Practice Midterm 1 โ Math 2153 (Section 10)
1. Give examples of the following. Be as explicit as possible. You do NOT need to justify your
answers.
(a) (2 points) Give an example of a continuous vector-valued function r(t)which is not differ-
entiable at t= 2.
(b) (2 points) Give an example of a vector-valued function which has constant curvature ฮบ6= 0.
(c) (2 points) Give equations for two different planes in R3which are parallel to the plane
z= 2xโ5y.
(d) (2 points) Give an example of a vector uโR2for which uยท๎cos ฯ
5,sin ฯ
5๎= 0
2. Let u=h3,2,2iand v=h1,โ4,6i.
(a) (2 points) Compute 2uโv.
(b) (2 points) Decide if the angle between uand vis acute, right or obtuse.
(c) (2 points) Compute projuv.
3. (5 points) A ball is thrown from an intial height of 20 m above with an initial speed of 10 m/s.and
initial angle of ฯ
3radians with the ground. Give the position vector of the ball r(t)at time t.
4. (5 points) A weight of 5 N is tied to two strings, both fastened to the ceiling. The first string is tied
to the weight at position h0,0iand is fastened to the ceiling at position h3,2iThe second string is
tied to the weight at position h0,0iand is fastened to the ceiling at position hโ1,2i. Compute the
force vectors which give the forces that the strings exhert on the weight.
5. (5 points) Compute the maximum and minimum curvature for the parametric curve
r(t) = h2 cos t, 3 sin tifor 0โคtโค2ฯ.
6. (5 points) Give an equation for the plane containing the line r(t) = h3 + t, 2,1โtiand the point
(2,โ3,1).
7. (5 points) Compute the normal and tangential components of acceleration (aNand aT) for a
particle whose position at time tis given by the vector-valued function
r(t) = ht2, t + 2i.
8. (5 points) Decide if the following limits exist, justifying your answer accordingly:
(i) lim
(x,y)โ(0,0)
6xy2
x2+ 7y4;(ii) lim
(x,y)โ(0,0)
sin(x2+ 4y2)
x2+ 4y2.
