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 |

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 |

[Definition.]

Correct answer : (1)

3.

The function $f$($x$) = | 5$x$ | 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 |

Right hand derivative of

[Definition.]

=

[Since | 5

= 5

Left hand side derivative of

[Definition.]

=

[Since | -5

= -5

Since right hand side derivative ≠ left hand side derivative at

So,

Correct answer : (4)

4.

If $f$($x$) = $x$($\sqrt{x}-\sqrt{x+16}$), 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 |

Since the domain of

[Definition of differentiability at the end point of an interval.]

=

Hence

Since the given function is differentiable at

Correct answer : (4)

5.

A function $f$ is defined by:

$f$($x$) = sin 2$x$ if 0 < $x$ ≤ $\frac{\pi}{4}$

= $a$$x$ + $b$ if $\frac{\pi}{4}$ < $x$ < 1 and is continuous and differentiable in its domain.

Find the values of $a$ & $b$.

Find the values of $a$ & $b$.

a. | $a$ = 0 , $b$ = 1 | ||

b. | $a$ = 0 , $b$ = - $\frac{\mathrm{\pi}}{2}$ | ||

c. | $a$ = 2, $b$ = 1 - $\frac{\mathrm{\pi}}{2}$ | ||

d. | $a$ = - 2, $b$ = $\frac{\mathrm{\pi}}{2}$ |

[Condition for continuity.]

sin (

f′(

[Condition for differentiability.]

a = 2cos (

b = 1 - 0 = 1

[Substitute the value of

Correct answer : (1)

6.

If a function $f$ is defined by,

$f$($x$) = $\frac{|x+4|}{\mathrm{tan}(x+4)}$ for $x$ ≠ - 4

= 4 for$x$ = - 4, then

= 4 for

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 |

[Evaluate.]

[Evaluate.]

Since

Similarly

[Check.]

Hence the function

Correct answer : (4)

7.

For what values of $x$ where the function $f$($x$) = |$x$^{2} - 13$x$ + 42| is not differentiable?

a. | - 7 & - 6 | ||

b. | 7 & - 6 | ||

c. | - 13 & - 42 | ||

d. | 7 & 6 |

[Check.]

Hence the function is not differentiable at

[Check.]

Hence the function is not differentiable at

So the function is not differentiable at

Correct answer : (4)

8.

A function $f$ defined by $f$($x$) | = $\frac{2{x}^{2}}{2}$, if 0 ≤ $x$ ≤ 1 |

= 2$x$^{2} - 2$x$ + 2( $\frac{1}{2}$), if 1 < $x$ ≤ 2 is |

a. | continuous at $x$ = 1 | ||

b. | differentiable at $x$ = 1 | ||

c. | discontinuous at $x$ = 1 | ||

d. | both A & B |

[Left hand limit.]

[Right hand limit.]

Since

[Left hand derivative.]

[Right hand derivative.]

Since left hand derivative = right hand derivative, the function

Correct answer : (4)

9.

If a function $f$ is defined by $f$($x$) = 3$x$^{3} - 3$\mathrm{kx}$^{2} + 3$x$, $x$ $\in $ R is an odd function, then find $k$.

a. | 1 | ||

b. | -1 | ||

c. | 2 |

[Condition for an odd function.]

- 3

[Substitute the values.]

Correct answer : (4)

10.

If $f$($x$) | = 1 for $x$ < 0 |

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 |

[Left hand limit.]

[Right hand limit.]

Since

Similarly,

[Check. ]

Hence, the function

Correct answer : (4)