Calculus: Early Transcendentals
4th Edition
ISBN: 9781319050740
Author: Jon Rogawski, Colin Adams, Robert Franzosa
Publisher: W. H. Freeman
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Chapter 14.5, Problem 10E
To determine
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Use the chain rule to find ∂w/∂t given w =√ x^2 + y^2 + z^2 and x = cos(st), y = −2s ln(t) and z = s − 3t.
A.) express dw/dt as a function of t, both by using the chain rule and by expressing w in terms of t and differentiating directly with respect to t. Then (b) evaluate dw dt at the given value of t.
Consider the following.w=x^2/y , x=t^2 , y=t+1 , t=3
(a) Find d2w/dt2 using the appropriate Chain Rule.
Chapter 14 Solutions
Calculus: Early Transcendentals
Ch. 14.1 - Prob. 1PQCh. 14.1 - Prob. 2PQCh. 14.1 - Prob. 3PQCh. 14.1 - Prob. 4PQCh. 14.1 - Prob. 5PQCh. 14.1 - Prob. 1ECh. 14.1 - Prob. 2ECh. 14.1 - Prob. 3ECh. 14.1 - Prob. 4ECh. 14.1 - Prob. 5E
Ch. 14.1 - Prob. 6ECh. 14.1 - Prob. 7ECh. 14.1 - Prob. 8ECh. 14.1 - Prob. 9ECh. 14.1 - Prob. 10ECh. 14.1 - Prob. 11ECh. 14.1 - Prob. 12ECh. 14.1 - Prob. 13ECh. 14.1 - Prob. 14ECh. 14.1 - Prob. 15ECh. 14.1 - Prob. 16ECh. 14.1 - Prob. 17ECh. 14.1 - Prob. 18ECh. 14.1 - Prob. 19ECh. 14.1 - Prob. 20ECh. 14.1 - Prob. 21ECh. 14.1 - Prob. 22ECh. 14.1 - Prob. 23ECh. 14.1 - Prob. 24ECh. 14.1 - Prob. 25ECh. 14.1 - Prob. 26ECh. 14.1 - Prob. 27ECh. 14.1 - Prob. 28ECh. 14.1 - Prob. 29ECh. 14.1 - Prob. 30ECh. 14.1 - Prob. 31ECh. 14.1 - Prob. 32ECh. 14.1 - Prob. 33ECh. 14.1 - Prob. 34ECh. 14.1 - Prob. 35ECh. 14.1 - Prob. 36ECh. 14.1 - Prob. 37ECh. 14.1 - Prob. 38ECh. 14.1 - Prob. 39ECh. 14.1 - Prob. 40ECh. 14.1 - Prob. 41ECh. 14.1 - Prob. 42ECh. 14.1 - Prob. 43ECh. 14.1 - Prob. 44ECh. 14.1 - Prob. 45ECh. 14.1 - Prob. 46ECh. 14.1 - Prob. 47ECh. 14.1 - Prob. 48ECh. 14.1 - Prob. 49ECh. 14.1 - Prob. 50ECh. 14.1 - Prob. 51ECh. 14.1 - Prob. 52ECh. 14.1 - Prob. 53ECh. 14.1 - Prob. 54ECh. 14.1 - Prob. 55ECh. 14.1 - Prob. 56ECh. 14.1 - Prob. 57ECh. 14.1 - Prob. 58ECh. 14.1 - Prob. 59ECh. 14.1 - Prob. 60ECh. 14.1 - Prob. 61ECh. 14.2 - Prob. 1PQCh. 14.2 - Prob. 2PQCh. 14.2 - Prob. 3PQCh. 14.2 - Prob. 4PQCh. 14.2 - Prob. 1ECh. 14.2 - Prob. 2ECh. 14.2 - Prob. 3ECh. 14.2 - Prob. 4ECh. 14.2 - Prob. 