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Unformatted text preview: MATH 138 Assignment 2 (2 pages) Winter 2007 Submit all problems marked * by 8:20 am. on Friday, January 19. 1. Find the partial fractions decomposition of each functiorr and hence ﬁnd / f (1‘) drr. , 101: + 2 :I: ~— 1 .1: >13 ,=' 1» ,,.: .'.»=____ Ufa) x3—512+;r—5 )) 'Hl) 2:3+:L' C) “1) r2—3:r+2 .772 1: — 2 1'4 + 31'3 + 212 + 1 *d I = * , ' = ' ; = ————' )ﬂ) ﬂ+% aha ﬂ_ﬁ aha ﬂ+h+2 2. Evaluate each antiderivative or deﬁnite integral.
173 ‘2 ' 1
*, 2 , . >1: ‘ 3 1 _ _ 1 . a) /—($ _ 2)2 dcr b) /0 111(9: + 4) d1 c) M (£1? (Let :1, 2 — 2sec6) '1 ' r 2 , ‘1“ l(] .
2 . "‘ — L] {‘T .21 ._
WU/xw—mm a/—i¥iL——e *a/ Eli—imamw=w~n
0  0 m3+2:1;2—m—2 e"'+8 3. *a) Suppose the cost of heating a eertian house is $C(t) per day, where f. = 0 corresponds to ~15
January 1, and t is measured in days. Find the units of the integral C(i)dt, and give
0
45
. ‘ t . 1
an inteipretation for 21—5 0 C(t)dt. *b] If the population of Mexico can be modeled by
Pa) = 67.38(1.026)", Where P is in millions of people and t is the time in years, with I, : (l on January 1, 1980,
ﬁnd the average population between then and New Yzar‘s 2000 c) Find all numbers b such that the average value of f(1') = 2 + 6:2? — 3.1;2 on [0, b] is 3. d) Poiseuille’s law oi" laminar ﬂow implies that, if we think ol‘ a, blood vessel as a cylindrical
tube of radius R and length it then the velocity “U of blood ﬂow depends on the distance 'r
from the central axis according to P : WU}? — r2) ‘ for 0 g 7' S R, e(r) where P is the pressure difference between the ends of the blood vessel and 7} is the viscos—
ity of the blood. Compare the average velocity with the maximum velocity over 0 S 7' S R. MATH 138  Winter 2007 Assignment #2 Page 2 of 2 4. *a) Explain how you know that, if f is continuous on [(1.15], then there is a c in ((1.1)) such that
f('3 '=) ftLLg the axerage value of f on [a b]. [H1NT2 Apply MVT tog(:L /f 1]).dt b) For the function f(:r)= (x— 2,ﬁnd f.lug on [2. 5] and then sketch y— — f(:1:) on the same
graph as a rectangle 011 [2, 5] Will] area equal to the area undei the grapl 1 lof f. 5. 3.) Sketch the graph of the straight line y = f[a:) between two points ($1,311) and (132,112), and
Show that the average value of f on [$1,162] is the average of y1 and y2, independent of the
values of $1 and 9:2. Give a graphical interpretation of this result. 1)) VVI'I‘IIOU'I‘ CALCULA'I‘ING ANY lN’l‘ICGRALS, explain how you know that. for ﬁre) = sinI
011 [Um/2]3 1/2 < fmw < l. *C) A metal bar has temperature f(t) = 20+98llG—0‘H °C', where t Z 0 is measured in minutes.
(i) Find favg over the first hour.
(ii) Find the average of the temperatures at the start and end of that hour. (iii) Explain the discrepancy between the averages in (i) and (ii) by sketching the graph of
y = f(t), and interpreting both geometrically. ~1r/4 ~1
6. a) Show that / V1 + 3:201:13 = / sectiﬁdﬁ (let :1: = tanﬁ). Then evaluate the integral by
0 using the reasoning in Example 8 on page :18? of your text.
*h) Find the area of the ellipse 4x2 + ”g? = 9. [HIN'i'z Use symmetry. and the substitution 3
a: — 5 sinﬁ plus suitable trigonometric identities] ...
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