MA1505 11S2 Additional Exercises-1

18
MA1505 Mid-Term Test 1. Let f (x) = (2 - cos x) x π . Find f 0 (π). (A) 1 π ln 3 (B) ln 3 (C) 1 π ln 27 (D) 3 π ln 2 (E) None of the above 3

Transcript of MA1505 11S2 Additional Exercises-1

Page 1: MA1505 11S2 Additional Exercises-1

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

3

AdminNUS
Text Box
Additional Exercises 1
matyapwy
Text Box
1
Page 2: MA1505 11S2 Additional Exercises-1

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

4

AdminNUS
Text Box
Additional Exercises 1
matyapwy
Text Box
2
Page 3: MA1505 11S2 Additional Exercises-1

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

5

AdminNUS
Text Box
Additional Exercises 1
matyapwy
Text Box
3
Page 4: MA1505 11S2 Additional Exercises-1

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

6

AdminNUS
Text Box
Additional Exercises 1
matyapwy
Text Box
4
Page 5: MA1505 11S2 Additional Exercises-1

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

7

AdminNUS
Text Box
Additional Exercises 1
matyapwy
Text Box
5
Page 6: MA1505 11S2 Additional Exercises-1

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

8

AdminNUS
Text Box
Additional Exercises 1
matyapwy
Text Box
6
Page 7: MA1505 11S2 Additional Exercises-1

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

9

AdminNUS
Text Box
Additional Exercises 1
matyapwy
Text Box
7
Page 8: MA1505 11S2 Additional Exercises-1

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

10

AdminNUS
Text Box
Additional Exercises 1
matyapwy
Text Box
8
Page 9: MA1505 11S2 Additional Exercises-1

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

11

AdminNUS
Text Box
Additional Exercises 1
matyapwy
Text Box
9
Page 10: MA1505 11S2 Additional Exercises-1

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

12

AdminNUS
Text Box
Additional Exercises 1
AdminNUS
Text Box
matyapwy
Text Box
10
Page 11: MA1505 11S2 Additional Exercises-1
AdminNUS
Text Box
Additional Exercises 1 Solutions
matyapwy
Text Box
11
Page 12: MA1505 11S2 Additional Exercises-1
matyapwy
Text Box
12
Page 13: MA1505 11S2 Additional Exercises-1
matyapwy
Text Box
13
Page 14: MA1505 11S2 Additional Exercises-1
matyapwy
Text Box
14
Page 15: MA1505 11S2 Additional Exercises-1
matyapwy
Text Box
15
Page 16: MA1505 11S2 Additional Exercises-1
matyapwy
Text Box
16
Page 17: MA1505 11S2 Additional Exercises-1
matyapwy
Text Box
17
Page 18: MA1505 11S2 Additional Exercises-1
matyapwy
Text Box
18