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The Mean Value Theorem

The Mean Value Theorem

Consider two police officers. One officer is parked with a radar gun next to mile marker 8, while the other is parked with a radar gun at mile marker 17. A car passes the first officer at 1:00 pm. The radar gun says the vehicle travels at 65 mph. Nine miles down, the car passes the second officer at 1:06 pm at 67 mph. The speed limit is 70 mph. Should the officers issue the vehicle a speeding ticket?

Why would the officers issue a speeding ticket if the car was clocked traveling below the speed limit at both mile markers? Let's focus on the more important question: How did the car travel 9 miles in 6 minutes? This would mean that the car travels with an average speed of 90 mph which is certainly over the speed limit. The Mean Value Theorem is a Calculus theorem that ensures the car could not possibly have an average speed of 90 mph without traveling at exactly 90 mph at least once between the two police officers.

The Mean Value Theorem highlights a link between the tangent and secant lines. Although the result may seem somewhat obvious, the theorem is used to prove many other theorems in Calculus.

What is the Calculus behind the Mean Value Theorem and its Formula?

The Mean Value Theorem states that if a function f is:

  • continuous on the closed interval [a, b]

  • differentiable on the open interval (a, b)

then there is a number c such that a < c < b and

In words, the theorem states that for at least one point on the curve between endpoints a and b, the slope of the tangent line (instantaneous rate of change) will be equal to the slope of the secant line (average rate of change) through the points (a, f(a)) and (b, f(b)).

The Mean Value Theorem visual representation of secant and tangent line relationship StudySmarterThe slope of the tangent line between two points must equal the slope of the secant line between those two points at least once according to the Mean Value Theorem - StudySmarter Original

In a real-world application, the Mean Value Theorem says that if you drive 40 miles in one hour, then at some point within that hour, your speed will be exactly 40 miles per hour.

Mean Value Theorem Proof

Let A represent the point (a, f(a)) and B represent the point (b, f(b)).

Notice how the right-hand side of the Mean Value Theorem is the slope of the secant line through points A and B. Using point A to form the equation for the secant line, we get

Let F(x) be the vertical distance between some point (x, f(x)) on the graph of f and the corresponding point on the secant line through A and B. Then, F is positive when the graph of f is above the secant line and negative otherwise. In mathematical terms,

The Mean Value Theorem proof of MVT with secant line StudySmarter

We want to show that F satisfies the conditions of Rolle's Theorem. For more information on this theorem, please view our article on Rolle's Theorem.

  • Is F continuous on [a, b]?
    • Yes - the difference between f(x) and a polynomial function is continuous.
  • Is F differentiable on (a, b)?
    • Yes - the difference between f(x) and a polynomial function is differentiable
  • At a and b, does F equal 0?
    • and

With the conditions of Rolle's Theorem satisfied, we know that some c in the interval (a, b) must exist such that .

When we differentiate F(x), we find

Using our conclusion from satisfying Rolle's Theorem,

So, there exists some c in the interval (a, b) such that

The Mean Value Theorem Examples

Example 1

Let's look at a real-world application of the Mean Value Theorem.

A ball is dropped from a height of 100 ft. Its position in seconds after it's dropped is modeled by the function .

  1. How long does the ball take to hit the ground after it's dropped?
  2. Find the ball's average velocity between when it is released and when it hits the ground.
  3. Then, find the time guaranteed by the Mean Value Theorem when the instantaneous velocity of the ball is equal to the ball's average velocity.

Step 1: Ensure that s(t) is continuous and differentiable

Before using the Mean value Theorem, we must ensure our function meets the theorem's requirements.

Since is a polynomial, we know it is continuous and differentiable on the entire interval!

Step 2: Find the number of seconds it takes for the ball to hit the ground

The ball is dropped from a height of 100 ft. Let the ground be at 0 ft. So, to find when the ball hits the ground, we can set and solve for !

As our time in seconds cannot be negative, the ball must hit the ground at , or 2.5, seconds.

Step 3: Find the average velocity of the ball

We will use the time the ball is released, , and the time the ball hits the ground (which is the total time that the ball is in free fall) to find the average velocity of the ball over the course of its fall. Averaging over both values...

Therefore, the average velocity of the ball is (down) during the time it is in the air. The sign of the velocity is negative here because the ball travels in the negative (in this case, down) direction.

Step 4: Apply the Mean Value Theorem

The Mean Value Theorem states that there is at least one point on seconds where the ball has an instantaneous velocity of (down).

We'll start by taking the derivative of the position function s(t).

To find the time the ball has a velocity of (down), we set s'(t) equal to -40.

So, the ball reaches a velocity of -40 ft per second at time , or 1.25, seconds.

Example 2

Assume that a function f(x) is continuous and differentiable on the interval [5, 15]. Given f(5) = 4 and f'(x)10. Find the largest possible value that f(15) can take on.

