Calculus 2 Practice Exam

18.02 Final Exam
No books, notes or calculators.
15 problems, 250 points.

Useful formula: cos2 (θ) = 12 (1 + cos(2θ))

Problem 1. (20 points)
a) (15 pts.) Find the equation in the form Ax + By + Cz = D of the plane P
which contains the line L given by x = 1 − t, y = 1 + 2t, z = 2 − 3t and the point
(- 1, 1, 2).
b) (5 pts.) Let Q be the plane 2x + y + z = 4. Find the component of a unit normal
vector for Q projected on a unit direction vector for the line L of part(a).

Problem 2. (15 points)
Let L denote the line which passes through (0,0,1) and is parallel to the line in the
xy-plane given by y = 2x.
a) (5 pts.) Sketch L and give its equation in vector-parametric form.
b) (5 pts.) Let P be the plane which passes through (0,0,1) and is perpendicular to
the line L of part(a). Sketch in P (above) and give its equation in point-normal form.
� (0, 0, 1) lies on L. Write down the method or
c) (5 pts.) Suppose that the point P =
formula you would use to find the point P ∗ which is: (i) on L; (ii) the same distance
away from the point (0,0,1) as P ; and is (iii) on the other side of P from P .

Problem 3. (20 points) ⎡ ⎤
1 0 3
Given the 3 × 3 matrix: A a = ⎣ −2 1 −1 ⎦:
−1 1 a
a) (5 pts.) Let a = 2: show that | A2 | = 0
⎡
⎤ ⎡ ⎤
x 0
b) (7 pts.) Find the line of solutions to A2 ⎣ y ⎦ = ⎣ 0 ⎦
z 0
⎡ ⎤
∗ ∗ ∗
c) (8 pts.) Suppose now that a = 1, and that A1 −1 = ⎣ −3 p 5 ⎦. Find p.
∗ ∗ ∗

Problem 4. (10 points)
Let r(t) = �cos(et ), sin(et ), et �.
r� (t)
a) (5 pts.) Compute and simplify the unit tangent vector T(t) = .
| r� (t) |
b) (5 pts.) Compute T� (t)


1

, Problem 5. (20 points)
� y
Consider the function F (x, y, z) = z x2 + y + 2 :
z
a) (10 pts.) The point P0 : (1, 3, 2) lies on the surface F (x, y, z) = 7. Find the
equation of the tangent plane to the surface F = 7 at P0 .
b) (5 pts.) If starting at P0 a small change were to be made in only one of the
variables, which one would produce the largest change (in absolute value) in F ? If
the change is this variable was of size 0.1, approximately how large would the change
in F be ?
c) (5 pts.) What distance from P0 in the direction ±�−2, 2, −1� will produce an
approximate change in F of size 0.1 units, according to the (already computed) lin­
earization of F ?

Problem 6. (15 points)
2
Let f (x, y) = x + 4y + .
xy
a) (10 pts.) Find the critical point(s) of f (x, y)
b) (5 pts.) Use the second-derivative test to test the critical point(s) found in part(a).

Problem 7. (10 points)
Let P be the plane with equation Ax + By + Cz = D and P0 = (x0 , y0 , z0 ) be a point
which is not on P.
Use the Lagrange multiplier method to set up the equations satisfied by the point
(x, y, z) on P which is closest to P0 . (Do not solve.)

Problem 8. (15 points) √
a) (10 pts.) Let F (x, y, z) be a smooth function of three variables for which �F (1, −1, 2) =
�1, 2, −2�.
∂F
Use the Chain Rule to evaluate at (ρ, φ, θ) = (2, π4 , − π4 ).
∂φ
(Use x = ρ sin φ cos θ, y = ρ sin φ sin θ, z = ρ cos φ.)
b) (5 pts.) Suppose f (x, y) is a smooth, non-constant function. Is it possible that, for
all points (x, y), the gradient of f at the point (x, y) is equal to the vector �−y, x� ?
Justify (briefly).

Problem 9. (10 points)
�� � 2 � 2√2x
f dA = f (x, y) dy dx .
R 0 x2
a) (5 pts.) Sketch the region R.
b) (5 pts.) Rewrite the double integral as an iterated integral with the order inter­
changed.



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