MTH101- Calculus and Analytical Geometry

FINALTERM EXAMINATION Spring 2010 MTH101- Calculus And Analytical Geometry Time: 90 min Marks: 60

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

For any number e > 0 if we can find an open interval (x0 , x 1 ) on the x-axis containing the point "a" such that L-e < f( x ) < L + e for each x in (x0 , x 1 ) except the possible x = a then we say limx a f ( x ) = ------

► L ► L -e ► L + e ► L +1

Question No: 2 ( Marks: 1 ) - Please choose one p is a ...... number

► Integer ► Rational ► Irrational ► Natural

Question No: 3 ( Marks: 1 ) - Please choose one How the series 1- 3 + 5 - 7 + 9 - 11 can be expressed in sigma notation?

k=5 ► (- 1)k (2k + 1) k=0 k=5 ► (- 1)k (2k + 1) k=1 k=5 ► (2k + 1) k=1 k=5 ► (2k + 1) k=1

Question No: 4 ( Marks: 1 ) - Please choose one According to Power-Rule of differentiation, if f( x ) = xn where n is a real number, then d [xn ] = dx ► xn-1 ► n xn-1 ► n xn+1 ► (n- 1) xn+1

Question No: 5 ( Marks: 1 ) - Please choose one d If there is some function F such that [F ( x )]= f ( x ) then any function of the dx form F( x ) + C is ------of f (x)

► Derivative ► Antiderivative ► Slope ► Maximum value

x2 Let y = . Find average rate of change of y with respect to x over the interval [3, 4] 2 25 ► 2 7 ► 2 25 ► 14 7 ► 14

For a function f(x) to be continuous on interval (a,b) the function must be continuous at ------► Only at pints a and b ► mid point of a and b ► all point in (a,b)

Question No: 8 ( Marks: 1 ) - Please choose one 1 Consider the indefinite integral dx x2 + 2 Let t= x2 + 2

Is the following substitution correct? 1 1 dx= dt 蝌x2 + 2 t ► Yes ► No

Question No: 9 ( Marks: 1 ) - Please choose one In the following figure, the area bounded on the sides by the lines are :

► x = 0 ► x = 2 ► x=0 and x = 2 ► x = 6 Question No: 10 ( Marks: 1 ) - Please choose one What is the sum of following series?

12+ 2 2 + 3 2 + 4 2 + ______+ n 2 n( n+ 1)(2 n + 1) ► 6 n(2 n )(2 n + 1) ► 6 (n+ 1)( n + 2) ► 2 (n+ 1)(2 n + 1) ► 6

2 2 3 Let y= x2 - 2 x ; 0＃ x 1 then which of the following is the length of the curve? 3 2 1 轾骣2 2 3 ► L=1 +犏琪 x2 - 2 x dx 琪 3 0 臌犏桫 2 1 轾d 骣2 2 3 ► L=犏琪 x2 - 2 x dx dx 琪 3 0 臌犏桫 2 轾骣 3 d 2 2 2 ► L=1 +犏琪 x - 2 x dx 臌犏dx 桫 3 2 1 轾d 骣2 2 3 ► L=1 +犏琪 x2 - 2 x dx dx 琪 3 0 臌犏桫

Question No: 12 ( Marks: 1 ) - Please choose one If Rolle's Theorem is valid for some function f , then on the curve of f there exist atleast one point where the slope of tangent line is ______► Zero ► 1 ► -1 ► Undefined

If f is continuous on (a, b] but does not have a limit from the right then the integral b b defined by f( x ) dx= lim f ( x ) dx is called : 蝌 l a a l

► Improper ► Proper ► Line

Question No: 14 ( Marks: 1 ) - Please choose one 1 u If e du , then which of the following is true? 0 ► e ► 1- e ► e +1 ► e -1

d x The result [f ( t ) dt ]= f ( x ) is derived from the: dx a

► Riemann Integral ► Definite Integral ► First Fundamental Theorem of Calculus ► Second Fundamental Theorem of Calculus

Question No: 16 ( Marks: 1 ) - Please choose one If f and g are continues function on an interval [a, b] and f( x )常 g ( x ) for a x b , then area is bounded by the lines parallel to:

► X -axis ► Y-axis ► Both X -axis and Y-axis

Question No: 17 ( Marks: 1 ) - Please choose one What is the length of each sub-interval, if the interval [a, b] is divided into n sub-intervals of equal length? b+ a ► n -1 b- a ► n b+ a ► n b- a ► n -1

|uk+1 | Let uk be a series with nonzero terms and suppose that r =limk > 1 |uk | then which of the following is true? ► The series uk converges

► The series uk diverges ► No conclusion can be drawn.