5ECh. 14.2 - Prob. 6ECh. 14.2 - Prob. 7ECh. 14.2 - Prob. 8ECh. 14.2 - Prob. 9ECh. 14.2 - Prob. 10ECh. 14.2 - Prob. 11ECh. 14.2 - Prob. 12ECh. 14.2 - Prob. 13ECh. 14.2 - Prob. 14ECh. 14.2 - Prob. 15ECh. 14.2 - Prob. 16ECh. 14.2 - Prob. 17ECh. 14.2 - Prob. 18ECh. 14.2 - Prob. 19ECh. 14.2 - Prob. 20ECh. 14.2 - Prob. 21ECh. 14.2 - Prob. 22ECh. 14.2 - Prob. 23ECh. 14.2 - Prob. 24ECh. 14.2 - Prob. 25ECh. 14.2 - Prob. 26ECh. 14.2 - Prob. 27ECh. 14.2 - Prob. 28ECh. 14.2 - Prob. 29ECh. 14.2 - Prob. 30ECh. 14.2 - Prob. 31ECh. 14.2 - Prob. 32ECh. 14.2 - Prob. 33ECh. 14.2 - Prob. 34ECh. 14.2 - Prob. 35ECh. 14.2 - Prob. 36ECh. 14.2 - Prob. 37ECh. 14.2 - Prob. 38ECh. 14.2 - Prob. 39ECh. 14.2 - Prob. 40ECh. 14.2 - Prob. 41ECh. 14.2 - Prob. 42ECh. 14.2 - Prob. 43ECh. 14.2 - Prob. 44ECh. 14.2 - Prob. 45ECh. 14.2 - Prob. 46ECh. 14.2 - Prob. 47ECh. 14.2 - Prob. 48ECh. 14.2 - Prob. 49ECh. 14.2 - Prob. 50ECh. 14.2 - Prob. 51ECh. 14.3 - Prob. 1PQCh. 14.3 - Prob. 2PQCh. 14.3 - Prob. 3PQCh. 14.3 - Prob. 4PQCh. 14.3 - Prob. 5PQCh. 14.3 - Prob. 1ECh. 14.3 - Prob. 2ECh. 14.3 - Prob. 3ECh. 14.3 - Prob. 4ECh. 14.3 - Prob. 5ECh. 14.3 - Prob. 6ECh. 14.3 - Prob. 7ECh. 14.3 - Prob. 8ECh. 14.3 - Prob. 9ECh. 14.3 - Prob. 10ECh. 14.3 - Prob. 11ECh. 14.3 - Prob. 12ECh. 14.3 - Prob. 13ECh. 14.3 - Prob. 14ECh. 14.3 - Prob. 15ECh. 14.3 - Prob. 16ECh. 14.3 - Prob. 17ECh. 14.3 - Prob. 18ECh. 14.3 - Prob. 19ECh. 14.3 - Prob. 20ECh. 14.3 - Prob. 21ECh. 14.3 - Prob. 22ECh. 14.3 - Prob. 23ECh. 14.3 - Prob. 24ECh. 14.3 - Prob. 25ECh. 14.3 - Prob. 26ECh. 14.3 - Prob. 27ECh. 14.3 - Prob. 28ECh. 14.3 - Prob. 29ECh. 14.3 - Prob. 30ECh. 14.3 - Prob. 31ECh. 14.3 - Prob. 32ECh. 14.3 - Prob. 33ECh. 14.3 - Prob. 34ECh. 14.3 - Prob. 35ECh. 14.3 - Prob. 36ECh. 14.3 - Prob. 37ECh. 14.3 - Prob. 38ECh. 14.3 - Prob. 39ECh. 14.3 - Prob. 40ECh. 14.3 - Prob. 41ECh. 14.3 - Prob. 42ECh. 14.3 - Prob. 43ECh. 14.3 - Prob. 44ECh. 14.3 - Prob. 45ECh. 14.3 - Prob. 46ECh. 14.3 - Prob. 47ECh. 14.3 - Prob. 48ECh. 14.3 - Prob. 49ECh. 14.3 - Prob. 50ECh. 14.3 - Prob. 51ECh. 14.3 - Prob. 52ECh. 14.3 - Prob. 53ECh. 14.3 - Prob. 54ECh. 14.3 - Prob. 55ECh. 14.3 - Prob. 56ECh. 14.3 - Prob. 57ECh. 14.3 - Prob. 58ECh. 14.3 - Prob. 59ECh. 14.3 - Prob. 60ECh. 14.3 - Prob. 