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Math 53, First Midterm
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Name:
Signature:
TA’s Name:
Discussion section:
Instructions: Show your work (but not to others). Unjustified answers will not
receive credit. (No justification is required in the True/False section, however.)
1. (a) Define carefully: f is differentiable at (a, b) if and only if. . .
(b) Given an example of a function f which is continuous at (0, 0), but not differentiable
at (0, 0). (No justification needed.)
(c) What hypotheses on f let you conclude (via a major theorem in the book) that f
is differentiable at (a, b)?
2. The temperature on a flat plate is given by T (x, y) = x sin y degrees Celsius. A bug is
located at (1, π/2).
(a) In what direction should the bug move for the most rapid decrease in temperature?
(b) What is the rate of change of temperature (per unit distance) in the direction
−2~i + ~j?
(c) In what direction can the bug move so that the temperature remains the same as
it is at (1, π/2)?
3. (a) Find the distance from the point P (4, −1, 2) to the plane 2x + 2y − z + 1 = 0.
(b) Find the cosine of the angle made by the two tangents to the curve
x = 2 cos(t) and y = sin(t) cos(t)
at the point (0, 0).
4. (a) Find an equation for the plane tangent to the surface xyz 2 = 8 at the point (1, 2, 2).
(b) Is there a point on the surface xyz 2 = 8 where the tangent plane is parallel to the
xy plane? (Justify.)
5. A particle moving in a central force field in 3-space has position vector ~r(t) satisfying
d2~r
= f (t)~r(t) at all times t, where f is a scalar valued function.
dt2
d~r
~
× ~r(t) (representing angular momentum) is constant in
(a) Show the vector L(t)
=
dt
time.
(b) The motion of such a particle is confined to a plane. Suppose that at t = 0, we
d~r
have ~r = h1, 2, 1i and
= h2, 0, 1i. Find an equation for this plane.
dt
6. Locate all local maxima, local minima, and saddle points for the function f (x, y) =
x3 − 9xy + y 3 .
6. True or False. (There is no penalty for guessing wrong.)
1. ~a · (~b × ~c) = (~a · ~b) × ~c for all ~a, ~b, ~c.
2. If all partials of F exist and are continuous everywhere, and ∇F (a, b, c) 6= ~0,
then the equation F (x, y, z) = F (a, b, c) defines a surface near (a, b, c).
3. Let R(x, y) = f (x, y) − (f (a, b) + fx (a, b)(x − a) + fy (a, b)(y − b)). Then f is
differentiable at (a, b) if and only if R(x, y) → 0 as (x, y) → (a, b).
4. ~a · (~b − proj~a (~b) = 0 for all ~a, ~b.
5. If all second partial derivatives of f exist and are continuous everywhere, then
fxy = fyx .
d~r d
6. If = 1 for all t, and f is differentiable at ~r(0), then the value of (f (~r(t)))
dt
dt
d~r .
at t = 0 is the derivative of f at ~r(0) in the direction of the vector
dt t=0