18.02A
Exam 1
December 16, 2020
You have 120 minutes to work on the test, and an additional 30 minutes to upload your answers
to Gradescope. You must submit your exam by 11:30am Eastern time. Submissions made after
that time will be designated as late.
Submission is made via Gradescope. The answers to each question should be written on separate
pages, and will be marked accordingly on Gradescope.
The exam is open book in the sense that you may consult any materials from this class. You
are not allowed to consult any person during the exam. The hope and expectation is that it will be
neither necessary nor efficient for you to use outside non-human sources or calculators. But if you
do use any such help, you must identify your sources clearly within the problem where you used
them. Do not conceal your sources.
For questions during the exam on content and wording, or in case of technical difficulties,
please write to both the Course Administrator Andrei Ionov at aionov@mit.edu and Professor
John Bush at bush@math.mit.edu. Any announcements made during the course of the exam, such
as corrections or clarifications of the problems, will appear as announcements on Canvas.
In all problems, explain how you obtain your answer, and detail your solutions. Stating an
answer is insufficient for full credit.
There are 12 problems on 13 pages. Point scores (not all equal) are indicated. The
total test score is 200.
Exam contents:
page 1.....Cover page
page 2.....1ab [10 pts: 5+5]
page 3.....2abc [15 pts: 5+5+5]
page 4.....3abc [20 pts: 10+5+5]
page 5.....4ab [15 pts: 8+7]
page 6.....5ab [20 pts: 5+15]
page 7.....6ab [20 pts: 10+10]
page 8.....7 [15 pts]
page 9.....8 [15 pts]
page 10.....9abcd [20 pts: 5+5+5+5]
page 11.....10ab [20 pts: 15+5]
page 12.....11 [15 pts]
page 13.....12 [15 pts]
1
Problem 1. (10pts: 5 + 5)
Consider x = u2 + v 2 and y = euv , where u = s2 + t and v = s − t2 .
a) Calculate
∂x
at the point (s, t) = (1, 0).
∂t
b) Calculate
∂y
at the point (s, t) = (2, 1).
∂s
2
Problem 2. (15pts: 5+5+5 ) Consider the system of linear equations
3x − y + 2z = 6
2y − 3z = 8
x + az = 10 .
a) Write the system in matrix form: AX = B.
b) Compute det A in terms of the constant a.
c) For which values of a will the system AX = 0 have more than one solution?
3
Problem 3. (20pts: 10+5+5)
√
A hockey puck of radius 1 slides along the ice in the xy−plane at a speed 10 2 in the direction
of the vector î + ĵ. As it slides, it spins in a clockwise direction at 2 revolutions per second. At
time t = 0, the puck’s center is at the origin (0, 0).
a) Find the parametric equations for the trajectory of the point P on the edge of the puck initially
at (1, 0).
b) Find the velocity v of the point P.
c) What is the minimum speed of the point P, and what is the direction of the velocity at the
corresponding time?
4
Problem 4. (15pts: 8+7)
Consider f (x, y) = 8x4 + xy − 13 y 3 .
a) Find all the critical points of f .
b) Identify these points as being local minima, local maxima or saddle points.
5
Problem 5. (20pts: 5 + 15)
Z 1 Z −x
Consider the double integral:
I =
0
2
e−y dy dx
−1
a) Sketch the domain of integration.
b) Evaluate the integral. (Hint: choice of vertical or horizontal strips is important.)
6
Problem 6. (20 points: 10 + 10)
Consider the integral
Z 2Z x
I =
1
0
x4
1
dy dx .
+ x2 y 2
a) Sketch the domain of integration and define its four boundaries in terms of polar coordinates.
b) Evaluate the integral by transforming to polar coordinates. Hint: Use radial strips.
7
Problem 7. (15 points)
Use Lagrange multipliers to find the minimum value of x2 +y 2 +3z 2 on the plane 4x+2y+6z = 1.
8
Problem 8. (15pts)
Consider a right circular cone with base of radius R and height 2R, sitting with its base in the
xy−plane, centered at the origin (0, 0). Use Lagrange multipliers to find the vertical cylinder (with
symmetry axis along the z-axis, radius r and height z) with the largest volume V = πr2 z that fits
inside this cone. What is this cylinder’s radius, height and volume?
9
Problem 9. (20pts: 5+5+5+5)
a) Find the gradient of the temperature field T (x, y, z) = x2 + xy + yz at the point P = (1, 1, 0).
b) Calculate the tangent plane to the surface of constant temperature T = 2 at the point P .
c) Find the directional derivative of T (x, y, z) in the î + 2ĵ direction at the point P .
d) A fly moves along the parameterized curve x = t , y = t2 , z = 0 , where t is time. Calculate
dT
experienced by the fly at time t = 1 using two different
the rate of change of temperature
dt
approaches: i) direct substitution of x(t), y(t) and z(t) into T (t), and ii) the Multivariable Chain
Rule.
10
Problem 10. (20 points: 15+5).
Consider the domain D in the xy−plane bounded by y = x2 and y =
0 ≤ x ≤ 1.
√
x in the region
a) Compute the geometric center of the domain D.
b) Calculate Iy , the moment of inertia about the y-axis of the domain D, if this region has uniform
density ρ = 1.
11
Problem 11. (15pts)
Consider a circle of radius a = 1 lying flat in the xy-plane with its center at (1, 0). Its density
(mass per unit area) increases linearly with distance from the origin: ρ(r) = r. Calculate the mass
of the circle.
12
Problem 12. (15 points)
Find the volume of intersection of two perpendicular cylinders of equal radius a, specifically,
the volume contained within both cylinders x2 + y 2 = a2 and x2 + z 2 = a2 .
13