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Math 20C : Midterm exam
Instructions. You are allowed to consult your textbook or ebook, your notes, and the
lecture videos. Show all of your work. No credit will be given for unsupported answers,
even if correct. You are not allowed collaborate or communicate with any other
humans while working on this exam.
This exam is worth 40 points.
1. (6 points) Find an equation for the line which passes through (1, âˆ’2, 0) and is parallel to
the line through (3, âˆ’4, 2) and (2, âˆ’3, 0).
2. (8 points) Find an equation for the plane which passes through (âˆ’2, 0, 1) and is parallel
to the lines L1 (t) = ht + 1, 2t âˆ’ 1, âˆ’t + 2i and L2 (s) = hâˆ’2s + 5, âˆ’3, s + 1i.
3. (6 points) Let f (x, y) =
x2 y 2
. Evaluate the limit
x2 + y 2
lim
f (x, y) or determine that it
(x,y)â†’(0,0)
does not exist.
4. (10 points) Denote by r(t) and v(t) the position and
âˆšvelocity vectors of a particle at time
t. Assume that r(0) = h1, 0, 1i and v(t) = hcos t, sin t, t + 2i.
(a) (5 points) Find r(t).
(b) (5 points) Find the length of the curve r(t) over the interval 0 â‰¤ t â‰¤ 1.
5. (10 points) Let f (x, y) = x2 y 3 .
âˆ‚f
âˆ‚f
and
.
âˆ‚x
âˆ‚y
(b) (6 points) Find the points on the graph of z = f (x, y) at which the vector n = h4, âˆ’3, 4i
is normal to the tangent plane.
(a) (4 points) Find the partial derivatives

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Explanation & Answer:

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equation for the plane

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