The problem specifies that f(x) is in fact continuous and differentiable. Thus, we can apply the Mean Value Theorem.

Step 1: Rearrange the Mean Value Theorem equation

We are looking for the value of f(15), or f(b). So, let's rearrange the equation to solve for f(b).

Step 2: Plug in known values

If f'(x) has a maximum value of 10, then f'(c) also has a maximum value of 10. So, we can substitute 10 in for f'(c) into the Mean Value Theorem.

So, the maximum value that f(15) can be is 104.

The Mean Value Theorem for Integrals

The Mean Value Theorem can also be applied specifically to integrals. Details on this can be found in our article The Mean Value Theorem for Integrals.

The Mean Value Theorem - Key takeaways

  • The Mean Value Theorem says that if a function is continuous on the interval [a, b] and differentiable on the interval (a, b), then there is a number c such that and
    • Make sure f(x) is continuous on the open interval and differentiable on the closed interval before applying the Mean Value Theorem
  • Geometrically, the Mean Value Theorem states that for at least one point on the curve between endpoints a and b, the slope of the tangent line will be equal to the slope of the secant line through the point (a, f(a)) and (b, f(b))

Frequently Asked Questions about The Mean Value Theorem

The Mean Value Theorem illustrates the like between the tangent line and the secant line. For at least one point on the curve between endpoints and b, the slope of the tangent line will be equal to the slope of the secant line through the point (a, f(a)) and (b, f(b)).

The Mean Value Theorem is used to prove a multitude of other theorems in Calculus.

The Mean Value Theorem can be verified by making sure the the derivative at some point is equal to the average value of the function on the interval (a, b)

Unknown values can be found by plugging in known values to the Mean Value Theorem formula and solving for the unknown value.

The Mean Value Theorem is used in applications of position and velocity. It can also be used to prove a multitude of other Calculus theorems.

Final The Mean Value Theorem Quiz

Question

State the geometric definition of the Mean Value Theorem.

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Answer

The Mean Value Theorem illustrates the like between the tangent line and the secant line; for at least one point on the curve between endpoints and b, the slope of the tangent line will be equal to the slope of the secant line through the point (a, f(a)) and (b, f(b))

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Question

What are the requirements to use the Mean Value Theorem?

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Answer

The function must be continuous on the closed interval and differentiable on the open interval

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Question

The Mean Value Theorem states that if a car travels 140 miles in 2 hours, then at one point within the 2 hours, the car travels at exactly ______ mph.

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Answer

70

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Question

State Rolle's Theorem.

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Answer

Rolle's Theorem says that if a function f is continuous on the closed interval [a, b], differentiable on the open interval (ab), and f(a) = f(b), then there is at least one value where f'(c) = 0.

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Question

Rolle's Theorem is a special case of the Mean Value Theorem where...

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Answer

f(b) - f(a) = 0 or f(b) = f(a)

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Question

How can we interpret Rolle's Theorem geometrically?

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Answer

If a function meets the requirements of Rolle's Theorem, then there is a point on the function between the endpoints where the tangent line is horizontal, or the slope of the tangent line is 0.

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Question

What is a corollary?

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Answer

A corollary is a consequence that follows from a theorem that has already been proven.

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Question

State Corollary 1 of the Mean Value Theorem.

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Answer

Assume that f is differentiable over an interval [a, b]. Corollary 1 says that if f'(x) = 0 over the entire interval  [a, b], then f(x) is a constant over [a, b].

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Question

State Corollary 2 of the Mean Value Theorem.

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Answer

If functions f and g are both differentiable over the interval [ab] and f'(x)g'(x) at every point in the interval [ab], then f(x)g(x)C where is a constant.

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Question

State Corollary 3 of the Mean Value Theorem.

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Answer

Let f be a function that is continuous over [ab] and differentiable over (ab). If f'(x) is positive on the entire interval (ab), then f is an increasing function over [ab]. If f'(x) is negative on the entire interval (ab), then is a decreasing function over [ab].

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Question

An increasing function's derivative is....

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Answer

f'(x) > 0

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Question

A decreasing function's derivative is...

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Answer

f'(x) < 0

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Question

Why is the Mean Value Theorem important?

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Answer

The Mean Value Theorem produces numerous implications with important results in Calculus.

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Question

What is the difference between a theorem and a corollary?

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Answer

A theorem is a definitive statement that is proven to be true. A corollary is a simple consequence of a proven theorem.

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Question

How is a corollary related to a theorem?

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Answer

A corollary is a simple consequence of a theorem.

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Question

The Mean Value Theorem establishes a relationship between the ___________ and the _____________.

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Answer

tangent line, secant line

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Question

The Mean Value Theorem can also be applied to integrals.

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Answer

true

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