Calculus - Finding the derivative at a point using a Ti-83 or 84 calcululator

MySecretMathTutor
15 Sept 201305:26
EducationalLearning
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TLDRThis tutorial video guides viewers on how to find the derivative of a function at a specific point using a TI-84 or TI-83 calculator. The host begins by explaining the importance of the derivative as the slope of the tangent line and demonstrates how to input a cubic function into the calculator. They then show how to use the calculator's built-in function to determine the derivative at certain points, such as positive four and negative three, highlighting the expected positive and negative values respectively. Additionally, the video touches on estimating where the derivative equals zero by using the calculator's trace and derivative functions, and suggests using the maximum or minimum function to find exact points where the derivative is zero. The video concludes by directing viewers to the website for more math-related content.

Takeaways
  • 📚 The video is a tutorial on finding the derivative of a function at a specific point using a calculator.
  • 🔢 The presenter uses a TI-84 calculator, but the process can also be applied to a TI-83 calculator.
  • 📈 The purpose of finding a derivative at a point is to determine the slope of the tangent line to the function at that point.
  • 💡 The TI calculator has a built-in function to calculate the slope of the tangent line, which is accessed through the 'calc' menu.
  • 📝 The first step is to enter the function into the calculator's 'y=' screen.
  • 📐 Ensure the calculator's window settings are appropriate for viewing the function, typically from -10 to 10 for both x and y axes.
  • 📊 After graphing the function, use the 'trace' button to navigate to specific points on the graph.
  • 🔍 To find the derivative at a specific x-value, select 'dy/dx' from the calculator's options and input the x-value.
  • 📌 The calculator provides an estimated numerical value for the derivative at the given point.
  • 🔎 If finding where the derivative equals zero, use the calculator's maximum or minimum function to estimate the x-value.
  • 🔑 The video provides a step-by-step guide on using the calculator to find derivatives and estimate points of interest, such as local maxima and minima.
Q & A

  • What is the main topic of the video?

    -The main topic of the video is finding the derivative of a function at a specific point using a calculator, specifically the TI-84 or TI-83.

  • Why is it important to find the derivative of a function at a single point?

    -Finding the derivative at a single point is important because it gives us the slope of the tangent line to the function at that point, which can be useful in various mathematical and real-world applications.

  • What is the built-in function on the TI-84 calculator that helps find the slope of the tangent line?

    -The built-in function on the TI-84 calculator that helps find the slope of the tangent line is the 'dy/dx' function.

  • How do you input a function into the calculator as per the video?

    -You input a function into the calculator by going to the 'y=' screen and typing in the function, for example, '1 8x + 7x + 1 and x - 4'.

  • What should the window settings be for the X and Y axes when graphing the function on the calculator?

    -The window settings should be from negative 10 to 10 for both the X and Y axes.

  • How do you find the derivative of a function at a specific value using the calculator?

    -To find the derivative at a specific value, you press the '2nd' button followed by the 'trace' button, select the 'dy/dx' option, and then enter the value at which you want to find the derivative.

  • What does the calculator estimate when it provides a derivative value?

    -The calculator estimates the limit that defines the derivative at the given point, which is the slope of the tangent line to the function at that point.

  • What does the video suggest when the derivative is equal to zero?

    -When the derivative is equal to zero, it suggests that the function may have a local maximum or minimum at that point.

  • How can you estimate where the derivative is equal to zero if the calculator doesn't have a direct function for it?

    -You can estimate where the derivative is equal to zero by observing the graph and using the 'dy/dx' function to check the derivative at points around the suspected location.

  • What additional tools can be used to find the exact value of where the derivative equals zero?

    -Additional tools such as finding the maximum or minimum of the function can be used to find the exact value of where the derivative equals zero.

  • Where can viewers find more math videos similar to this one?

    -Viewers can find more math videos by visiting the website mysecretmathtutor.com.

Outlines
00:00
📚 Finding Derivatives with a Calculator

This paragraph introduces a tutorial on using calculators, specifically the TI-84 or TI-83, to find the derivative of a function at a specific point. The focus is on understanding the slope of the tangent line to the function's graph at that point. The video demonstrates how to input a cubic function into the calculator, check the window settings, and graph the function. It then guides viewers on how to use the calculator's built-in function to find the derivative at a given point, such as x=4, and how to interpret the result. The process is repeated for another point, x=-3, and the video also touches on how to estimate where the derivative might be zero by observing the graph and using the calculator's derivative function to check at specific points.