61ECh. 14.3 - Prob. 62ECh. 14.3 - Prob. 63ECh. 14.3 - Prob. 64ECh. 14.3 - Prob. 65ECh. 14.3 - Prob. 66ECh. 14.3 - Prob. 67ECh. 14.3 - Prob. 68ECh. 14.3 - Prob. 69ECh. 14.3 - Prob. 70ECh. 14.3 - Prob. 71ECh. 14.3 - Prob. 72ECh. 14.3 - Prob. 73ECh. 14.3 - Prob. 74ECh. 14.3 - Prob. 75ECh. 14.3 - Prob. 76ECh. 14.3 - Prob. 77ECh. 14.3 - Prob. 78ECh. 14.3 - Prob. 79ECh. 14.3 - Prob. 80ECh. 14.4 - Prob. 1PQCh. 14.4 - Prob. 2PQCh. 14.4 - Prob. 3PQCh. 14.4 - Prob. 4PQCh. 14.4 - Prob. 5PQCh. 14.4 - Prob. 6PQCh. 14.4 - Prob. 1ECh. 14.4 - Prob. 2ECh. 14.4 - Prob. 3ECh. 14.4 - Prob. 4ECh. 14.4 - Prob. 5ECh. 14.4 - Prob. 6ECh. 14.4 - Prob. 7ECh. 14.4 - Prob. 8ECh. 14.4 - Prob. 9ECh. 14.4 - Prob. 10ECh. 14.4 - Prob. 11ECh. 14.4 - Prob. 12ECh. 14.4 - Prob. 13ECh. 14.4 - Prob. 14ECh. 14.4 - Prob. 15ECh. 14.4 - Prob. 16ECh. 14.4 - Prob. 17ECh. 14.4 - Prob. 18ECh. 14.4 - Prob. 19ECh. 14.4 - Prob. 20ECh. 14.4 - Prob. 21ECh. 14.4 - Prob. 22ECh. 14.4 - Prob. 23ECh. 14.4 - Prob. 24ECh. 14.4 - Prob. 25ECh. 14.4 - Prob. 26ECh. 14.4 - Prob. 27ECh. 14.4 - Prob. 28ECh. 14.4 - Prob. 29ECh. 14.4 - Prob. 30ECh. 14.4 - Prob. 31ECh. 14.4 - Prob. 32ECh. 14.4 - Prob. 33ECh. 14.4 - Prob. 34ECh. 14.4 - Prob. 35ECh. 14.4 - Prob. 36ECh. 14.4 - Prob. 37ECh. 14.4 - Prob. 38ECh. 14.4 - Prob. 39ECh. 14.4 - Prob. 40ECh. 14.4 - Prob. 41ECh. 14.4 - Prob. 42ECh. 14.4 - Prob. 43ECh. 14.4 - Prob. 44ECh. 14.4 - Prob. 45ECh. 14.4 - Prob. 46ECh. 14.4 - Prob. 47ECh. 14.5 - Prob. 1PQCh. 14.5 - Prob. 2PQCh. 14.5 - Prob. 3PQCh. 14.5 - Prob. 4PQCh. 14.5 - Prob. 5PQCh. 14.5 - Prob. 1ECh. 14.5 - Prob. 2ECh. 14.5 - Prob. 3ECh. 14.5 - Prob. 4ECh. 14.5 - Prob. 5ECh. 14.5 - Prob. 6ECh. 14.5 - Prob. 7ECh. 14.5 - Prob. 8ECh. 14.5 - Prob. 9ECh. 14.5 - Prob. 10ECh. 14.5 - Prob. 11ECh. 14.5 - Prob. 12ECh. 14.5 - Prob. 13ECh. 14.5 - Prob. 14ECh. 14.5 - Prob. 15ECh. 14.5 - Prob. 16ECh. 14.5 - Prob. 17ECh. 14.5 - Prob. 18ECh. 14.5 - Prob. 19ECh. 14.5 - Prob. 20ECh. 14.5 - Prob. 21ECh. 14.5 - Prob. 22ECh. 14.5 - Prob. 23ECh. 14.5 - Prob. 24ECh. 14.5 - Prob. 25ECh. 14.5 - Prob. 26ECh. 14.5 - Prob. 27ECh. 14.5 - Prob. 28ECh. 14.5 - Prob. 29ECh. 14.5 - Prob. 30ECh. 14.5 - Prob. 31ECh. 14.5 - Prob. 32ECh. 14.5 - Prob. 33ECh. 14.5 - Prob. 34ECh. 14.5 - Prob. 35ECh. 