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MATH1111 Quizzes

The Chain Rule Quiz
Web resources available Questions

This quiz tests the work covered in the lecture on the chain rule and corresponds to Section 14.6 of the textbook Calculus: Single and Multivariable (Hughes-Hallett, Gleason, McCallum et al.).
There is a useful explanation and extra examples at http://www.math.hmc.edu/calculus/tutorials/multichainrule/. There are interactive exercises at http://www.math.temple.edu/ cow/ but you need to go to Book III, Functions, Chain Rule to find the questions. Pressing help gives you an explanation of how to use the chain rule.

There are more web quizzes at Wiley, select Section 6. This quiz has 14 questions.

Suppose z = 2xy where x = t2 + 1 and y = 3 t. Use the chain rule to determine which of the following is dz dt . Exactly one option must be correct)
a)
dz dt = 6t 4t2 2
b)
dz dt = 4t3 + 6t 6
c)
dz dt = 6t 2
d)
dz dt = 12t 6t2 2

Choice (a) is incorrect
Try again, you may have lost a t.
Choice (b) is incorrect
Try again, check your partial derivatives.
Choice (c) is incorrect
Try again, carefully watch your signs.
Choice (d) is correct!
We compute z x = 2y, dx dt = 2t z y = 2x and dy dt = 1, so
dz dt =z x dx dt + z y dy dt
 =2y 2t 2x
 =2(3 t) 2t 2(t2 + 1)
 =12t 4t2 2t2 2 = 12t 6t2 2.
Suppose z = xy 2y2 where x = 3t + 1 and y = 2t. Use the chain rule to determine which of the following is dz dt . Exactly one option must be correct)
a)
dz dt = 4t + 2
b)
dz dt = 16t t
c)
dz dt = 18t + 2
d)
dz dt = 10t2 + 8t + 1

Choice (a) is correct!
We compute z x = y, dx dt = 3 z y = x 4y and dy dt = 2, so
dz dt =z x dx dt + z y dy dt
 =y 3 + (x 4y) 2
 =2x 5y
 =2(3t + 1) 10t = 4t + 2.
Choice (b) is incorrect
Try again, check your signs.
Choice (c) is incorrect
Try again, check your signs.
Choice (d) is incorrect
Try again, you may have not differentiated x and y with respect to t.
Which of the following is the derivative, dz dt , written in terms of t where z = x2y + y2x where x = cost and y = t2? Exactly one option must be correct)
a)
dz dt = 2t2 costsint + t4 sint + 2tcos2t + 4t3 cost
b)
dz dt = 2t2 costsint 2t2 sint + 4tcost + 4t3
c)
dz dt = 2t2 costsint t4 sint + 2tcos2t + 4t3 cost
d)
dz dt = (2xy + y2)(sint) + (x2 + 2xy)2t

Choice (a) is incorrect
Try again, you may not have differentiated cost correctly, check your signs.
Choice (b) is incorrect
Try again, you may not have found the correct partial derivatives.
Choice (c) is correct!
We compute z x = 2xy + y2, dx dt = sint z y = x2 + 2xy and dy dt = 2t. Therefore, dz dt = z x dx dt + z y dy dt = (2xy + y2) sint + (x2 + 2xy) 2t = (2t2 cost + t4) (sint) + (cos2t + 2t2 cost 2t = 2t2 costsint t4 sint + 2tcos2t + 4t3 cost.
Choice (d) is incorrect
Try again, this is dz dt but not in terms of t.
Suppose z = lnxy where x = t and y = t2. Use the chain rule to determine which of the following is dz dt . Exactly one option must be correct)
a)
dz dt = 1 t2 + 2t3 t
b)
dz dt = 3 2t
c)
dz dt = 2 t
d)
dz dt = 1 2t

Choice (a) is incorrect
Try again, you may have not have differentiated lnxy correctly.
Choice (b) is correct!
We compute z x = 1 x, dx dt = 1 2t z y = 1 y and dy dt = 2t. Therefore, dz dt = z x dx dt + z y dy dt = 1 x 1 2t + 1 y 2t = 1 t 1 2t + 1 t2 2t = 1 2t + 1 t = 3 2t.
Choice (c) is incorrect
Try again, check your derivatives.
Choice (d) is incorrect
Try again, carefully watch your signs when you differentiate.