Lesson test: The Mean Value Theorem

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The Mean Value Theorem

1.[3p]
Find the point $c$ guaranteed by the Mean Value Theorem for $f (x) = x^{3} - x$ on $[0, 2]$ .

CorrectNot quite: 1.155

2.[3p]
Find the point $c$ guaranteed by the theorem for $f (x) = \sqrt{x}$ on $[1, 4]$ .

CorrectNot quite: 2.25

3.[3p]
Which corollary of the Mean Value Theorem makes antidifferentiation possible?

Two functions with the same derivative on an interval differ by a constant

Correct

A continuous function on a closed interval attains a maximum

The answer is: Two functions with the same derivative on an interval differ by a constant

A function with a positive derivative is increasing

The answer is: Two functions with the same derivative on an interval differ by a constant

A differentiable function is continuous

The answer is: Two functions with the same derivative on an interval differ by a constant

4.[2p]
The Mean Value Theorem provides a method for finding the point $c$ .

True

The answer is: False

False

Correct

5.[2p]
Evaluate $lim_{x \to 0} \frac{1 - \cos x}{x^{2}}$ .

CorrectNot quite: 0.5

6.[2p]
Use the tangent line at $x = 4$ to estimate $\sqrt{4.1}$ .

CorrectNot quite: 2.025

7.[3p]
Applying L'Hôpital's rule to $lim_{x \to 1} (x^{2} + x) / (x + 1)$ gives $3$ , but the limit is $1$ . What went wrong?

The form is not indeterminate, so the rule does not apply

Correct

The functions are not differentiable at $x = 1$

The answer is: The form is not indeterminate, so the rule does not apply

The rule needs two applications here

The answer is: The form is not indeterminate, so the rule does not apply

The denominator is zero at $x = 1$

The answer is: The form is not indeterminate, so the rule does not apply

8.[3p]
Which of these are proved using the Mean Value Theorem?

Select all that apply

A function whose derivative is zero throughout an interval is constant on it

Correct

A function with a positive derivative is increasing

Correct

$| \sin q - \sin p | \leq | q - p |$

Correct

Every continuous function is differentiable

The answer is: A function whose derivative is zero throughout an interval is constant on it, A function with a positive derivative is increasing, $| \sin q - \sin p | \leq | q - p |$

9.[3p]
Put these results in the order the course establishes them, each resting on those before it.

The Mean Value Theorem
Rolle's theorem
Two functions with equal derivatives differ by a constant
The Extreme Value Theorem
Fermat's condition at an interior extreme

Show the answer

a, b, c, d, e

1.[3p] Find the point c guaranteed by the Mean Value Theorem for f(x)=x3−x on [0,2].

2.[3p] Find the point c guaranteed by the theorem for f(x)=x on [1,4].

3.[3p] Which corollary of the Mean Value Theorem makes antidifferentiation possible?

4.[2p] The Mean Value Theorem provides a method for finding the point c.

5.[2p] Evaluate limx→01−cosxx2.

6.[2p] Use the tangent line at x=4 to estimate 4.1.

7.[3p] Applying L'Hôpital's rule to limx→1(x2+x)/(x+1) gives 3, but the limit is 1. What went wrong?

8.[3p] Which of these are proved using the Mean Value Theorem?

9.[3p] Put these results in the order the course establishes them, each resting on those before it.

1.[3p]
Find the point $c$ guaranteed by the Mean Value Theorem for $f (x) = x^{3} - x$ on $[0, 2]$ .

2.[3p]
Find the point $c$ guaranteed by the theorem for $f (x) = \sqrt{x}$ on $[1, 4]$ .

3.[3p]
Which corollary of the Mean Value Theorem makes antidifferentiation possible?

4.[2p]
The Mean Value Theorem provides a method for finding the point $c$ .

5.[2p]
Evaluate $lim_{x \to 0} \frac{1 - \cos x}{x^{2}}$ .

6.[2p]
Use the tangent line at $x = 4$ to estimate $\sqrt{4.1}$ .

7.[3p]
Applying L'Hôpital's rule to $lim_{x \to 1} (x^{2} + x) / (x + 1)$ gives $3$ , but the limit is $1$ . What went wrong?

8.[3p]
Which of these are proved using the Mean Value Theorem?

9.[3p]
Put these results in the order the course establishes them, each resting on those before it.