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Unformatted text preview: Math 221 Final exam, December 17, 2007
Please note: There are seven problems (each worth 25 or 30 points). YOUR NAME: Circle the name of your TA and discussion: Eric Andrejko, DIS 326, TR 11:00—11:50, and DIS 328, TR 12:0512:55, Erkao Bao, DIS 321, TR 7:45—8:35, and DIS 322, TR 8:50—9:40, Sam Eckels, DIS 323, TR 8:50—9:40, and DIS 330, TR 1:20—2:10 p.m.,
Hongnian Huang, DIS 324, TR 9:55—10:45, and DIS 334, TR 3:304:20 p.m.,
Seyﬁ Turkelli, DIS 325, TR 9:55—10:45, and DIS 327, TR 11:00—11:50,
Dongning Wang, DIS 331, TRI1:20—2:10 p.m., and DIS 332, TR 2:25—3:15 p.m.,
Xu Yang, DIS 329, TR 12:05—12:55 p.m., and DIS 333, TR 2:25—3:15 pm. Scores:
N0. 1: (max 30 pts.)
No. 2: ' (max 30 pts.)
N0. 3: (max 30 pts.)
N0. 4: (max 30 pts.)
, No. 5: (max 25 pts.)
N0. 6: (max 25 pts.)
N0. 7: ‘ (max 30 pts.) Exam Score: 1. Compute the derivative of the following functions. (a) (5ptS)
f(:c) = 37(5262 ~ 2678). (b) (8 Pm)
g(a:) = arctan(ezx).
(c) (8 pts.) For a: > 0:
Mm) = :51”. since
[C(33) = / etzdt.
o (d) (8 ms) 3 2. (6 pts.) A function f is differentiable at the point a if Write down this
deﬁnition. ' 
(ii) (6 pts.) Use the deﬁnition of derivative (and possibly some additional manip—
ulation) to determine
ln m2 lim .
:c—d a: — 1
(iii) (9 pts.) Show that ln(:c) < a: — 1 for cc > 1. (iv) (9pts.) Show that ln($) > m —— 1 — (3:292— for x > 1. .3. Let
0 ifar<§
2 iflgz<3
ﬂit): . 2
110—5 1f3$$£27r 27r~5+sina7 if27r<m. (i) (15 pts.) Sketch the graph of f and compute the integral f0?” f (ii) (7 pts.) Find the area of the region bounded by the graph of f, the curve given
by y = —7 — m2 and the vertical lines ac = 1/2 and a: = 37r. ' (iii) (8 pts.) Let A(m) = I: f (t)dt. Determine where A is differentiable and at
those points write down the derivative A’ Also determine all points at which A
is not differentiable and give an explanation why A is not differentiable there. 5., 4. (a) (8 pts.) There is a function f deﬁned on (—00, 00) which has the property
that f (0) = 2 and f’ = 6972 for all :12. Express this function using an integral.
(b) (11 pts.) Find explicitly the expression 9(33) for which
9:4 sin(:£5) + 4:1:
3
(c) (11 pts.) Find emplz’cz'tly the expression h(m) for which
h’(:c) : 331nm and h(1) = 2. Hint for (c Use integration byvparts to compute the relevant integral. 9,03) = and 9(0) 2 1. 5. (10 pts.) Deﬁne the function f : R ——> R by f(a:) = :I: + 2x3 and show that f
has an inverse (do not attempt to compute it). Compute the derivative of the inverse
at the point 3/ = 3 = (ii) (8 pts.) Write an integral equal to the length of the part of the curve y = 6
between a: = 2 and a: = 4. (Do not evaluate this integral). (iii) (7 pts.) Find numbers a and b such that the length of the part of the curve
élnt between t = a and t 2 b is exactly. equal to the length of the curve in part 331: y 2
(ii). 6. (25 pts.) Determine where
ﬁx) 2 2:173 —15x2 + 36x — 7 (i) is increasing, (ii) is decreasing, (iii) has a local maximum, (iv) has a local minimum,
(V) is concave downward, (Vi) is concave upward, (Vii) has a point of inﬂection. Sketch
the graph of f. 7. A function y(x) satisﬁes y(0) :11 and is implicitly given by
' ey(x)’9°” — sin x + 31(3)) = 2 + 3:5. (i) (12pts.) Find y’(0) and write the equation for the tangent line through (0, 1).
(ii) (12 pts.) Find y”(0).
(iii) (6pts.) Sketch the graph of for x near 0. ...
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This note was uploaded on 08/08/2008 for the course MATH 221 taught by Professor Denissou during the Summer '07 term at Wisconsin.
 Summer '07
 Denissou
 Calculus, Geometry

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