Definition of the Derivative | MIT 18.01SC Single Variable Calculus, Fall 2010

MIT OpenCourseWare
7 Jan 201112:27
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TLDRIn this recitation, the professor guides students through graphing the function f(x) = 1 / (1 + x^2) and computing its derivative from the limit definition. The lesson begins with plotting points and observing the function's even symmetry and behavior as x approaches infinity. The derivative is then derived by manipulating the difference quotient, leading to the result f'(x) = -2x / (1 + x^2)^2. The professor also validates the derivative by relating it to the graph's properties, such as the horizontal tangent at x=0 and the function's odd symmetry.

Takeaways
  • 📚 The lecture focuses on computing derivatives from the limit definition of a derivative and includes graphing a function.
  • 🔍 The function defined is f(x) = 1 / (1 + x^2), and students are encouraged to graph it and compute its derivative.
  • 📈 To graph the function, the professor suggests plotting points for x = 0, x = 1, and x = 2 to get an initial idea of the curve.
  • 📉 The function f(x) is an even function, symmetric across the y-axis, which simplifies the graphing process.
  • 📊 The function's value is always positive and diminishes as x approaches infinity in both positive and negative directions.
  • 🔑 The maximum value of the function is at x = 0, where the denominator 1 + x^2 is at its minimum.
  • 🧩 To compute the derivative, the limit definition is used, which involves a difference quotient that simplifies to a 0/0 form.
  • 📝 The algebraic manipulation involves combining fractions and expanding binomials to simplify the expression for the derivative.
  • 🔍 After simplification, the derivative is found to be f'(x) = -2x / (1 + x^2)^2, which is an odd function.
  • 📉 The derivative's graph has a horizontal tangent at x = 0, and its sign changes at this point, which is consistent with the original function's graph.
  • 🔄 The symmetry of the derivative function confirms the correctness of the derivative computation by reflecting the original function's behavior.
Q & A

  • What is the function defined in the script?

    -The function defined in the script is f(x) = 1 / (1 + x^2).

  • What is the first step in graphing the function f(x) as described in the script?

    -The first step in graphing the function f(x) is to plot a few points by substituting values of x and calculating the corresponding f(x) values.

  • What is the value of the function f(x) when x equals 0?

    -When x equals 0, the value of the function f(x) is 1, because f(0) = 1 / (1 + 0^2) = 1.

  • How does the script describe the symmetry of the function f(x)?

    -The script describes the function f(x) as an even function, which means it has symmetry across the y-axis because f(-x) = f(x) due to the property of x^2.

  • What is the behavior of the function f(x) as x approaches positive or negative infinity?

    -As x approaches positive or negative infinity, the value of the function f(x) approaches 0, because the denominator 1 + x^2 becomes very large, making the fraction smaller.

  • What is the maximum value of the function f(x), and at what point does it occur?

    -The maximum value of the function f(x) is 1, and it occurs at x = 0, because that's when the denominator 1 + x^2 is at its minimum.

  • How does the script suggest simplifying the difference quotient for finding the derivative of f(x)?

    -The script suggests simplifying the difference quotient by combining the terms over a common denominator and then expanding and canceling terms to isolate a factor of delta x.

  • What is the final expression for the derivative f'(x) of the function f(x) according to the script?

    -The final expression for the derivative f'(x) is -2x / (1 + x^2)^2, obtained after canceling out the delta x and simplifying the limit as delta x approaches 0.

  • How does the script verify the derivative by looking at the graph of the function?

    -The script verifies the derivative by noting that the derivative is 0 when x is 0, which corresponds to the horizontal tangent line on the graph at that point, and by checking the sign of the derivative to match the increasing and decreasing behavior of the function.

  • What property of the derivative f'(x) does the script mention, and how does it relate to the function f(x)?

    -The script mentions that the derivative f'(x) is an odd function, meaning that f'(-x) = -f'(x), which corresponds to the symmetry of the function's graph with respect to the origin.

Outlines
00:00
📚 Introduction to Derivative Computation and Graphing

The professor begins the recitation by welcoming students and introducing the task of computing derivatives from the limit definition of a derivative. The function in question is f(x) = 1 / (1 + x^2). Students are encouraged to graph the function and compute its derivative, f'(x), before the professor proceeds to demonstrate the process. The initial approach involves plotting points for x = 0, 1, and 2 to get a basic understanding of the function's behavior. The function's even nature is highlighted, and its symmetry across the y-axis is discussed. The professor also explains how the function's value diminishes as x moves away from zero due to the increasing denominator, while the numerator remains constant.

