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Differentiable and Continuous Worksheet

Differentiable and Continuous Worksheet
  • Page 1
 1.  
Which of the following is true?
a.
Every continuous function need not be differentiable & every differentiable function is continuous
b.
Every continuous function need not be differentiable
c.
Every continuous function is differentiable
d.
Every differentiable function is continuous


Solution:

Every continuous function need not be differentiable and every differentiable function is continuous.


Correct answer : (1)
 2.  
Which of the following is correct for a function that is differentiable at a point x = 3 ?
a.
left hand side derivative = right hand side derivative at x = 3
b.
left hand side derivative may not be equal to the right hand side derivative at x = 3
c.
f(3) does not exist
d.
the function is not continuous at x = 3


Solution:

A function f(x) is said to be differentiable at a point x = 3 if both left, right hand side derivatives of f(x) are finite and equal at x = 3 .
[Definition.]


Correct answer : (1)
 3.  
The function f(x) = | 5x | is
a.
discontinuous at x = 0 and differentiable at x = 0
b.
an odd function
c.
continuous at x = 0 and not differentiable at x = 0
d.
continuous at x = 0 and differentiable at x = 0


Solution:

f(x) = | 5x |

Right hand derivative of f(x) at x = 0 is f ′(0+) = limh0 f(0 + h) - f(0)h
[Definition.]

= limh0 5h - 0h
[Since | 5h | = 5h.]

= 5

Left hand side derivative of f(x) at x = 0 is f ′(0-) = limh0 f(0 - 5h) - f(0)-h
[Definition.]

= limh0 5h-h
[Since | -5h | = 5h.]

= -5

Since right hand side derivative ≠ left hand side derivative at x = 0, the given function | x | is not differentiable at x = 0.

limx0+ f(x) = limx0+ | x | = 0

limx0- f(x) = limx0- | x | = 0 and f(0) = 0

So, limx0+ f(x) = limx0- f(x) = f(0), the given function f (x) = | x | is continuous at x = 0


Correct answer : (4)
 4.  
If f(x) = x(x -x + 16), then
a.
f is continuous but not differentiable at x = 0
b.
f is not continuous at x = 0
c.
f is differentiable but not continuous at x = 0
d.
f is differentiable at x = 0


Solution:

f(x) = x(x -x + 16)

Since the domain of f(x) = [0, ∞),

f ′(0+) = limh0+h(h -h + 16)h
[Definition of differentiability at the end point of an interval.]

= limh0+(h -h  +  16) = - 4

Hence f is differentiable at x = 0.

Since the given function is differentiable at x = 0, hence it is continuous at x = 0.


Correct answer : (4)
 5.  
A function f is defined by:

f(x) = sin 2x if 0 < xπ4
      = ax + b if π4 < x < 1 and is continuous and differentiable in its domain.
Find the values of a & b.
a.
a = 0 , b = 1
b.
a = 0 , b = - π 2
c.
a = 2, b = 1 - π 2
d.
a = - 2, b = π 2


Solution:

sin 2(π4) = a( π4) + b
[Condition for continuity.]

sin (π2) = a (π4) + b

f′(π4-) = f′(π4+)
[Condition for differentiability.]

limh0 f(π4 - h)-f(π4)-h = limh0 f(π4+h)-f(π4)h

limh0 sin 2(π4 - h) - sin 2(π4) - h = limh0[a(π4 + h) + b] - [a.π4 + b]h

2 cos 2. (π4) = a

a = 2cos (π2) = 0

b = 1 - 0 = 1
[Substitute the value of a in step 2.]


Correct answer : (1)
 6.  
If a function f is defined by,
f(x) = |x + 4|tan(x + 4) for x ≠ - 4

      = 4 for x = - 4, then
a.
f is continuous at x = - 4
b.
f is not continuous at x = - 4
c.
f is not differentiable at x = - 4
d.
both B & C


Solution:

limx-4+ f(x) = limx-4+(x + 4)tan (x + 4) = 1.
[Evaluate.]

limx-4- f(x) = limx-4--(x + 4)tan (x + 4) = -1
[Evaluate.]

Since limx-4+(x+4)tan (x + 4)limx-4--(x + 4)tan (x + 4) the function is not continuous at x = - 4.

Similarly f ′(- 4- ) ≠ f ′(- 4+)
[Check.]

Hence the function f(x) is not differentiable at x = - 4


Correct answer : (4)
 7.  
For what values of x where the function f(x) = |x2 - 13x + 42| is not differentiable?
a.
- 7 & - 6
b.
7 & - 6
c.
- 13 & - 42
d.
7 & 6


Solution:

f(x) = | x 2 - 13x + 42| = |(x - 7)(x - 6)|

f ′(7- ) ≠ f ′(7+)
[Check.]

Hence the function is not differentiable at x = 7

f ′(6- ) ≠ f ′(6+)
[Check.]

Hence the function is not differentiable at x = 6

So the function is not differentiable at x = 7 , 6


Correct answer : (4)
 8.  
A function f defined by f(x)= 2x22, if 0 ≤ x ≤ 1
= 2x2 - 2x + 2( 1 2), if 1 < x ≤ 2 is

a.
continuous at x = 1
b.
differentiable at x = 1
c.
discontinuous at x = 1
d.
both A & B


Solution:

limx1- f(x) = limx1- ( 2x22) = 2 / 2
[Left hand limit.]

limx1+ f(x) = limx1+ (2x2 - 2x + 2(1 / 2)) = 2 / 2
[Right hand limit.]

Since limx1- f(x) = limx1+ f(x) = f(1), the function f(x) is continuous at x = 1.

f ′(1-) = limh0f(1-h)-f(1)-h = 1
[Left hand derivative.]

f ′(1+) = limh0 f(1+h)-f(1)h = 1
[Right hand derivative.]

Since left hand derivative = right hand derivative, the function f(x) is differentiable at x = 1.


Correct answer : (4)
 9.  
If a function f is defined by f(x) = 3x3 - 3kx2 + 3x, x R is an odd function, then find k.
a.
1
b.
-1
c.
2


Solution:

f(- x) = - f(x)
[Condition for an odd function.]

- 3x3 - 3kx2 - 3x = - 3x3 + 3kx2 - 3x
[Substitute the values.]

k = 0


Correct answer : (4)
 10.  
If f(x) = 1 for x < 0
= 3sin 3x for 0 ≤ xπ2,
then which of the following is correct?
a.
f(x) is continuous at x = 0
b.
both A & B
c.
f(x) is differentiable at x = 0
d.
f(x) is discontinuous at x = 0


Solution:

limx0- f(x) = limx0-1 = 1
[Left hand limit.]

limx0+ f(x)= limx0+ 3sin 3x = 0
[Right hand limit.]

Since limx0- f(x) ≠ limx0+ f(x), the function f(x) is discontinuous at x = 0.

Similarly, f ′(0-) ≠ f ′(0+)
[Check. ]

Hence, the function f(x) is not differentiable at x = 0.


Correct answer : (4)

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