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Unformatted text preview: r——W Last Name: First Name: Student Number: University of Toronto
Faculty of Applied Science and Engineering MAT196F — Final Exam
Friday' December 8. 2006
Examiners: S. AbouWard, R. Kaplan Duration: 2 112 hours NO AIDS ALLOWED Use Onl I14
120
I17
I19 Page 1 of 9 pages 1. For each of the statements below, identify whether the statement is true or false. Justify
your answer with either an example or a counterexample. (a) When a continuous function is deﬁned over a ﬁnite interval, that function always
attains its absolute maximum and absolute minimum values on that interval. (2 marks)
T F [3:] (b) If f(x) does not exist at the point x = c, then f'(x) does not exist at x = c.
(2 marks) (c) Given a function f (x) on an open interval, with c in the domain of f , all
numbers 0 for which f ' (c) = 0 or f '(c) does not exist indicate either a
maximum, minimum or inﬂection point in the function f (x). (2 marks) Page 2 of 9 pages r—r—————f (d) Whenever f "(c) = 0 , with c in the domain of f , the function f (x) has an inﬂection point at x = c. (2 marks)
T F E3: (e) If f (x) = l on the interval [0, b] when a = —l , b =1, then there exists a number
x a < c < b, such that f'(c) = w is veriﬁed.
—a (2 marks) (t) The equation x3 + 9x2 + 33x —8 = 0 has exactly one real root.
(2 marks) (g) A function g , deﬁned for all positive real numbers, satisﬁed the following two 7
conditions 3(1) :1 and 8'02) = x3 for all .r > 0. Then 8(4) 2 4 7+6. (2 marks)
T F [:13 Page 3 of 9 pages 7—?— 3
I
2. Let f(x)= [ l6+t6dt.
l (a) Show that f has an inverse.
(6 marks) 0» Find (f")‘(0)
(8 marks) 3. Evaluate: 2x
(a) j
(5 marks) sin e‘ dx
82x 1 1
(b)! (5 marks) Page 4 of 9 pages (c) H'(3) if H(x)= [(22—3H'(:))d:
3 (5 marks) I—
(d) lime—J3
x—)l lnx (5 marks) Page 5 of 9 pages 4. The velocity of a weight suspended on a spring is given by V(t) = 38in! + 4cost , I Z 0.
At time I = 0, the weight is one unit below the equilibrium position. (a) Determine the position of the weight at any time :2 0 . /////// // / / / (7 marks) (b) What is the weight’s maximum displacement from the equilibrium position?
(Note: The motion of the weight is called the spring harmonic motion.) (10 marks) Page 6 of 9 pages r———— 5. (a) If the volume enclosed by the surface generated by revolving the region (2 between the curves y = x2 and y = 2—  x] about the/:2}: axis is El?— cubic units, ﬁnd the centroid of the region (2 using Pappus’s theorem.
(10 marks) Page 7 of 9 pages (b) A ball of radius Q is cut in two pieces by a horizontal plane (1 units above
the center of the ball. Determine the volume of the upper piece by using the Shell method.
(9 marks) Page 8 of 9 pages 6. Water ﬁlls a tank in the shape of a right~circular cone with top radius 3m and depth 4m.
How much work must be done (against gravity) to pump all the water out of the tank over the top edge of the tank?
(16 marks) Page 9 of 9 pages ...
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This test prep was uploaded on 04/17/2008 for the course MAT 197 taught by Professor Cohen during the Spring '08 term at University of Toronto.
 Spring '08
 COHEN
 Calculus

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