Mean Value Theorem Worksheet

**Page 1**

1.

We verify the "Mean Value theorem" for a function $f$ ($x$):

a. | If it is continuous on [$a$, $b$] | ||

b. | If it is differentiable on ($a$, $b$) | ||

c. | If it is continuous on ($a$ , $b$) | ||

d. | Both A and B |

Correct answer : (4)

2.

State whether the function $f$ ($x$) = 3$x$^{2} - 2 on [2, 3] satisfies the mean value theorem.

a. | no | ||

b. | yes |

[Given function.]

The given function is continuous on [2, 3] and differentiable in (2, 3).

So, there exists

[By mean value Theorem.]

[

6

Hence, the given function satisfies the mean value theorem.

Correct answer : (2)

3.

Which of the following functions satisfies the mean value theorem?

a. | $f$($x$) = $x$ ^{2} - 6 , $x$ $\in $ [7, 8] | ||

b. | $f$ ($x$) = log $x$, $x$ $\in $ [-7, 8] | ||

c. | $f$ ($x$) = [$x$], $x$ $\in $ [-7, 7] | ||

d. | $f$ ($x$) = |$x$|, $x$ $\in $ [-8, 8] |

[Consider the choice A.]

[Try to get

2

[

[There exists

So, the function in the choices A satisfies the mean value theorem

[Consider the choice B.]

[Consider the choice C.]

[Consider the choice D.]

Correct answer : (3)

4.

Determine the point on the parabola $f$ ($x$) = ($x$ - 2)^{2}, at which the tangent is parallel to the chord joining the points (2, 0) & (3, 1).

a. | ($\frac{5}{2}$, $\frac{1}{4}$) | ||

b. | ($\frac{3}{2}$, $\frac{1}{4}$) | ||

c. | ($\frac{3}{2}$, $\frac{9}{4}$) | ||

d. | ($\frac{3}{2}$, - $\frac{3}{2}$) |

Slope of the tangent to the curve at any point (

[By Mean Value Theorem.]

2(

Hence, the point where the tangent to the parabola is parallel to the given chord is (

Correct answer : (1)

5.

Find the point on the parabola $y$ = ($x$ + 3)^{2} , at which the tangent is parallel to the chord of the parabola joining the points (- 3, 0) & (- 4, 1).

a. | (- 4, 1) | ||

b. | (- $\frac{5}{2}$, $\frac{1}{4}$) | ||

c. | (- 2, 1) | ||

d. | (- $\frac{7}{2}$, $\frac{1}{4}$) |

Slope of the tangent to the curve at any point (

[By mean value theorem.]

2(

Hence, the point where the tangent to the parabola is parallel to the given chord is (

Correct answer : (4)

6.

Find the point on the graph of $f$ ($x$) = $x$^{3}, at which the tangent is parallel to the chord joining the points (1, 1) & (3, 27).

a. | (${(\frac{13}{3})}^{\frac{1}{2}}$, ${(\frac{13}{3})}^{\frac{3}{2}}$) | ||

b. | ($\frac{13}{3}$, $\frac{3}{13}$) | ||

c. | (1, 1) | ||

d. | (0, 0) |

Slope of the tangent to the curve at any point (

[By mean value theorem.]

3

Hence, the point where the tangent to the curve is parallel to the given chord is (

Correct answer : (1)

7.

Find the point at which the tangent to the curve $f$($x$) = $x$^{2} - 6$x$ + 1 is parallel to the chord joining the points (1, - 4) & (3, -8).

a. | (2, -7) | ||

b. | (2, 7) | ||

c. | (-2, -7) | ||

d. | (-2, 7) |

Slope of the tangent to the curve at any point (

[By Mean Value Theorem.]

2

Hence, the point where the tangent to the curve is parallel to the given chord is (

Correct answer : (1)

8.

Find the point on the curve $y$ = 12($x$ + 1)($x$ - 2) in the interval [-1, 2] , at which the tangent is parallel to the $x$ - axis.

a. | (- $\frac{1}{2}$, - 27) | ||

b. | ($\frac{1}{2}$, - 27) | ||

c. | (- $\frac{1}{2}$, 27) | ||

d. | ($\frac{1}{2}$, 27) |

[Given curve.]

= 12

Slope of the tangent to the curve at any point (

[By Mean Value Theorem.]

24

= 3 - 6 - 24

= - 27

Hence, the point where the tangent to the curve is parallel to

Correct answer : (2)

9.

State whether the function $f$($x$) = ln $x$ on [1, 2] satisfies the mean value theorem.

a. | yes | ||

b. | no |

[Given function.]

So, there exists

[By mean value Theorem.]

[

Hence, the given function satisfies the mean value theorem.

Correct answer : (1)

10.

State whether the function $f$ ($x$) = sin $x$ - sin 2$x$, $x$ $\in $ [0, $\pi $] satisfies the mean value theorem.

a. | yes | ||

b. | no |

[Given function.]

The given function is continuous on [0,

So, there exists

[By mean value theorem.]

cos

[

cos

[cos 2

4 cos

[Solve use the calculator.]

Hence, the function

Correct answer : (1)