Download SAMPLE MIDTERM 2

CALCULUS III, SPRING 2005
SAMPLE MIDTERM EXAM 2
YOU MUST JUSTIFY YOUR ANSWERS. NO CALCULATORS.
(1) Consider the curve r(t) = (t, t2 , t3 ). Let P = (1, 1, 1).
(a) Find the equation of the normal plane to the curve at P.
(b) Find the curvature κ at P.
(2) Reparametrize the following curve with respect to arclength,
starting at t = 0 in the direction of increasing t,
r(t) = (3t + 1, 2 − 5t, 4t − 3)
(3) (a) Decide if the function
(
f (x, y) =
x
x2 +y 2
0
is continuous at (0, 0)
(b) Show that the function
( 2
g(x, y) =
xy
x2 +y 2
0
(x, y) 6= (0, 0)
(x, y) = (0, 0)
(x, y) 6= (0, 0)
(x, y) = (0, 0)
is continuous at (0, 0).
∂g
(4) (a) A function g(u, v) satisfies: g(3, 4) = 10, ∂u
(3, 4) = 2,
∂g
(3, 4) = −4. Use this information to estimate g(2.8, 4.1).
∂v
(b) If f (x, y, z) = xyez and x = cos(t), y = sin(t), z = t2 , use
. Express your answer as a
the chain rule to compute df
dt
function of t.
(5) Consider two parametrizations of the same curve: r(t) and r(s),
where s = f (t). Show that if s0 = f (t0 ), then the vectors
(T(t0 ), N(t0 ), B(t0)) = (T(s0 ), N(s0), B(s0 ))
i.e., the TNB-frame is independent of parametrization.