14.5 - Prob. 36ECh. 14.5 - Prob. 37ECh. 14.5 - Prob. 38ECh. 14.5 - Prob. 39ECh. 14.5 - Prob. 40ECh. 14.5 - Prob. 41ECh. 14.5 - Prob. 42ECh. 14.5 - Prob. 43ECh. 14.5 - Prob. 44ECh. 14.5 - Prob. 45ECh. 14.5 - Prob. 46ECh. 14.5 - Prob. 47ECh. 14.5 - Prob. 48ECh. 14.5 - Prob. 49ECh. 14.5 - Prob. 50ECh. 14.5 - Prob. 51ECh. 14.5 - Prob. 52ECh. 14.5 - Prob. 53ECh. 14.5 - Prob. 54ECh. 14.5 - Prob. 55ECh. 14.5 - Prob. 56ECh. 14.5 - Prob. 57ECh. 14.5 - Prob. 58ECh. 14.5 - Prob. 59ECh. 14.5 - Prob. 60ECh. 14.5 - Prob. 61ECh. 14.5 - Prob. 62ECh. 14.5 - Prob. 63ECh. 14.5 - Prob. 64ECh. 14.5 - Prob. 65ECh. 14.5 - Prob. 66ECh. 14.5 - Prob. 67ECh. 14.5 - Prob. 68ECh. 14.5 - Prob. 69ECh. 14.5 - Prob. 70ECh. 14.5 - Prob. 71ECh. 14.5 - Prob. 72ECh. 14.5 - Prob. 73ECh. 14.5 - Prob. 74ECh. 14.5 - Prob. 75ECh. 14.5 - Prob. 76ECh. 14.5 - Prob. 77ECh. 14.6 - Prob. 1PQCh. 14.6 - Prob. 2PQCh. 14.6 - Prob. 3PQCh. 14.6 - Prob. 4PQCh. 14.6 - Prob. 5PQCh. 14.6 - Prob. 6PQCh. 14.6 - Prob. 1ECh. 14.6 - Prob. 2ECh. 14.6 - Prob. 3ECh. 14.6 - Prob. 4ECh. 14.6 - Prob. 5ECh. 14.6 - Prob. 6ECh. 14.6 - Prob. 7ECh. 14.6 - Prob. 8ECh. 14.6 - Prob. 9ECh. 14.6 - Prob. 10ECh. 14.6 - Prob. 11ECh. 14.6 - Prob. 12ECh. 14.6 - Prob. 13ECh. 14.6 - Prob. 14ECh. 14.6 - Prob. 15ECh. 14.6 - Prob. 16ECh. 14.6 - Prob. 17ECh. 14.6 - Prob. 18ECh. 14.6 - Prob. 19ECh. 14.6 - Prob. 20ECh. 14.6 - Prob. 21ECh. 14.6 - Prob. 22ECh. 14.6 - Prob. 23ECh. 14.6 - Prob. 24ECh. 14.6 - Prob. 25ECh. 14.6 - Prob. 26ECh. 14.6 - Prob. 27ECh. 14.6 - Prob. 28ECh. 14.6 - Prob. 29ECh. 14.6 - Prob. 30ECh. 14.6 - Prob. 31ECh. 14.6 - Prob. 32ECh. 14.6 - Prob. 33ECh. 14.6 - Prob. 34ECh. 14.6 - Prob. 35ECh. 14.6 - Prob. 36ECh. 14.6 - Prob. 37ECh. 14.6 - Prob. 38ECh. 14.6 - Prob. 39ECh. 14.6 - Prob. 40ECh. 14.6 - Prob. 41ECh. 14.6 - Prob. 42ECh. 14.6 - Prob. 43ECh. 14.6 - Prob. 44ECh. 14.6 - Prob. 45ECh. 14.6 - Prob. 46ECh. 14.6 - Prob. 47ECh. 14.6 - Prob. 48ECh. 14.6 - Prob. 49ECh. 14.6 - Prob. 50ECh. 14.6 - Prob. 51ECh. 14.6 - Prob. 52ECh. 14.6 - Prob. 53ECh. 14.6 - Prob. 54ECh. 14.6 - Prob. 55ECh. 14.7 - Prob. 1PQCh. 14.7 - Prob. 2PQCh. 14.7 - Prob. 3PQCh. 14.7 - Prob. 1ECh. 14.7 - Prob. 2ECh. 14.7 - Prob. 3ECh. 14.7 - Prob. 4ECh. 14.7 - Prob. 5ECh. 14.7 - Prob. 6ECh. 14.7 - Prob. 7ECh. 14.7 - Prob. 8ECh. 14.7 - Prob. 9ECh. 14.7 - Prob. 10ECh. 14.7 - Prob. 11ECh. 14.7 - Prob. 12ECh. 14.7 - Prob. 13ECh. 14.7 - Prob. 14ECh. 14.7 - Prob. 15ECh. 14.7 - Prob. 16ECh. 14.7 - Prob. 17ECh. 14.7 - Prob. 18ECh. 14.7 - Prob. 19ECh. 14.7 - Prob. 20ECh. 14.7 - Prob. 21ECh. 14.7 - Prob. 22ECh. 14.7 - Prob. 23ECh. 14.7 - Prob. 24ECh. 14.7 - Prob. 25ECh. 14.7 - Prob. 26ECh. 14.7 - Prob. 27ECh. 14.7 - Prob. 28ECh. 14.7 - Prob. 29ECh. 14.7 - Prob. 30ECh. 14.7 - Prob. 31ECh. 14.7 - Prob. 32ECh. 14.7 - Prob. 33ECh. 14.7 - Prob. 34ECh. 14.7 - Prob. 35ECh. 14.7 - Prob. 36ECh. 14.7 - Prob. 37ECh. 14.7 - Prob. 38ECh. 14.7 - Prob. 39ECh. 14.7 - Prob. 40ECh. 14.7 - Prob. 41ECh. 14.7 - Prob. 42ECh. 14.7 - Prob. 43ECh. 14.7 - Prob. 44ECh. 14.7 - Prob. 45ECh. 14.7 - Prob. 46ECh. 14.7 - Prob. 47ECh. 14.7 - Prob. 48ECh. 14.7 - Prob. 49ECh. 14.7 - Prob. 50ECh. 14.7 - Prob. 51ECh. 14.7 - Prob. 52ECh. 14.7 - Prob. 53ECh. 14.7 - Prob. 54ECh. 14.7 - Prob. 55ECh. 14.7 - Prob. 56ECh. 14.7 - Prob. 57ECh. 14.7 - Prob. 58ECh. 14.7 - Prob. 59ECh. 14.7 - Prob. 60ECh. 14.7 - Prob. 61ECh. 14.7 - Prob. 62ECh. 14.7 - Prob. 63ECh. 14.7 - Prob. 64ECh. 14.8 - Prob. 1PQCh. 14.8 - Prob. 2PQCh. 14.8 - Prob. 3PQCh. 14.8 - Prob. 1ECh. 14.8 - Prob. 2ECh. 14.8 - Prob. 3ECh. 14.8 - Prob. 4ECh. 14.8 - Prob. 5ECh. 14.8 - Prob. 6ECh. 14.8 - Prob. 7ECh. 14.8 - Prob. 8ECh. 14.8 - Prob. 9ECh. 14.8 - Prob. 10ECh. 14.8 - Prob. 11ECh. 14.8 - Prob. 12ECh. 14.8 - Prob. 13ECh. 14.8 - Prob. 14ECh. 14.8 - Prob. 15ECh. 14.8 - Prob. 16ECh. 14.8 - Prob. 17ECh. 14.8 - Prob. 18ECh. 14.8 - Prob. 19ECh. 14.8 - Prob. 20ECh. 14.8 - Prob. 21ECh. 14.8 - Prob. 22ECh. 14.8 - Prob. 23ECh. 14.8 - Prob. 24ECh. 14.8 - Prob. 25ECh. 14.8 - Prob. 26ECh. 14.8 - Prob. 27ECh. 14.8 - Prob. 28ECh. 14.8 - Prob. 29ECh. 14.8 - Prob. 30ECh. 14.8 - Prob. 31ECh. 14.8 - Prob. 32ECh. 14.8 - Prob. 33ECh. 14.8 - Prob. 34ECh. 14.8 - Prob. 35ECh. 14.8 - Prob. 36ECh. 14.8 - Prob. 37ECh. 14.8 - Prob. 38ECh. 14.8 - Prob. 39ECh. 14.8 - Prob. 40ECh. 14.8 - Prob. 41ECh. 14.8 - Prob. 42ECh. 14.8 - Prob. 43ECh. 14.8 - Prob. 44ECh. 14.8 - Prob. 45ECh. 14.8 - Prob. 46ECh. 14.8 - Prob. 47ECh. 14.8 - Prob. 48ECh. 14.8 - Prob. 49ECh. 14.8 - Prob. 50ECh. 14.8 - Prob. 51ECh. 14.8 - Prob. 52ECh. 14.8 - Prob. 53ECh. 14.8 - Prob. 54ECh. 14.8 - Prob. 55ECh. 14.8 - Prob. 56ECh. 14 - Prob. 1CRECh. 