April 10, 2008
Name_________________________
Quiz 4
Please show all work. Partial credit may be given, but only if all work is shown.
1. Find the curvature for the following curve and determine the value(s) of t where the
curvature is a maximum: x = t, y = t, z = 1 + t2
r' (t) = < 1,1,2t >
r' '(t) = < 0,0,2 >
r'(t) x r''(t) = <2, -2, 0>
r' (t) × r'' (t) = 2 2 , r' (t) = 2 + 4t 2
k(t) =
2 2
(2 + 4t2 )3/2
The curvature is a maximum when t = 0
2. A ball is thrown at an angle of 45 o to the ground. If the ball lands 90 m away, what
was the initial speed of the ball?
y = v0 (sin 45) t - (1/2) g t2 = 0 at time when ball hits the ground
Solving for t gives t = (2/g) v0 sin 45
Substitute this into the expression for x and set x equal to 90:
x = (2/g) v0 2cos 45 sin 45 = 90
Thus, vo2 = (90)(9.8)/ sin(90) = 882 since sin 90 = 1
v 0 = 882 = 21 2 m/ s
y − x2
3. Find and sketch the domain of the function: f(x,y) =
1− x2
x2 -1 ≠ 0 and y - x2 ≥ 0 so the domain is the
region inside the parabola but not the
dashed lines.