Ryan and Nick do L'hopital's rule

Chad Gilliland
30 Jan 201404:03
EducationalLearning
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TLDRIn this educational video, Ryan Dewey and Nicholas Cromwell demonstrate the application of L'Hôpital's Rule to solve an indeterminate limit problem. They tackle the limit as x approaches one of the function (3/(natural log of x - 2))/(x - 1), showing that substituting x with one results in an indeterminate form. They then simplify the expression and differentiate both the numerator and the denominator to resolve the issue, leading to a new indeterminate form. The video concludes with the final step of differentiation, which still yields an indeterminate result, emphasizing the iterative nature of L'Hôpital's Rule in such scenarios.

Takeaways
  • 📚 The video is an educational tutorial on La Hopital's Rule by Ryan Dewey and Nicholas Cromwell.
  • 🔍 The initial problem presented is to find the limit as x approaches 1 from the right side of the function (3 / (natural log of x - 2)) / (x - 1).
  • 🤔 The problem is recognized as an indeterminate form of 0/0 when x is substituted with 1.
  • 📝 The script suggests combining the fractions by cross-multiplying to simplify the expression.
  • 🔄 After simplification, the expression is rewritten to show the indeterminate form more clearly.
  • 📉 The video demonstrates the process of taking the derivative of both the numerator and the denominator to apply La Hopital's Rule.
  • 📚 The derivative of the numerator is 3 - 2/x, while the derivative of the denominator uses the product rule resulting in ln(x) + 1.
  • 🔢 Upon plugging in x = 1 into the derivatives, the numerator results in 3 - 2, and the denominator results in 0 (since ln(1) = 0 and 1 - 1 = 0).
  • 🚫 The final answer of 1/0 is stated as indeterminate, which is incorrect as the correct answer should be 1, as the numerator simplifies to 1 and the denominator to 0.
  • 👨‍🏫 The video concludes with a reminder that La Hopital's Rule is used to solve indeterminate forms like 0/0 or ∞/∞.
  • 📝 There is a minor error in the final calculation where the result of 3 - 2 is mistakenly stated as 'three minus two' instead of the correct 'one'.
Q & A

  • What is the main topic of the video?

    -The main topic of the video is demonstrating an example of L'Hôpital's Rule.

  • Who are the presenters in the video?

    -The presenters in the video are Ryan Dewey and Nicholas Cromwell.

  • What is the initial problem presented in the video?

    -The initial problem is to find the limit as x approaches one from the right side of the expression (3 / (natural log of x - 2)) over (x - 1).

  • Why is L'Hôpital's Rule necessary for this problem?

    -L'Hôpital's Rule is necessary because when x is plugged in as one, the expression results in an indeterminate form of 0/0.

  • What is the first step taken in solving the problem using L'Hôpital's Rule?

    -The first step is to rewrite the expression by cross-multiplying to simplify the fraction before differentiating.

  • What happens when x equals one in the original expression?

    -When x equals one, both the numerator and the denominator of the original expression become zero, resulting in an indeterminate form.

  • What is the derivative of the numerator in the simplified expression?

    -The derivative of the numerator, after simplification, is 3 - 2/x.

  • What is the derivative of the denominator in the simplified expression?

    -The derivative of the denominator, using the product rule, results in ln(x) + 1/x.

  • What is the final result after applying L'Hôpital's Rule in the video?

    -The final result after applying L'Hôpital's Rule is an indeterminate form of 1/0, which is undefined.

  • Why is the final answer considered indeterminate in the video?

    -The final answer is considered indeterminate because any number divided by zero is undefined.

  • What mistake is made in the video during the explanation of the derivatives?

    -A mistake is made when the presenter incorrectly states the derivative of the numerator as 'three minus two' instead of correctly calculating the derivative of each term.

  • How does the video demonstrate the process of L'Hôpital's Rule?

    -The video demonstrates the process by showing the steps of identifying the indeterminate form, rewriting the expression, taking the derivatives of the numerator and denominator, and then re-evaluating the limit.

Outlines
00:00
📚 Introduction to L'Hôpital's Rule

Ryan Dewey and Nicholas Cromwell introduce the concept of L'Hôpital's Rule in this educational video. They begin by explaining the rule's application to the limit problem as x approaches one from the right side, specifically focusing on the expression (3/(natural log of x - 2))/(x - 1). They emphasize that despite the initial complexity, the problem can be simplified by combining fractions and applying L'Hôpital's Rule, which involves taking derivatives to resolve indeterminate forms like 0/0.

🔍 Simplifying the Fraction and Applying L'Hôpital's Rule

The video continues with a detailed walkthrough of simplifying the given fraction by cross-multiplying and rewriting the expression. The hosts demonstrate that substituting x = 1 results in an indeterminate form of 0/0, which is a clear indication that L'Hôpital's Rule is necessary. They proceed to take the derivatives of the numerator and the denominator, using basic differentiation rules and the product rule, to resolve the indeterminate form.

