Math 251, Practice Midterm #2
Rutgers University, Dr. Molnar, Fall 2020
Name:
On my honor, I have neither received nor given any unauthorized assistance on this examination.
Signature:
Section:
Question
Points
Score
1
8
2
8
3
18
4
18
5
13
6
12
7
12
Total:
89
This practice exam is based on sections 14.5–14.8 and 15.1–15.4 in Thomas, 14th ed.
Math 251 Fake Exam 2 Page 2 of 7
1. [8 points] A dorito, D, is bounded by the x-axis and the lines y “ 2x, y “ 8 ́ 2x. The density of cheese dust on D is given by cpx, yq “ .03p8 ́ 2x ́ yq grams of processed cheese particles per square inch. Calculate the total quantity of delicious “cheese” on D. (Hint: go horizontally.)
ż 1{2 ż 1{4 ˆ2
2. [8 points] Evaluate the double integral sec2 0x2 3
̇
y3{2
dy dx.
Math 251 Fake Exam 2 Page 3 of 7 3. [18points] Considerfpx,yq“4×2 ́xy`4y2 onthedomainD“tpx,yq:x2`y2 ď8u.
a) Find the critical point of f on the interior of D, and classify it using the second derivative test.
b) Use Lagrange multipliers to find the extreme values of f on the boundary of D. Combined with your answer to part a), report the minimum and maximum values of f on all of D.
Math 251 Fake Exam 2 Page 4 of 7 4. [18 points] Let fpx,y,zq “ x , and let S “ tpx,y,zq P R3 : fpx,y,zq “ 2u. Note that
y ́z
a) In the direction of which unit vector u does DufpPq take on its maximum value, and what is
P “ p2,2,1q is on S. that value?
b) Find an equation for the tangent plane to S at P .
Math 251 Fake Exam 2 Page 5 of 7
5. [13 points] Let D be the region in the first quadrant bounded by the x-axis, the circle x2 ` y2 “ 4, ij
a
and the line x “ 1. Use polar coordinates to calculate
x2 ` y2 ́ 1 dA.
D
Math 251 Fake Exam 2
6. [12points] Letfpx,yq“px`1qpy ́2q.
a) Find the linearization Lpx, yq of f at the point p0, 0q.
Page 6 of 7
b) Determine whether the limit
px,yqÑp0,0q
exists, that is, whether f is differentiable at (0, 0). Justify your conclusion.
fpx,yq ́Lpx,yq lim a
x2 ` y2
Math 251 Fake Exam 2 Page 7 of 7
7. [12 points] A piece of kryptonite located at the origin gives off radiation. The amount of radiation is given by the function Rpx, y, zq “ e ́px2`y2`z2q, where x, y, and z are measured in astronomical units and the radiation is measured in Grays.
a) Describe the level surfaces of R.
b) Calculate ∇R.
c) If a space guy travels along a line from p1, ́2, 6q toward p4, 4, 4q at a constant speed of one a.u. per day, what will the rate of change in the amount of radiation be at the starting point?