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Unformatted text preview: Math 101 Exam April 1991 1. Evaluate the following integrals. b) /___ 41:2  4d
:c" + 232d
1r/2
c) / sin3 :1: cos3 2: dz
0 d) [stan'l :1: d2: . Find the area of the inﬁnitely long region that lies between the yaxis and the graph of y: ln 1, and below the xaxis. 3. Let R be the ﬁnite region bounded by the hyperbola :52 — 3/2 = 1 and the line 3 — 2. Find the volume of the solid generated by rotating R
about the yaxis. 4. Find the centroid of the plane region OSySI—cosz
05252" 5. Find the area of the (polar) region that lies inside the cardioid r = 1 + c039 and to the right of the line 1' = gsec 0. 6. Find the length of the parametric curve sin t cos t
cos2 t I 1r
y (05‘5‘2‘) 7. a) Show that the Maclaurin polynomial of degree 6 for f(:1:) = sin a: is Z 3
Pa(Z)=$§!+y b) Using the Lagrange remainder for R, = f(z)  Po(x), estimate
the maximum value of 1: for which the magnitude of the error in the approximation
:3 1:5
sin :1:~ ~ 1' —  +5 — 31
is less than 0.01. 8. 8) Using the identity
cos3 0 = %(cos 30 + 3cos 9),
ﬁnd the Maclaurin series for f (z) = cos3 1:. b))Using the Maclaurin series for ﬂex) = cos3 3, ﬁnd f“)(0) and
f“ (0) 1 i 9. Let y = E. Show that the improper integral from/1 + (31’)2 dz converges if p > 1, and diverges if p 5 1. ...
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 Spring '08
 Broughton
 Math

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