🧩 Derivatives and Final Calculation

After differentiating both the numerator and the denominator, the video shows the process of plugging in x = 1 into the new expressions. The derivatives simplify to 3 - 2 for the numerator and 0 for the denominator, leading to an indeterminate form of 1/0. The hosts conclude that this result is also indeterminate, indicating that further steps or a different approach might be needed to find a definitive answer.

Mindmap
Keywords
💡La Hopital's Rule
La Hopital's Rule is a mathematical principle used to find the limit of a function when it results in an indeterminate form, such as 0/0 or ∞/∞. In the video, the rule is applied to a complex fraction that initially results in an indeterminate form when x approaches one from the right side. The rule states that the limit of the ratio of two functions as x approaches a certain value is equal to the limit of the ratio of their derivatives. The script demonstrates the application of this rule to resolve the indeterminate form and find the limit of the given function.
💡Limit
In calculus, a limit is the value that a function or sequence approaches as the input approaches some value. The script discusses finding the limit of a function as x approaches one from the right side, which means considering the behavior of the function as x gets arbitrarily close to one but remains greater than one. The limit is a fundamental concept in calculus and is central to the theme of the video.
💡Indeterminate Form
An indeterminate form occurs when the limit of a function results in a form that does not provide a clear answer, such as 0/0 or ∞/∞. In the script, the initial attempt to find the limit results in 0/0, which is an indeterminate form. The concept is crucial as it triggers the need to apply La Hopital's Rule to find the actual limit of the function.
💡Derivative
A derivative in calculus represents the rate at which a function changes with respect to its variable. The script involves taking the derivative of both the numerator and the denominator of the original function to apply La Hopital's Rule. The derivative is a key component in resolving indeterminate forms and finding the limit of a function.
💡Cross Multiplying
Cross multiplying is a mathematical technique used to solve equations involving fractions by multiplying the numerator of one fraction by the denominator of the other and vice versa, then setting the products equal to each other. In the script, cross multiplying is used to combine the two fractions into a single fraction, which is a step towards simplifying the expression and applying La Hopital's Rule.
💡Natural Logarithm
The natural logarithm, often denoted as ln(x), is the logarithm of a number to the base e, where e is an irrational constant approximately equal to 2.71828. In the video, the natural logarithm is part of the original function whose limit is being sought. The natural logarithm is a transcendental function and is used in various mathematical contexts, including the function in the script.
💡Product Rule
The product rule is a fundamental calculus rule used to find the derivative of a product of two functions. In the script, when taking the derivative of the denominator, which is a product of (x - 1) and ln(x), the product rule is implicitly used. The rule states that the derivative of a product is the derivative of the first function times the second function plus the first function times the derivative of the second function.
💡Constant
In the context of calculus, a constant is a value that does not change during the differentiation process. In the script, when differentiating the numerator, the constant 3 is mentioned, which comes out to zero when differentiated because constants have a derivative of zero. This simplifies the expression and contributes to finding the limit.
💡Plug In
To 'plug in' a value in mathematics means to substitute that value into an expression to evaluate it. In the script, the term is used when the value of x is substituted as one to determine the behavior of the function. This action leads to the realization that the function results in an indeterminate form, prompting the use of La Hopital's Rule.
💡Simplify
Simplification in mathematics involves making a complex expression easier to understand or solve. In the script, simplification is part of the process to make the function more manageable before applying La Hopital's Rule. The simplification process includes combining fractions and reducing the expression to a form where the rule can be effectively applied.
💡Undefined
A mathematical expression or result is considered undefined when it does not have a meaningful value in the context it is being evaluated. In the script, the final answer of 'one over zero' is mentioned as undefined because division by zero is not defined in mathematics. This illustrates the importance of correctly applying mathematical rules and concepts to avoid such outcomes.
Highlights

Introduction to the video by Ryan Dewey and Nicholas Cromwell.

Explanation of La Hopital's Rule with an example problem.

Problem statement: Limit as x approaches one from the right side of the graph.

Initial expression given: (3/(natural log of x - 2)) / (x - 1).

Simplification of the expression by combining fractions.

Cross multiplication to simplify the equation.

Identification of the indeterminate form 0/0 after plugging in x=1.

Application of La Hopital's Rule to resolve the indeterminate form.

Derivative of the numerator: 3 - 2/x.

Use of the product rule for the derivative of the denominator.

Final calculation after taking the derivative and plugging in x=1.

Result of the final calculation is an indeterminate form 1/0.

Conclusion of the video with a summary of the result.

Misstep in the calculation process corrected during the video.

Explanation of the derivative of the natural log function.

Clarification of the derivative of the bottom part of the expression.

Final answer is presented as an indeterminate form, emphasizing the rule's application.

Transcripts

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Related Tags
La Hopital's RuleMathematicsEducationalLimit CalculationDerivativeIndeterminate FormCross MultiplyProduct RuleRyan DeweyNicholas Cromwell