05:01
📈 Analyzing the Function's Graph and Properties

Continuing from the previous discussion, the professor delves deeper into the function's graphical representation. The function's even nature is reiterated, and the points are mirrored across the y-axis to complete the graph. The function's positivity and its behavior as x approaches infinity are discussed, with the function value approaching zero but never reaching it. The maximum value at x = 0 is identified, and the function's flattening out as x increases is noted. The professor then transitions to the process of computing the derivative, emphasizing the limit definition and the need for algebraic manipulation to evaluate it.

10:04
🔍 Derivative Computation Using Limit Definition

The professor embarks on the derivative computation process for f(x) = 1 / (1 + x^2). The limit definition of the derivative is introduced, and the difference quotient is set up. The challenge of evaluating the limit as delta x approaches zero is addressed, with the professor highlighting the need for algebraic manipulation to simplify the expression. The process involves combining the fractions over a common denominator and expanding the binomials in the numerator. Through cancellation and simplification, the expression is reduced to a form where delta x can be factored out and canceled with the delta x in the denominator, leading to a solvable limit expression for the derivative.

Mindmap
Keywords
💡Derivative
A derivative in calculus is a measure of how a function changes as its input changes. It is the slope of the tangent line to the function's graph at a certain point. In the video, the professor is teaching students how to compute derivatives of a function from the limit definition, which is foundational for understanding the concept of rates of change.
💡Limit Definition
The limit definition of a derivative is a formal way to express the derivative using the concept of limits in mathematics. It is defined as the limit of the difference quotient as the change in the input (denoted as delta x) approaches zero. The professor uses this definition to guide students through the process of finding the derivative of the given function.
💡Function
A function in mathematics 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. In the script, the function f(x) = 1 / (1 + x^2) is the main focus, and understanding its properties and behavior is central to the lesson.
💡Graphing
Graphing is the process of plotting a function's values on a graph to visualize its behavior. The professor instructs students to graph the function f(x) to understand its shape and key features, such as its maximum value and symmetry.
💡Even Function
An even function is a function that satisfies the condition f(-x) = f(x) for all x in its domain. The professor notes that the function f(x) is even, which implies that its graph is symmetric with respect to the y-axis.
💡Symmetry
Symmetry in the context of functions refers to the property of a function's graph that allows it to be mirrored across an axis. The professor discusses the symmetry of the function f(x) across the y-axis, which is a key aspect of its graphical representation.
💡Difference Quotient
The difference quotient is an expression used to estimate the slope of the secant line between two points on a function's graph, and it is the basis for the limit definition of a derivative. The professor uses the difference quotient to set up the process of finding the derivative of the function.
💡Algebraic Manipulation
Algebraic manipulation refers to the process of rearranging and combining terms in an algebraic expression to simplify it or to make it more recognizable for further analysis. The professor demonstrates this by manipulating the difference quotient to find a form that can be evaluated as delta x approaches zero.
💡Horizontal Tangent Line
A horizontal tangent line to a curve at a point indicates that the derivative of the function at that point is zero, meaning the slope of the curve is horizontal. The professor mentions that the function has a horizontal tangent line at x = 0, which is a key feature derived from the function's derivative.
💡Odd Function
An odd function is a function that satisfies the condition f(-x) = -f(x) for all x in its domain. The professor concludes by noting that the derivative of the original function is an odd function, which is evident from its changing sign with the change in the sign of x.
💡Tangent Line
A tangent line to a curve at a point is a straight line that touches the curve at that point without crossing it. The concept is crucial in understanding derivatives, as the slope of the tangent line at any point on the curve is given by the derivative of the function at that point. The professor uses the concept of tangent lines to explain the behavior of the function and its derivative.
Highlights

Introduction to the recitation session focusing on computing derivatives from the limit definition.

Definition of the function f(x) as 1 over (1 plus x squared).

Instruction to graph the function y equals f(x) and compute its derivative.

Graphing the function by plotting points at x = 0, x = 1, and x = 2.

Observation that the function is even and symmetric across the y-axis.

Analysis of the function's behavior as x approaches infinity, tending towards zero.

Identification of the function's maximum value at x = 0.

Explanation of the process to compute the derivative using the limit definition.

Description of the difference quotient and its algebraic manipulation.

Strategy to simplify the limit expression by factoring out delta x.

Expansion and simplification of the numerator to facilitate cancellation with delta x.

Final computation of the derivative as the limit of a simplified expression.

Derivative result of -2x over (1 plus x squared) squared.

Verification of the derivative's properties by analyzing the graph.

Observation that the derivative is zero at x = 0 and changes sign thereafter.

Identification of the derivative as an odd function with respect to x.

Double-checking the work by correlating the derivative's properties with the graph's characteristics.

Transcripts

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Graphing TechniquesDerivative CalculationMath RecitationFunction AnalysisLimit DefinitionAlgebra ManipulationEven FunctionPositive ValuesHorizontal TangentEducational Content