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Unformatted text preview: Each of these multiplechoice questions has exactly ONE correct
answer.
Complete this Assessment On—line via Blackboard 1. Consider the following two linear equations in x and y : 4x+3y = 2
3x+2y—5 = 0. Which of the following statements is correct? A. There is no solution.
B. There is a unique solution.
C. There are inﬁnitely many solutions. D. There are exactly two solutions. 2. Consider the following two equations in a: and y : y = 233+1
y = $2—6JJl—3 Which of the following statements is correct? A. There is no solution.
B. There is a unique solution.
C. There are inﬁnitely many solutions. D. There are exactly two solutions. 3. Consider the following equation in x : e” 2 use + b where a and b are parameters. A suﬂicient condition for a solution to exist is: A.b>1 B
C.a>0
D .b<1 4. Consider the following function What is its DOMAIN? A.x>0
B. 33<0
C. x2>0
D. 37%1 5. Consider the following function What is its RANGE? A. (1,00)
B. (0,00)
C. [0,00)
D. [0,1)
33: . .
6. Let y = f(a:) = 1 +x’ 3: 2 0. What 1s the inverse of ﬂay)?
A _ 9
9(9) — a, y 6 [03)
B — y >
9(9) — E: y _ 0
_ y
1+y
D g y = —, y 2 0
< > 3y LL‘ ,x > —1.Fi ure 2 lots
1+x g p 7. The Figure 1 is a graph of the function f (x) =
f (x) and another function g(x), the “dashed—line”. Figure 1 Figure 2 The function g(x) is: 2 A. a: ,x>—1
1+x
2x B. >—12
1+2x’x / C. x ,x>—2
2+x
a: D. —12
1+2$,x> / 8. Let f (x) be a continuous function deﬁned on an interval D = [(1,1)], such that
f (a) = f (b) = 0, say, and f’ (a) > 0 and f’ (b) > 0. Which of the following statements must necessarily be true?
A. f(x) is a cubic
B. The equation f (as) = c has only two roots in D, namely :0 = a and a: = b. C. There is a value 360 E (a, b) , such that ﬂew) = c. D. ﬂap) 2 c for all :0 E D. 9. Consider the following function:
f(a:) = 3x4 — 2x2 + 6x + 4.
What is its derivative?
A. f’(w) = 9x3 — 45L“ + 6
B. fl(fL‘) = 33:3 — 2x —— 6 C. f’(x) = 4x3 — 4x —— 6
D. f’(x) = 12x3 — 4x + 6 10. Let f (as) = ln(a:) + 332, x > 0. What is the equation of the linear approximation
to f(x) at x = 1? A. yzx—l
B. y=3$—2
C. y=3x—3 D. y—1=3a:—1 11. Let f (as) = ln(a:) + :52, a: > 0. What is the second derivative of f (x)? A. f”(x) = i + 2x
B. f”(x) _ g + 2
C f”(m) — —% D f”(a:) = 2 — é LL‘ , x > 0. What is the second derivative of f (x)? 12. Let f(x) = 1 A. f”(rr) = “3—2).
B. f”(m) = ﬂ
C. f”(x) = —(1—2—x)2
D. f”(x) = % 13. The stationary points of f (x) = x3 — 6x2 + 12% + 18, can be characterised as: A. IL“ 2 2 is a point of inﬂection
B. x = 2 is an upward turning point and x = 0 is a downward turning point
C. x = 2 is an downward turning point and x = 0 is a upward turning point D. x = 2 can not be characterised because f” (2) = 0. 14. Consider the function
f(:L') = 333 — 6352 +1235 + 18.
Which of the following statements is correct? A. f x) is convex for x g 2 and concave for x Z 2.
B. f :r) is convex for :r g 2 and concave for :r > 2.
C. f x) is (globally) strictly increasing.
D. f x) is globally convex. 15. What is the global maximum and what is the global minimum of the following function:
f(a:) 2 3326—3”, 3: > 0 7 A. Minimum value of 0 at :r 2 00 and maximum value of 46—2 at x = 2.
B. Minimum value of 0 at x = 0 and maximum value of 46—2 at :L' = 2.
C. No maximum, but a minimum value of 0 at a: = 0. D. No minimum, but a maximum value of 46—2 at a: = 2. 5 16. Let y = y(x) be a differentiable function of as, satisfying x2 — 2953; + 3/2 = 4.
What is y’(x) at the point (x, y) = (1, 2)? A. 0
B. 4
C. 2
D. 1
17. Given the difference equation xt+1 = gent — 6, x0 = 0, the particular solution is: A. 3t — 3, t = 0, 1,2,
B. 37*2 — 9, t = 0, 1, 2,
C. 3t—3, t=0,1,2,... D. 32—75 — 9, t = 0, 1, 2, 18. Suppose the proportion of people in the population who will have seen “2012”
( ) by the end of the ﬁrst week, following its release, is
m > 0. Thereafter, the proportion of people in the population who will have seen
it by the end of week t + 1, following its release, is given by nt+1 = omf, t = 1, 2, ...,
where 0 < a < 1 and 0 < 6 < 1 are parameters. Taking natural logarithms we
obtain a linear difference equation in xt = ln nt as xt+1 = lna + Bait, t = 1, 2, .
Following a long run of the ﬁlm at the cinemas, approximately what proportion of
the population will have seen it? A.a lna 31—5 19. Consider the following (2 X 1) vectors: 1 —2 1 —1 0
1(2),b:(1>,.=(0>,d:(_2).:(1>
Which of the following statements is correct? A. a 85 b are orthogonal and c 85 e are orthogonal B. a 85 b are orthogonal, c 85 e are orthogonal and a 85 d are orthogonal
C. a 85 d are orthogonal and c 85 e are orthogonal D. Just 0 85 e are orthogonal. 20. For the (2 X 1) vectors aTb
calculate
llall Nb“
1
A. —
x/E
4
B. —
V65
1
C. —
V65
4
D. —
65 THE END ...
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 Fall '15

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