fy(2,4) = 3 2

Choice (a) is correct!
The point (2,4) is on the +6 contour. As x increases f(x,y) decreases so fx(x,y) is negative. The contours are evenly spread and as x increases by 1 f(x,y) decreases by 3, so fx(2,4) = 3.
Similarly fy(x,y) is positive as we see f(x,y) increases as y is increases. The contours are evenly spread and as y increases by 2 f(x,y) increases by 3, so fx(2,4) = 3 2.
Choice (b) is incorrect
Try again, you seem to have confused your x and y values.
Choice (c) is incorrect
Try again, you have the correct value for f(2,4).
Choice (d) is incorrect
Try again, you do no have the correct value for f(2,4).