Xii Continuity And Differentiablity Assignment

ASSIGNMENT CLASS XII CONTINUITY AND DIFFERENTIABILITY Important Formulas 1. A function f ( x ) is continuous at x  a iff lim f ( x )  lim f ( x )  f (a) . x a

xa

2. A function f ( x ) is differentiable at x  a iff lim x a

lim h 0

f ( x)  f ( a ) exists finitely ie. xa

f ( a  h)  f ( a ) f (a  h)  f (a )  lim . h0 h h

1   x sin , 1. Show that the function f ( x )   x  0

x0

is continuous at x  0 .

x 0

 sin x  cos x,  2. Show that the function f ( x)   x  2

x0

is continuous at x  0 .

x0

when 0  x 1 5 x  4 3. Show that the function f ( x)   3 is continuous at x 1 . 4 x  3 x when 1  x  2 

4. Show that the function f ( x)  2 x  | x | is continuous at x  0 .  x | x | , x  2

5. Show that the function f ( x)  

| x  4 |

6. If f is defined as f ( x )   x  4  

,

0

x  0 is discontinuous at x  0 .

x 4

x0

. Show that f is everywhere continous except at x  4 .

x 4

 | sin x | , 7. Show that the function f ( x)   x  1 

x 0

is discontinuous at x  0 .

x0

8. Find the value of k so that the function f is continous at the indicated point: 1  cos 2 x (a) f ( x)   2 x 2 ,  k

x0 x0

at x  0 .

 x 2  25 (b) f ( x)   x  5 ,  k 

x5

at x  5 .

x 5

if x  2  x 1 9. Show that the function f ( x )   is not differentiable at x  2 . 2 x  3 if x  2 10. Discuss the continuity and differentiability of f ( x)  | x 1|  | x  2 | . ANSWERS 8.(a) 1 (b) 10 10. continous but not differentiable at x  1, 2