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Differential Quotient JavaScript Simulation Applet HTML5

 

 

 

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Intro Page

Difference quotient/ differential  quotient

The transition from the difference quotient to the differential quotient (first derivative) is demonstrated for the sine function.

y = sin(x)

Difference quotient (x1 ) = (y2 - y1)/(x2 -x1)

Differential quotient dy/dt (x1 ) = Limit for x2→ x1 of (y2 - y1)/(x2 -x1)

The zero ordinate line y=0 is drawn in red. The blue model point x1 , y1, is defined with the slider. Drawing the red point with the mouse defines the second point x2 , y2. The line connecting both points (secant) is drawn in black, and is prolongated beyond the second point in green.

The difference of the ordinates (y2 - y1) is shown as a red arrow, the difference of the abscissa (x2 -x1) as a blue one.

The difference quotient (y2 - y1)/(x2 -x1), which is the tangent of the angle between both components at the model point, is shown as a magenta colored point.

The analytic first derivative function cos(x) = d(sin(x))/dx is drawn as a beige line.

Choose any arbitrary model point with the slider. Draw the red point closer and closer to the blue model point. In the limit of this process the secant turns into a tangent at the model point, and the difference quotient becomes a point on the derivative = the differential quotient at the model point.

1st and 2nd

First and second derivative

The simulation demonstrates the limit process in deriving the first derivative of a sine function with a superimposed linear term (a straight line) . For this case the second derivative should be independent of the linear term.

y = sin(x) +c x

dy/dt(x1) = x2➙x1 lim (y2 - y1 )/(x2 - x1) = cos(x1)+c

d 2y/dt2(x1) = -sin(x1)

A slider defines the blue model point x1 , y1 . Drawing the red point along the function defines the second point x2, y2 in the limit process. The parameter (gradient) c of the linear term is defined by drawing the magenta colored rectangle.

The secant between the 2 points is drawn in black, and is prolongated beyond the second point in green.

The ordinate difference (y2 - y1) is shown as a red arrow, the abscissa difference

(x2 -x1) as a blue one.

The difference quotient (y2 - y1)/(x2 -x1), which corresponds to the tangent of the angle between both components, is shown as a magenta colored point.

The analytic first derivative cos(x) +c = d(sin(x)+cx)/dx is drawn as a beige line, the second derivative sin(x) =d(cos(x)+c)/dx. as a blue one.

Choose an arbitrary model point with the slider. Draw the blue point towards the model point. With this limit process the secant will become the tangent in the model point, the difference quotient will become a point on the beige line ~ a differential quotient.

Change c by drawing the magenta rectangle. The analytic first derivative will be shifted parallel in the y- direction, while the analytic second derivative will stay unaffected.

Experiments

E1: Choose a model point with the slider. Draw the red point to the blue model point (limiting process with vanishing difference).

The secant turns into a tangent, the magenta colored difference quotient becomes a point on the beige first derivative = a differential quotient.

E2: Move the model point near a maximum or minimum of the sine function and study the context.

E3: Position the model point close to the steepest point of the curve and study the process when you cross the inflexion point.

E4: Try several model points of the sine and convince yourself that the limiting process will always lead to a point on the cosine.

E5:  After this experience, how would you design an algorithm for the numerical calculation of the first derivative of an arbitrary, continuous function?

E6: How would you design an algorithm for the second derivative?

Experiment2

E1: Choose a model point with the slider. Draw the red point towards the model point (limit process with vanishing distance). What do you observe?

E2: Choose a model point near to an extremum (maximum or minimum) of the function and study the transition.

E3: Choose a model point near a point of extreme steepness (inflexion point). Study the behavior when the model point transgresses the inflexion point.

E4: Try with several model points if the beige curve is the limit of all of them.

E5:  Change c (by drawing the magenta rectangle):  What happens?

E6:  On the basis of your observation, how would you construct an algorithm for numerical calculation of the first derivative of a continuous function?

E7: How for the second derivative??

 

Translations

Code Language Translator Run

Credits

dieter.roess@t-online.de; Fremont Teng; Loo Kang Wee

This briefing document summarizes the key features, functionalities, and educational value of the "Differential Quotient JavaScript Simulation Applet HTML5" provided by Open Educational Resources / Open Source Physics @ Singapore.

