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Unformatted text preview: u 1. u a. Math 101 Exam Apr. 1992 a) Integrate / sin2 :5 cos3 :1: dx. 2 .
b) Evaluate/ xv 2:5 — $2 (Ix.
1 m(w+ 1) (m — 4W + 4) ‘1‘“ c) Integrate d) Find a reduction formula for I = /:L'(ln.1:)" (1:: (that is, a formula for In in terms of 1",} . Use the Simpson’s Rule approximation 52 (based on two subintervals) 2 d2:
— , to show that for the integral
1 :1: and estimate the error K(b — a) <
_ 18011“ fabf(m)dz—Sn in this approximation, where K is the maximum of |f(4)(:r)| on the
interval a S a: S b. Find the area of the plane region bounded by the line a: + y = 1 and
the curve x/E + J37 = 1. 4. Find the area of the surface obtained by rotating the inﬁnitely long curve y = e", (0 S x < 00) about the x-axis. You might need the
formula /\/a2+:l;2 (lx=%x a2+x2+éln|$+Va2+xZI+C 5. a) Sketch the polar graph r = sin(29). b) Find the area of the polar region that lies inside of the circle 1' =
1/\/§ and outside r = sin(29). 6. Do only one of the problems i) and ii) below. Clearly indicate which one you select. i) Let C be the curve
a: = e‘ cost
y = e‘ sint where t 2 0. Show that the lentgh of C from A = (1,0) to any point P
on C is equal to W times the distance from P to the nearest point Q
on the circle through A centered at the origin: arc(AP) = ﬂ m4 " ‘4‘ .0) ii) Find the centroid of the plane region that lies inside the cricle
x2 + y2 = 1, outside the circle 11:2 + y2 = x, and above the z-axis. y 7. 21 Find the Maclaurin polynomial P3 .1‘ of de ree three, and the La-
grange remainder R3(.r), for the function f(x) 2 V1 + 3:. b) Estimate the error if the polynomial P3(z) from part a) is used to
approximate 1.2. tan‘1 t
t sins: — F(a:) 8. Calulate lim dt. :—>0 1:3 , where F(z) = /
0 9. Let f be a function that is strictly increasing and differentiable on the
interval 0 S :0 S 1, and that satisﬁes f(0) = 0 and f(1) = 1. Given that 1 1
/ dm => A, find / f"1(a:) dx, where f'1 is the inverse function
of f. ...
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This note was uploaded on 01/30/2011 for the course MATH 101 taught by Professor Broughton during the Spring '08 term at UBC.
- Spring '08