05:02
🔍 Estimating Derivative Zero Points and Additional Tools

The second paragraph continues the tutorial by discussing how to estimate where the derivative of a function equals zero, which typically corresponds to local maxima or minima. It explains that while calculators can't directly find where the derivative is zero, they can be used to estimate these points by observing the graph and checking the derivative at nearby points. The video provides an example of how to use the calculator's 'calc' function to estimate a local minimum, which involves setting bounds and a guess for the calculator to work with. It concludes by suggesting that for more precise calculations, other mathematical tools or methods may be necessary and invites viewers to visit the website for more math-related content.

Mindmap
Keywords
💡Derivative
A derivative in calculus is a measure of how a function changes as its input changes. In the context of the video, the derivative represents the slope of the tangent line to the curve of a function at a specific point, which is crucial for understanding the rate of change of the function. The script discusses finding the derivative of a function at a point using a calculator, specifically for a cubic function.
💡Calculator
A calculator is an electronic device used to perform arithmetic operations and more complex calculations. In the video, the TI-84 calculator is used to find the derivative of a function at specific points. The script demonstrates how to input a function into the calculator and use its built-in functions to calculate the derivative, showing the practical application of technology in mathematical problem-solving.
💡Tangent Line
A tangent line is a straight line that touches a curve at a single point without crossing it. In the video, the concept of a tangent line is used to describe the instantaneous rate of change of a function at a particular point, which is found by calculating the derivative. The script emphasizes the importance of the tangent line's slope in determining whether the function is increasing or decreasing at that point.
💡Slope
Slope is a measure of the steepness of a line, indicating the ratio of the vertical change to the horizontal change between two points on the line. In the video, the slope of the tangent line is synonymous with the derivative of the function at a specific point. The script uses the slope to determine whether the function is increasing (positive slope) or decreasing (negative slope).
💡Function
In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. The video focuses on finding the derivative of a specific function, which is a cubic polynomial represented as '1 8x^3 + 7x^2 + x - 4'. The function is the core element in the process of finding its rate of change at different points.
💡Cubic Function
A cubic function is a polynomial of degree three. It has the general form f(x) = ax^3 + bx^2 + cx + d, where a, b, c, and d are constants and a ≠ 0. In the video, the cubic function '1 8x^3 + 7x^2 + x - 4' is used as an example to demonstrate how to find its derivative at specific points, showcasing the properties and behavior of cubic functions.
💡Graph
In the context of the video, a graph is a visual representation of a function, showing the relationship between the input and output values. The script instructs viewers to graph the function on the calculator to visually identify points of interest, such as where the function may be increasing or decreasing, which is essential for understanding the behavior of the function.
💡Estimating
Estimating in mathematics involves making an approximate calculation or judgment about a value or quantity. The video script mentions that the calculator uses a numerical method to estimate the derivative at a point. This process is demonstrated when finding the derivative at x = 4 and x = -3, where the calculator provides an approximate value for the slope of the tangent line.
💡Numerical Method
A numerical method is a mathematical technique used to find approximate values of mathematical functions, especially where exact solutions are not feasible. In the video, the calculator's process of finding the derivative is described as a numerical method, which involves estimating the limit to determine the slope of the tangent line at a specific point on the function.
💡Maximum and Minimum
In calculus, a maximum or minimum refers to a point on the graph of a function where the function reaches its highest or lowest value, respectively. The video script discusses using the calculator to find local maxima and minima of the function, which are points where the derivative is zero. This is demonstrated when the tutor attempts to estimate where the derivative equals zero by observing the graph and using the calculator's maximum/minimum function.
Highlights

Introduction to a tutorial on finding the derivative of a function at a point using calculators.

Use of the TI-84 calculator for demonstration, with compatibility for TI-83 as well.

Explanation of the concept of a derivative as the slope of a tangent line at a single point.

Built-in function on the calculator to find the slope of a tangent line.

Step-by-step guide to input a function into the calculator.

Setting the window for the graph on the calculator.

Graphing the function to visualize the curve.

Finding the derivative at a specific point using the TRACE button.

Selecting the dy/dx option to calculate the derivative.

Demonstration of finding the derivative at x = 4 and getting a positive value.

Estimation of the derivative value using numerical methods.

Finding the derivative at x = -3 and expecting a negative value.

Using the calculator to estimate where the derivative equals zero.

Estimating the local maximum or minimum by observing the graph.

Using the second calc feature to find where the derivative is approximately zero.

Explanation of the limitation of calculators in finding exact points where the derivative equals zero.

Using the maximum or minimum function to find more accurate estimates.

Final advice on using the calculator's derivative function for direct computation and other tools for reverse problems.

Invitation to visit the website for more math videos.

Transcripts

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