Rolle’s Theorem and the Mean Value Theorem

Rolle’s Theorem and the Mean Value Theorem

By Sahel Otoom

Rolle’s Theorem and the Mean Value Theorem

By Sahel Otoom

Multiple Choice

For Rolle's theorem function must be continues on

• Closed interval

• open interval

• semi closed interval

• all of the above

Multiple Choice

For Rolle's theorem function must be differentiable on

• Closed interval

• Open interval

• Semi closed interval

• Function need bot be differentiable

Multiple Choice

Rolle's theorem tells about the

• Existence of a point c where derivative of a function becomes zero

• Existence of a point c where derivative of a function becomes positive

• Existence of a point c where derivative of a function becomes negative

• Existence of a point c where derivative of a function becomes monotonically increasing

Multiple Choice

Does Rolle's theorem is applicable if f(a) is not the same as f(b)

• Yes

• No

• Under particular condition

• May be

Multiple Choice

Does Rolle's theorem is applicable

• Yes

• No

Multiple Choice

Does Rolle's theorem is applicable

• Yes

• No

Multiple Choice

The function, F, above satisfies the conclusion of Rolle's Theorem in the interval [a,b] because:

I. F is continuous

II. F is differentiable on (a,b)

III. F(a) = F(b) = 0

• All 3 statements are true

• Only 2 and 3 are true

• Only 1 is true

• None of the statements are true

Multiple Choice

Which of the functions given below satisfy all three conditions of Rolle's theorem?a) f(x) = cos(x) , for x in [0 , 2π]

b) g(x) = |x - 2| , for x in [0 , 4]

c) h(x) = 1 / x² , for x in [-1 , 1]

• a)f(0) = 1 and f(2π) = 1 therefore f(0) = f(2π)f is continuous on [0 , 2π]Function f is differentiable in (0 , 2π)Function f satisfies all conditions of Rolle's theorem

• b)function g has a V-shaped graph with vertex at x = 2 and is therefore not differentiable at x = 2.Function g does not satisfy all conditions of Rolle's theorem

• c)Function h is undefined at x = 0.Function h does not satisfy all conditions of Rolle's theorem.

Open Ended

Find all numbers c that satisfy Rolls theorem if

$$f\left(x\right)=x^2-5x+3\$$  on the closed interval [0,5]

Multiple Choice

$$Find\ the\ value\ of\ c\ for\ the\ function\ f\left(x\right)=x^{2\ }\ in\ \left[-1,1\right]$$  

• 0

• 1

• -1

• 1/2

Multiple Choice

Check that function f(x) = x ² - 4 x + 3 on the interval [1 , 3] satisfies all conditions of Rolle's theorem and then find all values of x = c such that f '(c) = 0.

• c=2

• c=4

• c=8

Open Ended

Find the two x-intercepts of

         f(x) = x² – 6x + 8

and show that $$\frac{dy}{dx}$$   = 0 at some point between the two x-intercepts. 

Open Ended

The value of c guaranteed to exist by the MVT for  $$f\left(x\right)=x^2$$  on the interval [0,3] is:

Multiple Choice

What is the average rate of change of the function below over the indicated interval?
f(x) = x² - 3x - 28,  [-4,7]

• 0
• 11
• 6
• 66

Multiple Choice

What is the derivative of
f(x) = x² - 3x - 28 ?

• 2x - 3
• 2x + 3
• x - 3
• 2x² - 3x - 28

Multiple Choice

What is x-value at which the function below has the same slope as the average rate of change over the indicated interval?
f(x) = x² - 3x - 28,  [-4,7]

• 1.5
• -1.5
• 0.5
• -0.5

Multiple Choice

What is the average rate of change of the function below over the indicated interval?
f(x) = x² + 5x + 5,  [-2,6]

• 9
• 1
• 72
• 0

Multiple Choice

What is the derivative of
f(x) = x² + 5x + 5 ?

• 2x + 5
• x² + 5
• 5x + 5
• 2x - 5

Multiple Choice

What is x-value at which the function below has the same slope as the average rate of change over the indicated interval?
f(x) = x² +5x +5,  [-2,6]

• 2
• -2
• 7
• -7

EMSAT

Some text here about the topic of discussion

Multiple Choice

The solution set to the equation: $$4^{x^2-4x}=2^{-6}$$  

• {1, 3}

• {1, -3}

• {-1, 3}

• {-1, -3}

Multiple Choice

For what value of x is the equation  $$2^{\left(2x+7\right)}\ =\ 2^{15}$$  true?

• 2

• 4

• 16

• 44

Multiple Choice

In the standard (x,y) coordinate plane, what is the slope of the line given by the equation 4x = 7y + 5 ?

• $$-\frac{4}{7}$$

• $$\frac{4}{7}$$

• $$\frac{7}{4}$$

• 4

• 7

Multiple Choice

What is the slope of the line through (−2,1) and (2,−5) in the standard (x,y) coordinate plane?

• $$\frac{3}{2}$$

• 1

• -1

• $$-\frac{3}{2}$$

• -4

Multiple Choice

 Let the function f be defined as   $$5x^{2\ }-7\left(4x+3\right)$$ What is the value of f(3) ? 

• -18

• -26

• -33

• -60

• -75