Page 27 - Engineering Mathematics Workbook_Final

Calculus
)
f x = + )log (1 x on [0,1] the 113. For the function ( ) x = x , the mean
+
( ) (1 x
f
value of c  ( ) is value theorem does not held in the
0,1
interval….
4 1
(a) (b)   1  

−
e e (a)  1,0 (b) 0, 2  



−
−
4 e 1 e
(c) (d)  1 


e e (c) 0, 2   (d)  1,1
−


110. By applying mean value theorem, for the  
=
=
function ( ) x = f x − 2 4 on [2,4], the 114. If x r cos , y r sin then the
)
value of c  (2,4 is value of  2  +  2  =
 x 2  y 2
(a) 6 (b) 8
(a) 0 (b) 1
(c) 2.5 (d) 3.14 x y
(c) (d)
111. By applying Rauchy’s mean value r r
2
theorem for ( ) x = x , ( ) x = x
3
g
f
115. The Taylor series expansion of sin x
over [1,2] the value of c  ( ) is 
1,2
about x = is
6
14 13
(a) (b)
9 7 1 3      1    2


(a) +   x −   −   x −   + ......
2 2  6  4  6 
2 3
(c) (d)
3 2 x 3 x 5
(b) x − 3! + 5! − .......
=
3
112. A curve ‘C’ is defined as x a cos  ,

y = a sin  in 0, / 2 . What will be 1

3
(c)
the point P on curve C where the tangent 2
to the curve is parallel to the chord 3
−
−
joining points (a,0) & (0, a) x  / 6 (x  / ) 6
(d) − + .....
 a a  1! 3!
(a) (a, a) (b)   ,    
 2 2  116. The Taylor series expansion of
sin x
 a a   a a  f ( ) x = x  at x = is
−
(c)   ,     (d)   ,    
 2 2   2 2 2 2 

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