14 - Prob. 2CRECh. 14 - Prob. 3CRECh. 14 - Prob. 4CRECh. 14 - Prob. 5CRECh. 14 - Prob. 6CRECh. 14 - Prob. 7CRECh. 14 - Prob. 8CRECh. 14 - Prob. 9CRECh. 14 - Prob. 10CRECh. 14 - Prob. 11CRECh. 14 - Prob. 12CRECh. 14 - Prob. 13CRECh. 14 - Prob. 14CRECh. 14 - Prob. 15CRECh. 14 - Prob. 16CRECh. 14 - Prob. 17CRECh. 14 - Prob. 18CRECh. 14 - Prob. 19CRECh. 14 - Prob. 20CRECh. 14 - Prob. 21CRECh. 14 - Prob. 22CRECh. 14 - Prob. 23CRECh. 14 - Prob. 24CRECh. 14 - Prob. 25CRECh. 14 - Prob. 26CRECh. 14 - Prob. 27CRECh. 14 - Prob. 28CRECh. 14 - Prob. 29CRECh. 14 - Prob. 30CRECh. 14 - Prob. 31CRECh. 14 - Prob. 32CRECh. 14 - Prob. 33CRECh. 14 - Prob. 34CRECh. 14 - Prob. 35CRECh. 14 - Prob. 36CRECh. 14 - Prob. 37CRECh. 14 - Prob. 38CRECh. 14 - Prob. 39CRECh. 14 - Prob. 40CRECh. 14 - 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- Use the Chain Rule to find dz / dt or dw/ dt.(odd) z = xy3 - x2y, x = t2 + 1, y = t2 -1arrow_forward(a) express dw/dt as a function of t, both by using the Chain Rule and by expressing w in terms of t and differentiating directly with respect to t. Then (b) evaluate dw/ dt at the given value of t. 1. w = x2 + y2, x = cos t, y = sin t; t = paiarrow_forward(a) express dw/dt as a function of t, both by using the Chain Rule and by expressing w in terms of t and differentiating directly with respect to t. Then (b) evaluate dw/ dt at the given value of t. w = ln (x2 + y2 + z2), x = cos t, y = sin t, z = 4sqrt(t) ; t =3arrow_forward
- Exercises 35–36, find the linearization L(x, y) of the function ƒ(x, y) at P0. Then find an upper bound for the magnitude |E| of the error in the approximation ƒ(x, y) ≈ L(x, y) over the rectangle R. 35. ƒ(x, y) = x2 - 3xy + 5 at P0(2, 1), R: |x - 2| <=0.1, | y - 1| <=0.1 36. ƒ(x, y) = (1/2)x2 + xy + (1/4)y2 + 3x - 3y + 4 at P0(2, 2), R: |x - 2| <=0.1, |y - 2|<= 0.1arrow_forwardLet f (x, y) = (x − y)e^x . Use the Chain Rule to calculate ∂f/∂u and ∂f/∂v (in terms of u and v), where x = u − v and y = u + v.arrow_forwardUse the chain rule to find dz/dt z=xy^5-x^2y x=t^2+1 y=t^2-1arrow_forward
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