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FINALTERM EXAMINATION 2009

Calculus & Analytical Geometry-I

Gulshan Ali

(Hafizabad Campus) gulshanvu@yahoo/gmail.com www.vuzs.net

Time: 120 min Marks: 80

Question No: 1 ( Marks: 1 ) - Please choose one

If f is a twice differentiable function at a stationary point

x

and

f ''( x ) >

0 then f

has relative …………. At

0

0

x

0

► Minima

► Maxima

► None of these

Question No: 2 ( Marks: 1 ) - Please choose one

In the notation ∫

f ( xdx ) = F ( x ) + C C represents

► A polynomial

► A Constant

► A Variable

► None of these

Question No: 3 ( Marks: 1 ) - Please choose one

According to Power-Rule of differentiation, if

f ( x ) =

x n where

n

is a real number, then d dx

[ x

n

] =

►

nx − 1 ► nx n − 1 ► nx n +

1

Wis

---

1 ( n − 1) x n + Question No: 4 ( Marks: 1 ) - Please choose one

If

►

dy dx

=

► 2

► -2

► 0

► -3

Question No: 5 ( Marks: 1 ) - Please choose one

30

2 x − y = −

3 then

0

= ________

►

π

►

3

π

►

4

π

►

6

π 2

Question No: 6 ( Marks: 1 ) - Please choose one

If a function g is differentiable at a point x and a function f is differentiable at a point g(x), then the ________ is differentiable at point x .

► Composition (f o g)

► Quotient ( f / g )

► Product (f . g)

► Sum (f + g)

Question No: 7 ( Marks: 1 ) - Please choose one

Let a function

f

be defined on an interval, and let x

1

and x

2

denote points in that

---

( interval. If

f x 1 ) <

f ( x 2

) whenever

x 1 <

x 2

then which of the following statement is correct?

►

f

is an increasing function.

►

f

is a decreasing function.

►

f

is a constant function.

Question No: 8 ( Marks: 1 ) - Please choose one

If

f ′′ ( x ) <

0 on an open interval (a,b) then which of the following statement is correct?

►

f

is concave up on (a, b).

►

f

is concave down on (a, b)

►

f

is linear on (a, b).

Question No: 9 ( Marks: 1 ) - Please choose one

The sum

∑ n

f ( x *

k )

∆ x

k k

=

1

is known as:

► Riemann Sum

► General Sum

► Integral Sum

► Geometric Sum

Question No: 10 ( Marks: 1 ) - Please choose one

What does 'n' represent in Riemann Sum

∑ n

f ( x *

k )

∆ x

k k

=

1

?

► No. of Circles

► No. of Rectangles

► No. of Loops

► No. of Squares

Question No: 11 ( Marks: 1 ) - Please choose one

What is the area of the region in the following figure?

---

2 A = ∫

⌈ ⌊ ( x + 6 ) −

( x 2

)

⌉ ⌋ dx 0

►

►

∫ 2

( 6

) ( 2 )

x

►

A = ⌈ ⌊ x + −

x ⌉ ⌋ dx 2 ∫

( ) ( 2

)

0

►

A = ⌈ ⌊ x + 6 +

x ⌉ ⌋ dx A = ∫

x

⌈ ⌊ ( x + ) −

( x 2

)

⌉ ⌋ dx 0

Question No: 12 ( Marks: 1 ) - Please choose one

If

6

4

1

4 ∫

f ( x ) dx = 2 and

∫ 1

g ( x ) dx = 10 then which of the following is value of 4 ∫ [3 f ( x ) − g ( x )] dx 1

?

► 16

► 12

► -4

► -8

Question No: 13 ( Marks: 1 ) - Please choose one

∫ 1

2 x ( x 2 + 4) dx = ____________ 0

Wis

---

9 2

►

►

5 2

►

2 5

►

−

9 2

Question No: 14 ( Marks: 1 ) - Please choose one

Let f is a smooth function on [0, 3]. What will be the arc length L of the curve y = f(x) from

x = 0 to x = 3?

►

3 L = ∫

1 + [ f ( x )] 2 dy 0

►

b L = ∫

+ f x a

►

1 [ '( )] 2 3 L = ∫

1 + [ f '( x )] 2 dy 0

►

3 L = ∫

1 + [ f '( x )] 2 dx 0

Question No: 15 ( Marks: 1 ) - Please choose one

Let f be a smooth, nonnegative function on [1, 3]. What is the surface area S generated by revolving the portion of the curve y = f(x) between x = 1 and x = 3 about the x-axis?

---

2 S = ∫

2 1 + [ f ( x )] dx 0

►

►

3 S = ∫

2 π f ( x ) 1 + [ f '( x )] dx 0

►

2 S = ∫

2 1 + [ f '( x )] dx 0

►

S = ∫ 3

2 π f ( x ) 1 + [ f '( x )] 2 dx 1

Question No: 16 ( Marks: 1 ) - Please choose one

Let an object is displaced 2m by a force of 2N. What is the work done W?

