Study Guide
241F Test #2
10/22/04______1_
Do any three problems.
You may do any of the remaining problems for extra credit.
Problem #1 Given a vector valued function of the form r(t) = x(t)i + y(t)j :
(a)-[5 pts.] Find the velocity vector v(t) and the speed |v(t)|.
(b)-[5 pts.] Find the unit tangent vector T(t).
(c)-([5 pts.] Find the curvature .
(d)-[5 pts.] Find the principal unit normal vector N(t).
(e)-[5 pts.] Calculate the length of the curve traced out by r(t) as t increases from t0 to t1.
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Problem #2 Given a vector valued function of the form r(t) = x(t)i + y(t)j + z(t)k:
(a)-[5 pts.] Find the velocity vector v(t).
(b)-[5 pts.] Find the acceleration vector a(t).
(c)-[5 pts.] Find the tangential component aT and the normal component aN of the acceleration vector .
(d)-[5 pts.] Find the principal normal vector N(t) and the binormal vector B(t).
(e)-[5 pts.] Find the curvature .
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Problem #3
Given the acceleration vector a(t)  a1 (t)i  a2 (t)j  a3 (t)k , the initial velocity v(0) ,
and the initial position vector r (0) :
(a)-[10 pts.] Find the velocity vector v (t) , and the speed v(t) .
(b)-[8 pts.] At what time is the speed a minimum, and what is the minimum speed?
(c)-[7 pts.] Find the position vector r (t)
(See prob. # 15 on page 879, and prob. 18 on page 883.)
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Problem #4
(a)-[5 pts.] Sketch some level curves of a given function z = f(x, y).
(b)-[5 pts.] Describe the level surfaces of a given function f(x, y, z).
(c)-[5 pts.] Given z = f(x, y), find the limit at (x 0 , y0 ) if it exists, or
show that it does not exist.
(c)-[5 pts] Given f(x, y, z), find the limit at (x 0 , y0 , z 0 ) if it exists, or
show that it does not exist.
(d)-[5 pts.] Determine the set of points in the x-y plane where z = f(x, y) is continuous.
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Problem #5 Given the function z = f(x, y):
(a)-[5 pts.] Calculate fx(x, y) and fy(x, y). Evaluate these partial derivative at a given point (x0, y0).
(b)-[10 pts.] Calculate fxx(x, y), fxy(x, y), fyx(x, y), fyy(x, y). Evaluate these partial derivatives at a
specified point (x 0 , y0 ) .
Given the function w = f(x, y, z):
(c)-[5 pts.] Calculate fx(x, y, z), fy(x, y, z) and fz(x, y, z). Evaluate these partial derivative at a
given point (x 0 , y0 , z0 ) .
(d)-[5 pts.] Calculate fxz(x, y, z), fyz(x, y, z) and fzz(x, y, z).
Evaluate these partial derivative at a given point (x 0 , y0 , z0 ) .
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