Main Theme:

The central theme of this resource is to visually and interactively demonstrate the fundamental concept of the derivative in calculus, specifically the transition from the difference quotient to the differential quotient (the first derivative). It achieves this through a JavaScript-based simulation applet.

Key Ideas and Facts:

  • Focus on the Limit Process: The core concept illustrated is how the difference quotient, which represents the slope of a secant line between two points on a curve, approaches the differential quotient (the derivative), which represents the slope of the tangent line at a single point, as the distance between the two points approaches zero.
  • The description explicitly states: "The transition from the difference quotient to the differential quotient (first derivative) is demonstrated for the sine function."
  • The differential quotient is defined as: "Differential quotient dy/dt (x1 ) = Limit for x2→ x1 of (y2 - y1)/(x2 -x1 )"
  • Sine Function as the Primary Example: The simulation initially focuses on the function y = sin(x) to illustrate the derivative concept. The analytic first derivative, cos(x) = d(sin(x))/dx, is also displayed for comparison.
  • Visual Representation: The applet utilizes visual elements to aid understanding:
  • A blue point (x1, y1) is user-defined via a slider.
  • A red point (x2, y2) is draggable by the mouse.
  • A black line (secant) connects the two points, extended in green.
  • Red and blue arrows represent the difference in ordinates (y2 - y1) and abscissa (x2 - x1), respectively.
  • A magenta point visualizes the difference quotient (y2 - y1)/(x2 - x1).
  • A beige line represents the analytic first derivative.
  • Interactive Exploration: Users can actively engage with the simulation to understand the limit process:
  • They can choose a model point (x1, y1) using a slider.
  • They can drag the second point (x2, y2) closer to the first point to observe how the secant line approaches the tangent line and how the difference quotient approaches the derivative.
  • The instructions encourage users to "Draw the red point closer and closer to the blue model point. In the limit of this process the secant turns into a tangent at the model point, and the difference quotient becomes a point on the derivative = the differential quotient at the model point."
  • Extension to First and Second Derivatives with a Linear Term: The simulation also demonstrates the first and second derivatives for a function with a sine and a linear term: y = sin(x) +c x.
  • The first derivative is given as: *dy/dt(x1) = x2➙x1 lim (y2 - y1 )/(x2 - x1) = cos(x1 )+c
  • The second derivative is given as: *d 2y/dt2(x1) = -sin(x1 )
  • This section highlights that the second derivative is independent of the linear term (c).
  • Parameter Control: The parameter c (the gradient of the linear term) can be adjusted by dragging a magenta colored rectangle, allowing users to observe its effect on the first derivative (vertical shift) while the second derivative remains unchanged.
  • Guided Experiments: The resource includes a set of suggested experiments to guide learning:
  • Observing the limiting process as the two points get closer.
  • Analyzing the derivative at extrema (maxima and minima).
  • Studying the behavior around inflection points.
  • Verifying that the limiting process consistently leads to the cosine function for y = sin(x).
  • Prompting users to think about designing algorithms for numerical calculation of first and second derivatives.
  • Target Audience: The resource is designed for learners of calculus and educators teaching the concept of derivatives. The inclusion of "Sample Learning Goals" and a section "For Teachers" further reinforces this.
  • Technical Implementation: The applet is built using JavaScript and HTML5, making it embeddable in web pages via an iframe. This ensures accessibility across different devices and platforms without requiring specific plugins.
  • Related Resources: The page provides links to a vast array of other interactive JavaScript simulation applets covering various topics in physics and mathematics, indicating a broader commitment to open educational resources in these fields. Examples include simulations on mechanics, electromagnetism, waves, quantum physics, and more advanced mathematical concepts like Fourier and Taylor series. The sheer number of resources (over 1193 articles) highlights the extensive nature of the project.
  • Licensing: The content is licensed under a Creative Commons Attribution-Share Alike 4.0 Singapore License, promoting open access and sharing. Commercial use of the underlying EasyJavaScriptSimulations Library requires a separate license.

Educational Value:

This simulation applet offers significant educational value by:

  • Visualizing an Abstract Concept: The derivative, often defined abstractly through limits, is made tangible through visual representation and interactive manipulation.
  • Facilitating Active Learning: Users are not passive recipients of information but actively engage in exploring the relationship between the difference quotient and the differential quotient.
  • Promoting Conceptual Understanding: By observing the limit process dynamically, learners can develop a deeper conceptual understanding of what the derivative represents.
  • Supporting Inquiry-Based Learning: The suggested experiments encourage students to explore different scenarios and draw their own conclusions.
  • Bridging Theory and Practice: The experiments prompting algorithmic design connect the theoretical concept of the derivative to its practical application in numerical methods.