► - 4

► 4

► 2

► 0

Question No: 17 ( Marks: 1 ) - Please choose one

Consider the improper integral

+∞

l ∫ a

= l

→∞

∫ α

if the limit exists then which of the following can be occured?

► Diverges

► Converges

► Test fail

Question No: 18 ( Marks: 1 ) - Please choose one

If f is continuous on (a, b] but does not have a limit from the right then the integral

defined by

f ( x ) dx lim f ( x )

dx

∫ b a f ( x ) dx = lim l → a

∫ b

l

f ( x )

dx

is called :

► Improper

---

► Proper

► Line

Question No: 19 ( Marks: 1 ) - Please choose one

− < For a sequence

then the sequence is known as :

► Increasing

► Decreasing

► Nondecreasing

► Nonincreasing

Question No: 20 ( Marks: 1 ) - Please choose one

For a sequence

{ a

n

}

if the difference between successive terms

a n +

1

a

n

0

a

a

+ >

then the sequence is known as:

► Increasing

► Decreasing

► Nondecreasing

► Nonincreasing

Question No: 21 ( Marks: 1 ) - Please choose one

Which of the following is true for the sequence

n 1 1 { a

n

}

if the ratio of successive terms

n

n ∞

=

?

► Nonincreasing

► Nondecreasing

► Increasing

► Decreasing

Question No: 22 ( Marks: 1 ) - Please choose one

If

{ }

n

0

f ( n )

=

a n

is the nth term of the sequence and f is differentiable and

f '( n ) ≤

0 then the sequence will be :

► Increasing

► Decreasing

► Nondecreasing

---

► Nonincreasing

Question No: 23 ( Marks: 1 ) - Please choose one

If Newton's Method is used to approximate the real solutions of the equation x 3 + x − 3 =

0 and the first guess x 1

=

1 , What is x

2

?

►

5 4

►

1 4

►

−

1 2

►

3 4

►

3 2

Question No: 24 ( Marks: 1 ) - Please choose one

Suppose that we apply Newton’s Method to approximate the real solutions of the

equation

x =

2 , then which of the following is value of

2

x 3 − 2 x 2 − 1 =

0 . If we start at

1

x

?

► 6

► 2.25

► 0

► 2

Question No: 25 ( Marks: 1 ) - Please choose one

If the sequence of partial sum of a series converges then what will the series show itself ?

► Diverges

► Converges

► Gives no information

Question No: 26 ( Marks: 1 ) - Please choose one

The series

u ∑

u k

ρ =

lim →∞

k

+ 1 k

u

k ρ be a series with positive terms and suppose that

>

1 , then which of the following is true?

if

---

► Converges

► Diverges

► May converges or diverges

► Gives no information

Question No: 27 ( Marks: 1 ) - Please choose one

The series

=

ρ =

1 , then which of the following is true?

if

► Converges

► Diverges

► May converges or diverges

► Gives no information

Question No: 28 ( Marks: 1 ) - Please choose one

The series

∑

u k

be a series with positive terms and suppose that

ρ lim u k

→∞

k

+ 1 u k

∑

u k

be a series with positive terms and suppose that

1 ρ

= lim k →∞ k

u k =

lim( k →∞

u k

)k if

ρ =

1 , then which of the following is true?

► Converges

► Diverges

► May converges or diverges

► Gives no information

Question No: 29 ( Marks: 1 ) - Please choose one

For an alternating series to be convergent which of the following condition must be satisfied?

►

=

►

lim k →∞

a

k

1

a > a > a > a >

►

1 2 3

...... k

....

a 1 ≤ a 2 ≤ a 3

...... ≤ a k

≤

....

► Gives no information

W]

---

Question No: 30 ( Marks: 1 ) - Please choose one

For an alternating series to be convergent which of the following condition must be satisfied?

►

a ≥ a ≥ a ...... ≥ a k

≥

....

►

1 2 3

lim k →∞

a

k

=

0

►

a ≤ a ≤ a ...... ≤ a k

≤

....

►

1 2 3

lim k →∞

a

k

=

1

Question No: 31 ( Marks: 1 ) - Please choose one

What is the base of natural logarithm?

► 2.71

► 10

► 5

► Any real number

Question No: 32 ( Marks: 1 ) - Please choose one

A function

F

is called an antiderivative of a function

f

on a given interval if _______ =

f ( x ) , for all

x

in that interval. F '( x ) ►

F ( x ) ► f '( x ) ► f ′′ ( x ) ►

Question No: 33 ( Marks: 1 ) - Please choose one

log b

ac =

_______

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