Hands-­‐On Activity Instructions To be done in groups of 2. 1. First Round: • One person takes two of the colored pieces of paper (i.e., two homework submissions). The other person takes the other two colored pieces of paper. • Compare the two homework submissions in your hand. • Once you have determined a winner, make two piles on your table: one for those who have won this round and one for those who have lost this round. 2. Second Round: • One person takes the pile of submissions that won the first round. The other person takes the pile of submissions that lost the first round. • Compare the two homework submissions in your hand as you did in the first round. • Order the assignments accordingly: o First: won the first round, won the second round o Second: won the first round, lost the second round o Third: lost the first round, won the second round o Fourth: lost the first round, lost the second round • If you finish before the allotted time, please look over the questions on the worksheet entitled “Questions” and discuss them with each other. Homework Assignment: Griffiths Problem 1.12 The height of a certain hill (in feet) is given by ℎ 𝑥, 𝑦 = 10 2𝑥𝑦 − 3𝑥 ! − 4𝑦 ! − 18𝑥 + 28𝑦 + 12 , where y is the distance (in miles) north, and x the distance east of South Hadley. (a) Where is the top of the hill located? Contact Info: Nicole Michelotti ([email protected]) Jared Tritz ([email protected]) Dave Winn ([email protected]) Physics 405 HW 1
January 13, 2014
1. Griffiths 1.12
(a) The height of the hill (in feet) is given by:
3x2
h(x, y) = 10(2xy
4y 2
18x + 28y + 12)
To find the top of the hill, we need to find where rh = 0 (i.e. the location of the function extrema)
rh = 10(2y
6x
18)x̂ + 10(2x
) 2y
6x
) 2x
8y + 28 = 0
8y + 28)ŷ = 0
18 = 0
We can solve this system of equations by solving the second equation for x and substituting x into
the first equation to obtain y
) 2y 24y + 84 18 = 0
)y=3
Substituting y back in to find x we obtain
)x=
2
Therefore the top of the hill is located at (x, y) = ( 2, 3)
or equivalently 3 miles north and 2 miles west of South Hadley
(b) Substituting the (x, y) = ( 2, 3) coordinate of the top of the hill into the height function, we can
obtain the height of the hill
h( 2, 3) = 10(2( 2)(3)
3( 2)2
4(3)2
18( 2) + 28(3) + 12) = 720 feet
(c) The steepness of the slope at (x, y) = (1, 1) is equivalent to the magnitude of rh evaluated at (1,1)
rh(1, 1) = 10(2(1)
6(1)
18)x̂ + 10(2(1) 8(1) + 28)ŷ = 220x̂ + 220ŷ = 220( x̂ + ŷ)
p
|rh| = 2202 + ( 220)2 ⇡ 311 feet/mile
The direction of the steepest slope at this point is in the direction of the gradient, which we
found to be in the direction of ( x̂ + ŷ) which is exactly northwest. Therefore our final answer is
311 feet/mile, northwest .
Questions From an instructor’s perspective, what are some of the benefits of having students do peer review on quantitative problems? From an instructor’s perspective, what are some of the drawbacks of having students do peer review on quantitative problems? What are some suggestions you have? Contact Info: Nicole Michelotti ([email protected]) Jared Tritz ([email protected]) Dave Winn ([email protected])