Quotes:

  • "The transition from the difference quotient to the differential quotient (first derivative) is demonstrated for the sine function."
  • "Differential quotient dy/dt (x1 ) = Limit for x2→ x1 of (y2 - y1)/(x2 -x1 )"
  • "In the limit of this process the secant turns into a tangent at the model point, and the difference quotient becomes a point on the derivative = the differential quotient at the model point."

Conclusion:

The "Differential Quotient JavaScript Simulation Applet HTML5" is a valuable open educational resource for teaching and learning the fundamental concept of the derivative. Its interactive nature, clear visual representations, and guided experiments effectively illustrate the limit process and the relationship between the difference quotient and the differential quotient. The inclusion of extensions to the second derivative and prompts for algorithmic thinking further enhances its educational utility. Its accessibility through web browsers and the broader context of a rich collection of similar simulations make it a significant contribution to open source physics and mathematics education.

 

 

Quiz

  1. What is the difference quotient formula for a function y = f(x) between two points (x1, y1) and (x2, y2)? Explain what this quotient represents geometrically in relation to the function's graph.
  2. According to the text, what is the differential quotient (first derivative) defined as in mathematical terms, particularly in relation to the difference quotient? How does this relate to the concept of a limit?
  3. Describe how the provided JavaScript simulation demonstrates the transition from the difference quotient to the differential quotient for the sine function, y = sin(x). Mention the roles of the different colored points and lines in the visualization.
  4. Explain how the simulation extends to illustrate the first and second derivatives for the function y = sin(x) + cx. What is significant about how the linear term (cx) affects the first and second derivatives?
  5. In the context of the simulation, what happens to the secant line and the magenta colored difference quotient point as the red point is drawn closer and closer to the blue model point? What does this visually represent?
  6. According to Experiment E2 for the sine function, what should you observe when the model point is moved near a maximum or minimum of the curve and the limiting process is performed? What does this imply about the derivative at these points?
  7. Experiment E5 asks how you would design an algorithm for the numerical calculation of the first derivative of an arbitrary, continuous function based on the simulation. Outline a basic approach for such an algorithm.
  8. How does the simulation allow you to adjust the gradient (c) of the linear term in the function y = sin(x) + cx, and what is the effect of this adjustment on the first and second derivatives as shown in the beige and blue lines?
  9. What does the beige line represent in both parts of the simulation (for y = sin(x) and y = sin(x) + cx)? What does its graphical relationship to the original function indicate?
  10. The text mentions that the difference quotient corresponds to the tangent of the angle between which two components at the model point? How does this relate the difference quotient to the slope of a line?

Answer Key

  1. The difference quotient is given by the formula (y2 - y1) / (x2 - x1) or Δy / Δx. Geometrically, this represents the slope of the secant line connecting the two points (x1, y1) and (x2, y2) on the graph of the function.
  2. The differential quotient (first derivative) dy/dt (x1) is defined as the limit of the difference quotient as x2 approaches x1: Limit for x2→ x1 of (y2 - y1) / (x2 - x1). This means the derivative represents the instantaneous rate of change of the function at a specific point, which is the slope of the tangent line at that point.
  3. For y = sin(x), the blue point (x1, y1) is fixed, and the red point (x2, y2) can be moved. The black line (prolonged in green) connects these points, representing the secant. The magenta point shows the difference quotient. As the red point approaches the blue point, the secant becomes the tangent, and the magenta point converges to a point on the beige line, which is the first derivative cos(x).
  4. For y = sin(x) + cx, the simulation again shows the secant (black/green line) and the difference quotient (magenta point). The first derivative, cos(x) + c, is shown as a beige line, and the second derivative, -sin(x), is shown as a blue line. The linear term cx shifts the first derivative vertically by c but does not affect the second derivative.
  5. As the red point is drawn closer to the blue model point, the black secant line pivots and approaches the tangent line at the blue model point. Simultaneously, the magenta colored difference quotient point moves along a path and ultimately lands on the beige line representing the first derivative at that point.
  6. Near a maximum or minimum of the sine function, as the red point approaches the blue model point, the secant line approaches a horizontal tangent line. This indicates that the derivative (the slope of the tangent line) is zero at these extremum points.
  7. A basic algorithm could involve taking a point x and calculating the difference quotient using another point x + h, where h is a very small number. The value of this difference quotient would approximate the first derivative at x. To improve accuracy, one could use progressively smaller values of h and observe the trend.
  8. The gradient c of the linear term can be adjusted by drawing the magenta colored rectangle in the simulation. Changing c shifts the beige line (the first derivative cos(x) + c) vertically upwards or downwards, parallel to the y-axis. The blue line (the second derivative -sin(x)) remains unchanged.
  9. The beige line represents the analytic first derivative of the function being displayed. For y = sin(x), it is cos(x), and for y = sin(x) + cx, it is cos(x) + c. The graphical relationship shows that the derivative function gives the slope of the tangent line to the original function at every point.
  10. The difference quotient corresponds to the tangent of the angle between the abscissa difference (x2 - x1, shown as a blue arrow) and the ordinate difference (y2 - y1, shown as a red arrow) at the model point. This tangent value is equivalent to the slope of the secant line connecting the two points.

