Math 53 – Practice Midterm 1 A
z
Problem 1. (8 points)
A unit cube lies in the first octant, with a vertex at the origin (see figure). ¡ Q¡¡
¡
−−→ −−→ qR y
a) Express the vectors OQ (a diagonal of the cube) and OR O
¡ ¡
(joining O to the center of a face) in terms of ı̂, ̂, k̂. x¡ ¡
¡
b) Find the cosine of the angle between OQ and OR.
Problem 2. (7 points)
√
Find the area enclosed by a loop of the curve given by the polar equation r = sin 2θ.
Problem 3. (15 points)
a) Find the area of the space triangle with vertices P0 : (2, 1, 0), P1 : (1, 0, 1), P2 : (2, −1, 1).
b) Find the equation of the plane containing the three points P0 , P1 , P2 .
~ = h1, 1, 1i and passing
c) Find the intersection of this plane with the line parallel to the vector V
through the point S : (−1, 0, 0).
Problem 4. (15 points)
a) Let ~r = x(t)ı̂ + y(t)̂ + z(t)k̂ be the position vector of a path. Give a simple intrinsic formula
d
for (~r · ~r) in vector notation (not using coordinates).
dt
b) Show that if ~r has constant length, then ~r and ~v are perpendicular.
c) let ~a be the acceleration: still assuming that ~r has constant length, and using vector differ-
entiation, express the quantity ~r · ~a in terms of the velocity vector only.
Problem 5. (10 points)
Let f (x, y) = xy − x4 .
a) Find the gradient of f at P : (1, 1).
b) Give an approximate formula telling how small changes ∆x and ∆y produce a small change
∆w in the value of w = f (x, y) at the point (x, y) = (1, 1).
Problem 6. (10 points)
On the topographical map below, the level curves for the height function h(x, y) are marked (in
feet); adjacent level curves represent a difference of 100 feet in height. A scale is given.
a) Estimate to the nearest .1 the value at the point P of the directional derivative Dû h, where
û is the unit vector in the direction of ı̂ + ̂.
∂h ∂h
b) Mark on the map a point Q at which h = 2200, = 0 and < 0. Estimate to the
∂h ∂x ∂y
nearest .1 the value of at Q.
∂y
2200
2100
P
2000
1900
1000
Problem 7. (5 points)
Find the equation of the tangent plane to the surface x3 y + z 2 = 3 at the point (−1, 1, 2).
Problem 8. (5 points)
∂w ∂w
Let w = f (u, v), where u = xy and v = x/y. Express and in terms of x, y, fu and fv .
∂x ∂y
Problem 9. (15 points)
A rectangular box is placed in the first octant as shown, with one corner at the origin and the
three adjacent faces in the coordinate planes. The opposite point P : (x, y, z) is constrained to lie
on the paraboloid x2 + y 2 + z = 1. Which P gives the box of greatest volume? z
a) Show that the problem leads one to maximize f (x, y) = xy − x3 y − xy 3 , and
write down the equations for the critical points of f . ¡ P¡
¡ p¡
y
b) Find a critical point of f which lies in the first quadrant (x > 0, y > 0).
¡ ¡
¡
x¡
c) Determine the nature of this critical point by using the second derivative test. ¡
d) Find the maximum of f in the first quadrant (justify your answer).
Problem 10. (10 points)
In Problem 9 above, instead of substituting for z, one could also use Lagrange multipliers to
maximize the volume V = xyz with the same constraint x2 + y 2 + z = 1.
a) Write down the Lagrange multiplier equations for this problem.
b) Solve the equations (still assuming x > 0, y > 0).
