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Midterm

MATH 32A Study Guide - Midterm Guide: Square Root, Quotient Rule, Talking Lifestyle 1278Exam


Department
Mathematics
Course Code
MATH 32A
Professor
All
Study Guide
Midterm

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MATH 32A (Butler)
Practice for Midterm II (Solutions)
1. A particle moves through three dimensional space with velocity
v(t) = hsec2t, 2 sec ttan t, tan2ti.
At time t= 0 the particle is at h0,1,2i, find the position function of the particle for
π/4tπ/4.
If r(t) is the position function then r0(t) = v(t). So taking antiderivatives we
have
r(t) = Zv(t)dt =Zsec2t dt, Z2 sec ttan t dt, Ztan2t dt
=tan t+C, 2 sec t+D, Z(sec2t1) dt
=tan t+C, 2 sec t+D, tan tt+E.
Two of the three integrals are straightforward. The last one is the trickiest
but this follows by relating tan2t(something which we cannot directly inte-
grate) to sec2t(something which is easy to integrate). Now all that is left is
to determine the constants C, D, E. We have
r(0) = hC, 2 + D, Ei=h0,1,2i
giving the constants we need. So we have that the position function of the
particle is
r(t) = tan t, 2 sec t1,tan tt+ 2.
(The condition for π/4tπ/4 is not needed directly, it only is used to
guarantee that we stay away from the vertical asymptotes (solutions cannot
be pushed past vertical asymptotes, something you will learn in your future
math classes).)

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2. Find the cumulative length function s(t) (starting from a= 1) of the parametric
curve hln t, 2t, 1
2t2i.
We have that the cumulative arc length function will be
s(t) = Zt
ar0(u)du.
We are told that a= 1 and we now compute the derivative. We have
r0(t) = 1
t,2, t.
Therefore
s(t) = Zt
11
u,2, u
du
=Zt
1s1
u2
+22+u2du
=Zt
1r1
u2+2+u2du
=Zt
1s1
u+u2
du
=Zt
11
u+udu
=ln u+1
2u2
t
1
= ln t+1
2t21
2.
(Frequently in this type of problem the functions will be chosen so that the
terms on the inside of the square root “miraculously” combine into a perfect
square. Of course this is because they have been rigged to do so and would
not happen by coincidence.)
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