Essay Format Questions

  1. Discuss the conceptual relationship between the difference quotient and the differential quotient. How does the idea of a limit bridge these two concepts, and why is this transition fundamental to calculus?
  2. Explain how the interactive simulation visually demonstrates the concept of the derivative. Analyze the roles of the various graphical elements (points, lines, colors) in conveying the idea of the limit of the difference quotient.
  3. Consider the experiment involving the function y = sin(x) + cx. Analyze how the linear term cx affects the first and second derivatives, both analytically and as observed in the simulation. What does this illustrate about the derivative operation?
  4. Based on your experience with the simulation, describe the process of approximating the derivative of a function numerically. What are the key considerations and potential limitations of such numerical methods?
  5. Reflect on the pedagogical value of using interactive simulations, like the one described, for teaching and learning calculus concepts such as derivatives. What are the advantages and potential disadvantages of this approach compared to traditional methods?

Glossary of Key Terms

  • Difference Quotient: The ratio of the change in the function's value (Δy) to the change in the independent variable (Δx) between two distinct points on the function's graph, given by (y2 - y1) / (x2 - x1). It represents the slope of the secant line connecting these two points.
  • Differential Quotient (First Derivative): The limit of the difference quotient as the change in the independent variable approaches zero. It represents the instantaneous rate of change of the function at a specific point and is geometrically the slope of the tangent line to the curve at that point. Notation: dy/dx or f'(x).
  • Limit: A value that a function or sequence "approaches" as the input or index approaches some value. In the context of derivatives, it's the value the difference quotient gets closer and closer to as the two points become infinitesimally close.
  • Secant Line: A straight line that intersects a curve at two or more points. Its slope is represented by the difference quotient.
  • Tangent Line: A straight line that touches a curve at a single point and has the same slope as the curve at that point. Its slope is given by the derivative.
  • Ordinate: The y-coordinate of a point in a Cartesian coordinate system, representing the vertical distance from the x-axis.
  • Abscissa: The x-coordinate of a point in a Cartesian coordinate system, representing the horizontal distance from the y-axis.
  • Analytic Derivative: The exact derivative of a function obtained through the rules of differentiation. In the simulation, these are represented by the beige and blue lines.
  • Linear Term: A term in an equation or function that has a variable raised to the power of one (e.g., cx). Its graph is a straight line.
  • Superimposed: Placed or laid over something else. In the context, the linear term is added to the sine function.
  • Gradient: The slope of a line. In y = sin(x) + cx, c represents the gradient of the linear term.
  • Extremum (Maximum or Minimum): A point on a function where it reaches a local or global peak (maximum) or valley (minimum) value. At these points, the first derivative is typically zero.
  • Inflection Point: A point on a curve at which the concavity changes from upwards to downwards or vice versa.
  • Algorithm: A step-by-step procedure for solving a problem or accomplishing a task, especially by a computer.
  • Numerical Calculation: Approximating a mathematical value using numerical methods, often involving discrete steps or iterations.