Which operation can not be applied on the functions?

► Subtraction ► Cross product ► Addition ► Composition

For a sequence {an } if the difference between successive terms an+1 - a n 0then the sequence is known as:

► Nondecreasing ► Nonincreasing ► Decreasing ► Increasing

If the geometric series a+ ar + ar2 + ar 3 +... + ark- 1 + ... where ( a 0) , r 1 then which of the following is true for the given series? ► Converges ► Diverges ► Gives no information

|uk+1 | Let uk be a series with nonzero terms and suppose that r =limk = 1 , |uk | then which of the following is true? ► The series uk converges

► The series uk diverges ► No conclusion can be drawn.

If f is continuous on [a, b) but does not have a limits from the left then the integral b l defined by f( x ) dx= lim f ( x ) dx is called : 蝌 l b a a

► Line ► Proper ► Improper

Which of the following is the maclaurin series for Sin( x ) ?

x3 xk ► 1+x + + ... + + ... 3!k ! x3 x 5 x 7 x 2n+ 1 ► 1+x - + - ... + ( - 1)n + ... 3! 5! 7! (2n + 1)! x3 x 5 x 7 x 2n+ 1 ► x - + -... + ( - 1)n + ... 3! 5! 7! (2n + 1)! x3 xk ► 1-x - - ... - - ... 3!k !

Question No: 25 ( Marks: 1 ) - Please choose one Which of the following is area of the surface generated by revolving the curve y= 4 x ; 1＃ x 4 about the x-axis?

4 2 ► 2p 4x 1+ 轾 4 x dx ( ) 臌( ) 1 4 轾 ► 2p ( 4x) 1+ 犏( 4 x) dx 1 臌 2 4 轾 ► 2p + 1 + 犏( 4 x) dx 1 臌 2 4 轾 ► 2p ( 4x) 1+ 犏( 4 x) dx 1 臌

Question No: 26 ( Marks: 1 ) - Please choose one Which of the following is surface area S generated by revolving 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=2p f ( x ) 1 + [ f '( x )] dx 0 2 ► S=2 1 + [ f '( x )] dx 0 3 ► S=2p f ( x ) 1 + [ f '( x )]2 dx 1

Question No: 27 ( Marks: 1 ) - Please choose one ...... is the special case of Tylor's theorem. ► Roll's Theorem ► Picard's Method ► Integration ► Maclaurin's Theorem

Question No: 28 ( Marks: 1 ) - Please choose one The graph x= y2 is symmetric about ------axis

► X-axis ► Y-axis ► Origin

Question No: 29 ( Marks: 1 ) - Please choose one 0 5 5 If f( x ) dx = 2 and f( x ) dx = 1then which of the following is value of f( x ) dx ? 1 0 1 ► -3 ► -1 ► 1 ► 3

Question No: 30 ( Marks: 1 ) - Please choose one Which of the following is the work done W for an object moves 2m by a force of 2N? ► 2 N.m ► 0 N.m ► - 4 N.m ► 4 N.m

Question No: 31 ( Marks: 2 ) 4 Find {3f( x) - g ( x )} dx if 1 4 4 蝌f( x) dx=2 and g( x) dx = 10 1 1

Question No: 32 ( Marks: 2 ) x2 - 9 Use L’Hospital rule to evaluate the limit: lim x 3 x - 3

Question No: 33 ( Marks: 2 )

Find a definite integral indicating the arc length of the curve y= 2 x2 from x=0 to x = 1. But do not evaluate the integral.

Question No: 34 ( Marks: 3 ) 3 Evaluate the integral (1- x ) dx 2

Question No: 35 ( Marks: 3 ) 2k -1 Show that the series (- 1)k -1 converges absolutely. k =2 (k - 1)!

Question No: 36 ( Marks: 3 ) Let S be a solid bounded by two parallel planes perpendicular to the y- axis at y = 0 and y= h . If for each y in[0,h] , the cross-section perpendicular to the y- axis is a a square. If s=( h - y) denote the length of a side of this square then find A( y ) and h formulate the volume of the solid but do not evaluate the integral.

Question No: 37 ( Marks: 5 ) .

Determine whether the sequence {an }converges or diverges; if it converges then find its limit; 3n+ ( - 1) n where a = n 3n+1+ ( - 1) n + 1

Question No: 38 ( Marks: 5 ) 2 3 Find the arc length of the curve y=( x -1)2 from x=0 to x = 1. 3

Question No: 39 ( Marks: 5 )

(1+ x )3 Evaluate the integral: dx by using a proper substitution x