ma1505 11s2 additional exercises-1
TRANSCRIPT
MA1505 Mid-Term Test
1. Let f (x) = (2− cos x)xπ . Find f ′ (π).
(A) 1π ln 3
(B) ln 3
(C) 1π ln 27
(D) 3π ln 2
(E) None of the above
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MA1505 Mid-Term Test
2. A curve (called a deltoid) has parametric equations
x = 2 cos t + cos 2t
y = 2 sin t− sin 2t,
where 0 ≤ t ≤ 2π. Let L denote the tangent line to this curve at
the point where t = π4 . Find the x-coordinate of the point of
intersection of L with the line y = −1.
(A) 2 +√
2
(B) 2√
2 + 2
(C) 2−√2
(D) 2√
2− 2
(E) None of the above
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MA1505 Mid-Term Test
3. In a certain problem, two quantities x and y are related by the
equation
y = 20x2 − x3 + 1505.
It is known that x is increasing at a rate of 3 units per second.
Find the rate of change of y when x is equal to 10 units.
(A) Increasing at 300 units per second
(B) Increasing at 330 units per second
(C) Increasing at 200 units per second
(D) Increasing at 250 units per second
(E) None of the above
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MA1505 Mid-Term Test
4. Let a be a positive constant. Let M and m denote the absolute
maximum value and absolute minimum value respectively of
the function
f (x) = x2 +2a3
x,
in the domain[
a2,
4a3
]. Find M
m .
(A) 5954
(B) 153118
(C) 2116
(D) 1712
(E) None of the above
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MA1505 Mid-Term Test
5. Evaluate ∫ π3
0
| cos3 2x| dx
(A) 2π9 − 5
√3
24
(B) 23 −
√3
16 π
(C) 23 − 11
√3
56
(D) 23 − 3
√3
16
(E) None of the above
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MA1505 Mid-Term Test
6. Find the area of the finite region bounded by the curves
y2 + 4x = 0 and 2x + y + 4 = 0.
(A) 223
(B) 9
(C) 7
(D) 253
(E) None of the above
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MA1505 Mid-Term Test
7. Find ∫1√
1 + exdx.
(A) 12 ln
√1+ex+1√1+ex−1 + C
(B) 12 ln
√1+ex−1√1+ex+1 + C
(C) ln√
1+ex+1√1+ex−1 + C
(D) ln√
1+ex−1√1+ex+1 + C
(E) None of the above
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MA1505 Mid-Term Test
8. A finite region R is bounded by the curves y = 2 − x2 and
y = x2. Find the volume of the solid formed by revolving R one
complete round about the x-axis.
(A) 16π3
(B) 64π15
(C) 15π8
(D) 3π2
(E) None of the above
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MA1505 Mid-Term Test
9. Let f (x) = ln(1 + x + x2 + x3
)and
∞∑n=0
cnxn
be the Taylor series of f at x = 0. Then the value of c2009+c2010
is
(A) 12009 + 1
2010
(B) 12009 − 1
2010
(C) − 12009 + 1
2010
(D) − 12009 − 1
2010
(E) None of the above
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MA1505 Mid-Term Test
10. Find the radius of convergence of the power series
∞∑n=1
(5n + (−1)n
n3
)(x− 2)n .
(A) 6
(B) 13
(C) 12
(D) 5
(E) None of the above
END OF PAPER
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