Sample Learning Goals

[text]

For Teachers

 

Instructions

Combo Box

Toggling the combo box will choose their respective functions
(sin(x) selected)


(sin(x)+x*c selected)
 

Draggable Points

 
Drag the two points to get the difference quotient.
 
 
The last magenta box in the far right can adjust the c of the equation

Toggling Full Screen

Double click anywhere to toggle full screen.

Research

[text]

Video

[text]

 Version:

Other Resources

[text]

quotient?

The difference quotient for a function, such as y = sin(x), between two points (x1, y1) and (x2, y2) is defined as (y2 - y1) / (x2 - x1). This represents the slope of the secant line connecting these two points. The differential quotient, also known as the first derivative, dy/dx (x1), is the limit of the difference quotient as x2 approaches x1. Geometrically, this means the secant line becomes the tangent line at the point (x1, y1), and the difference quotient becomes the slope of this tangent.

How does the provided simulation demonstrate the transition from the difference quotient to the differential quotient?

The JavaScript simulation allows users to interactively visualize this transition. A blue point (x1, y1) is fixed or can be moved with a slider. A red point (x2, y2) can be dragged closer and closer to the blue point. As the red point approaches the blue point, the black secant line connecting them pivots and approaches the tangent line at the blue point. Simultaneously, the magenta point, representing the value of the difference quotient, moves towards the beige line, which represents the analytic first derivative (the differential quotient). This visually demonstrates the limiting process.

What does the simulation show regarding the first and second derivatives of a function with a linear term, such as y = sin(x) + cx?

The simulation extends to demonstrate the first and second derivatives of a function with a superimposed linear term. It shows that the first derivative, dy/dx = cos(x) + c, is affected by the gradient 'c' of the linear term; changing 'c' shifts the first derivative curve vertically. However, the second derivative, d²y/dx² = -sin(x), remains independent of the linear term 'c'. This is visually represented by the beige line (first derivative) shifting when the magenta rectangle (controlling 'c') is adjusted, while the blue line (second derivative) stays unchanged.

How can the simulation be used to explore the properties of derivatives at different points on a curve?

The "Experiments" section suggests several ways to use the simulation for this purpose. By moving the blue model point (x1, y1) to different locations on the sine curve, such as near a maximum, minimum, or a point of steepest slope (inflexion point), users can observe how the secant line approaches the tangent line and how the difference quotient approaches the differential quotient in these specific contexts. Experimenting with various model points reinforces the understanding that the limiting process consistently yields a point on the derivative curve (cosine for the sine function).

What insights does the simulation offer for designing an algorithm for numerical differentiation?

By observing the limiting process where the difference (x2 - x1) approaches zero, the simulation suggests a fundamental idea for numerical differentiation. An algorithm could approximate the first derivative at a point by calculating the difference quotient using a second point very close to the first. Similarly, for the second derivative, one might consider a process involving three closely spaced points or by numerically differentiating the first derivative approximation. The experiments explicitly ask users to consider how they would design such algorithms after interacting with the simulation.

What is the purpose of the draggable points and the magenta box in the simulation interface?

The draggable points (blue and red) are crucial for visualizing the difference quotient and the limiting process. The blue point (x1, y1) is often set by a slider, while the red point (x2, y2) can be dragged by the user to vary the interval (x2 - x1) and observe the secant line. For the function y = sin(x) + cx, the last magenta box on the far right allows the user to adjust the value of 'c', the gradient of the linear term, and see its effect on the first derivative while the second derivative remains unchanged.

What are some of the other interactive simulations available on the Open Educational Resources / Open Source Physics @ Singapore platform?

The provided text includes an extensive list of other JavaScript HTML5 applet simulations available on the platform. These cover a wide range of topics in mathematics and physics, including mechanics (e.g., projectile motion, oscillations), electromagnetism (e.g., magnetic fields, circuits), waves (e.g., reflection, superposition), thermodynamics (e.g., heat conduction), quantum physics (e.g., bound states), and even games and interactive tools for primary and junior college levels.

Under what licensing terms are the resources on this platform available?

The contents of the Open Educational Resources / Open Source Physics @ Singapore platform are licensed under the Creative Commons Attribution-Share Alike 4.0 Singapore License. For commercial use of the EasyJavaScriptSimulations Library, users are directed to read the specific license terms available at the provided URL and to contact fem@um.es directly for inquiries.

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Parent Category: Pure Mathematics
Category: 5 Calculus
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