Mean Value Theorem | MIT 18.01SC Single Variable Calculus, Fall 2010

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7 Jan 201106:05

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TLDRIn this recitation, Christine Breiner guides students through proving that the tangent function is greater than the variable x for x-values between 0 and pi/2, using the Mean Value Theorem. She explains the theorem's application to the tangent function, which is continuous and differentiable in the given range. The key step involves showing that secant squared of a variable 'c' between 0 and x is greater than 1, thus confirming that tangent x is indeed greater than x in the specified interval.

Takeaways

📚 The session is a recitation focused on proving the inequality \( \tan(x) > x \) for \( x \) values between 0 and \( \frac{\pi}{2} \) using the Mean Value Theorem.
📈 The Mean Value Theorem is introduced in the form \( f(x) = f(a) + f'(c)(x - a) \), where \( c \) is between \( a \) and \( x \).
🔍 The region of interest for the problem is from 0 to \( x \), where \( x < \frac{\pi}{2} \).
👉 The function \( f(x) = \tan(x) \) is continuous and differentiable in the region of interest, satisfying the conditions for applying the Mean Value Theorem.
🧩 The output at \( a = 0 \) is \( f(0) = \tan(0) = 0 \), as sine and cosine of 0 are equal.
📝 The derivative of \( \tan(x) \) is \( \sec^2(x) \), which is needed to apply the Mean Value Theorem.
🔑 To prove \( \tan(x) > x \), it's sufficient to show that \( \sec^2(c) > 1 \) for \( c \) in the interval (0, \( \frac{\pi}{2} \)) excluding 0.
📉 The value of \( \cos(c) \) decreases from 1 to 0 as \( c \) increases from 0 to \( \frac{\pi}{2} \), making \( \sec(c) = \frac{1}{\cos(c)} \) always greater than 1 in this interval.
🔄 The reciprocal of a value less than 1 is greater than 1, which applies to \( \sec(c) \) since \( \cos(c) < 1 \) in the given interval.
🎯 The conclusion is that for any \( x \) in the interval (0, \( \frac{\pi}{2} \)), \( \tan(x) \) is greater than \( x \), which is illustrated by the Mean Value Theorem.
🏁 The session concludes with the successful demonstration of the inequality using the Mean Value Theorem, highlighting its application in graphing functions like \( \tan(x) \) and \( \arctan(x) \).

Q & A

What is the main topic of the recitation session?
-The main topic of the recitation session is to prove that for any x between 0 and pi over 2, the tangent of x is greater than x, using the Mean Value Theorem.
Why is the Mean Value Theorem relevant to this discussion?
-The Mean Value Theorem is relevant because it is used to show that the tangent function grows faster than the linear function in the given interval, which helps in proving the inequality.
What is the form of the Mean Value Theorem used in the script?
-The form of the Mean Value Theorem used is f(x) = f(a) + f'(c) * (x - a), where c is between a and x.
What function is the focus of the discussion in the script?
-The focus of the discussion is the tangent function, denoted as f(x) = tangent x.
What are the conditions for applying the Mean Value Theorem to the tangent function in this context?
-The conditions are that the tangent function must be continuous and differentiable between 0 and any value less than pi over 2.
What is the value of f(0) in the context of the script?
-The value of f(0) is 0, as tangent 0 equals sine 0 divided by cosine 0, which is 0.
What is the derivative of the tangent function with respect to x?
-The derivative of the tangent function with respect to x is secant squared x.
Why is it necessary to show that secant squared c is greater than 1?
-It is necessary to show that secant squared c is greater than 1 to prove that the right-hand side of the Mean Value Theorem equation is greater than x, thus proving that tangent x is greater than x.
How does the value of cosine between 0 and pi over 2 relate to the secant function?
-Since cosine is always less than 1 in the interval between 0 and pi over 2, the secant function, which is the reciprocal of cosine, will be greater than 1.
What conclusion is reached by the end of the script?
-The conclusion reached is that for any value of x between 0 and pi over 2, the tangent of x is indeed greater than x.
What is the significance of the region of interest in the script?
-The region of interest, from 0 to x where x is less than pi over 2, is significant because it defines the interval within which the Mean Value Theorem is applied to prove the inequality.

Outlines

00:00

📚 Introduction to Proving Inequality with Mean Value Theorem

Christine Breiner introduces a mathematical problem to prove that for any x between 0 and pi/2, the tangent of x is greater than x, using the mean value theorem. She recalls the theorem's form and sets the stage for the proof by discussing the function f(x) = tangent x and its properties in the given interval. The focus is on showing that the derivative of tangent x, which is secant squared x, is greater than 1 in the interval of interest.

