MATH 32A Study Guide - Midterm Guide: Square Root, Quotient Rule, Talking Lifestyle 1278Exam
Course CodeMATH 32A
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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
If r(t) is the position function then r0(t) = v(t). So taking antiderivatives we
r(t) = Zv(t)dt =Zsec2t dt, Z2 sec ttan t dt, Ztan2t dt
=tan t+C, 2 sec t+D, Z(sec2t−1) dt
=tan t+C, 2 sec t+D, tan t−t+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
r(t) = tan t, 2 sec t−1,tan t−t+ 2.
(The condition for −π/4≤t≤π/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
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2. Find the cumulative length function s(t) (starting from a= 1) of the parametric
curve hln t, √2t, 1
We have that the cumulative arc length function will be
s(t) = Zt
We are told that a= 1 and we now compute the derivative. We have
r0(t) = 1
s(t) = Zt
= ln t+1
(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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