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Unformatted text preview: Last Name: First Name: Student Number: University of Toronto
Faculty of Applied Science and Engineering MAT196F — Final Exam
Thursday, December 15, 2005
Examiners: S. AbouWard, J. Sylvestre, Y. Bakhtin Duration: 2 1/2 hours NO AIDS ALLOWED 1,. (a) Evaluate the limit . . (1+x)”—1 <0 m ——,;—
(4) .. . t9 (H) Ill{101 tan92t
(4) , (b) Use the 8—5 definition of the limit to show that lirg Jx  = 3.
(10) ' 2. Consider the region bounded above by y = x2 —2 and below y = x. (a) Find the area between the two curves. (5) (b) Find the volume of the solid of revolution resulting of rotating the region in (a) above
about the line x = 2 . ‘ (9) (C) Is it possible to use pappus theorem to find the centroid of the region in (a) above?
Justify your answer. (2) 3. (a) Two electrons repel each other with a force that varies inversely as the square of the
distance between them. If one electron is fixed at the point (2,4), ﬁnd the work done from (—2,4) to (1,4).
(7) (b) Evaluate the following integrals: — /4
” x—l (i) ”7‘4 cos2 xdx
(5)
1 x
" ! T?” (5) 4. A man is in a boat 2 kilometers from the nearest point on the coast. He is to go to a point Q,
3 kilometers down the coast and 1 kilometer inland. If he can row at 2 kilometers per hour and
walk at 4 kilometers per hour. Toward what point on the coast should he row in order to reach point Q in the least time? (Hint: the fourth degree equation has only one real root.)
(10) llx x
1 1 . .
5. (a) Let f(x)= dt+ dt If x>0 then f(x) IS
J t2 +1 5' t2+1 (8) (b) Show that f (x) = I (1+ sin2 t)dt has an inverse and find (f‘1)'(0).
”/2 (10) 6. Make use of domain, range, symmetry, asymptotes, intercepts, relative extrema, or points of
inflection to obtain an accurate graph of the function: f(x) = 2x“3 5x“3
(a) f'(x)=
(6)
f ”(x) =
critical numbers: x= , x= , x: Work Sgace (b) Fill out the following table:
(10) (c) Sketch the graph.
(4) ...
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 Spring '08
 COHEN
 Calculus

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