05:01

🔍 Demonstrating the Inequality with Secant Squared

The second paragraph continues the proof by focusing on the derivative of the tangent function, secant squared x. It is established that secant squared c, where c is between 0 and x (less than pi/2), is always greater than 1 because the cosine of c in this interval is always less than 1. This leads to the conclusion that tangent x is indeed greater than x for the specified range, completing the proof by showing that the right-hand side of the mean value theorem's equation is greater than x itself.

Mindmap

Keywords

💡Recitation

Recitation in the context of this video refers to a teaching method where students are guided through a problem-solving process, often in a classroom setting. It is central to the video's theme as it is the method by which the instructor, Christine Breiner, engages with the audience to explore the inequality involving the tangent function.

💡Inequality

Inequality in mathematics is a relation that holds between two expressions, indicating that one expression is greater than, less than, or not equal to another. In the video, the inequality of interest is \( \tan(x) > x \) for \( x \)-values between 0 and \( \frac{\pi}{2} \), which is the main focus of the discussion.

💡Mean Value Theorem

The Mean Value Theorem is a fundamental concept in calculus that provides a relationship between the average rate of change of a function over an interval and its instantaneous rate of change at some point within that interval. In the video, the theorem is used to prove the inequality involving the tangent function.

💡Tangent Function

The tangent function, represented as \( \tan(x) \), is a trigonometric function that relates the angle of a right triangle to the lengths of its sides. In the video, it is the subject of the inequality that the instructor aims to prove using the Mean Value Theorem.

💡Graphing

Graphing is the process of visually plotting data points on a coordinate plane. In the video, the instructor mentions that the inequality was used by a student named Joel to graph the tangent and arctangent functions correctly on the same coordinate plane.

💡Differentiable

A function is said to be differentiable at a point if it has a derivative at that point, meaning it has a well-defined rate of change. In the context of the video, the tangent function is confirmed to be differentiable between 0 and any value less than \( \frac{\pi}{2} \), which is a prerequisite for applying the Mean Value Theorem.

💡Derivative

The derivative of a function is a measure of the rate at which the function's value changes with respect to changes in its input variable. In the video, the derivative of the tangent function, which is \( \sec^2(x) \), plays a crucial role in applying the Mean Value Theorem to prove the inequality.

💡Secant Function

The secant function, denoted as \( \sec(x) \), is the reciprocal of the cosine function. It is used in the video to express the derivative of the tangent function. The secant squared function, \( \sec^2(x) \), is shown to be greater than 1 in the interval of interest, which is key to proving the inequality.

💡Cosine Function

The cosine function, represented as \( \cos(x) \), is another fundamental trigonometric function that describes the ratio of the adjacent side to the hypotenuse in a right-angled triangle. In the video, the behavior of the cosine function between 0 and \( \frac{\pi}{2} \) is discussed to understand the values of the secant function.

💡Reciprocal

A reciprocal of a number is the value that, when multiplied by the original number, yields a product of one. In the video, the reciprocal of the cosine function is used to define the secant function, and the understanding of reciprocals is essential to grasp why \( \sec^2(x) \) is greater than 1 in the given interval.

Highlights

Introduction to the recitation session with a focus on proving the inequality tan(x) > x for 0 < x < π/2 using the mean value theorem.

Connection to previous work by Joel, who used the inequality to graph tan(x) and arctan(x) correctly.

The challenge to show that tan(x) is greater than x for the specified range using the mean value theorem.

Explanation of the mean value theorem's necessary form for the problem.

Clarification that c must be between a and x in the theorem's application.

Identification of the function f(x) as tan(x) and the region of interest for the theorem's application.

Verification of the function's continuity and differentiability in the region of interest.

Evaluation of the function at a, which is tan(0) = 0.

Derivation of the derivative of tan(x) as sec^2(x).

Substitution of known values into the mean value theorem equation to progress towards the solution.

Realization that the proof hinges on showing sec^2(c) > 1 in the region of interest.

Analysis of the values of secant in the specified range, relating it to the reciprocal of cosine.

Graphical representation of the cosine function between 0 and π/2 to understand secant's behavior.

Conclusion that secant is always greater than 1 in the region, leading to the proof of tan(x) > x.

Final confirmation of the inequality tan(x) > x for any x between 0 and π/2.

Transcripts

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Takeaways

Q & A

What is the main topic of the recitation session?

Why is the Mean Value Theorem relevant to this discussion?

What is the form of the Mean Value Theorem used in the script?

What function is the focus of the discussion in the script?

What are the conditions for applying the Mean Value Theorem to the tangent function in this context?

What is the value of f(0) in the context of the script?

What is the derivative of the tangent function with respect to x?

Why is it necessary to show that secant squared c is greater than 1?

How does the value of cosine between 0 and pi over 2 relate to the secant function?

What conclusion is reached by the end of the script?

What is the significance of the region of interest in the script?