2011 Calculus BC free response #6b | AP Calculus BC | Khan Academy

Khan Academy
14 Sept 201109:59
EducationalLearning
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TLDRThe video script discusses the Taylor series for the cosine function about x equals 0, outlining the first four non-zero terms. It then combines this with the sine of x squared series to approximate the function f(x) = sine of x squared plus cosine of x. The process involves calculating derivatives, evaluating them at x=0, and constructing the polynomial approximation, resulting in a detailed explanation of the mathematical concepts and steps involved.

Takeaways
  • 📌 The Taylor series for cosine of x about x equals 0 is derived using the Taylor series formula centered at x=0.
  • 🔢 The first four non-zero terms of the cosine Taylor series are: 1 (constant term), 0 (first derivative at 0), -x^2/2!, and x^4/4!.
  • 🌀 The derivatives of cosine function cycle through 0, -1, 1, and then repeat, which influences the coefficients of the Taylor series terms.
  • 📈 The Taylor series for sine of x squared is used in conjunction with the cosine series to approximate the function f(x) = sine of x squared plus cosine of x.
  • 🔍 The first four non-zero terms of the Taylor series for f(x) are determined by combining the terms from both the cosine and sine of x squared series.
  • 👉 The lowest degree term in the approximation of f(x) is 1, which corresponds to the constant term from the cosine series.
  • 📊 The second term is 1/2 * x^2, which results from combining the x^2 terms of both the cosine and sine of x squared series, accounting for their coefficients.
  • 🔑 The third term is x^4/4!, taken from the cosine series, as there are no third-degree terms in the sine of x squared series.
  • 🧩 The fourth term is a combination of the sixth-degree terms from both series, resulting in -121x^6/6! after simplification.
  • 🎯 The process of deriving the Taylor series for f(x) demonstrates the application of calculus concepts and series approximation techniques.
Q & A

  • What is the Taylor series centered at x equals 0 for the cosine function?

    -The Taylor series for cosine of x centered at x equals 0 is given by the sum of the terms: 1 (the function evaluated at 0), -x^2/2!, 0 (for the third derivative at 0), and x^4/4!.

  • What is the first non-zero term in the Taylor series for cosine of x?

    -The first non-zero term in the Taylor series for cosine of x is the constant term, which is 1, since the cosine function evaluated at 0 is 1.

  • What is the second non-zero term in the Taylor series for cosine of x?

    -The second non-zero term in the Taylor series for cosine of x is -x^2/2!. This term arises from the second derivative of the cosine function, which is -sine of x, evaluated at 0.

  • How is the Taylor series for sine of x squared derived in the script?

    -The Taylor series for sine of x squared is derived by squaring the Taylor series for sine of x. It includes terms like x^2, -x^6/3!, x^10/5!, -x^14/7!, and so on.

  • What is the Taylor series for the function f(x) = sine of x squared plus cosine of x?

    -The Taylor series for the function f(x) = sine of x squared plus cosine of x, centered at x equals 0, is approximately 1 + 1/2 x^2 + x^4/4! - 121x^6/6!.

  • What is the significance of the Taylor series in calculus?

    -The Taylor series is significant in calculus as it provides a way to approximate functions as polynomials. This is useful for computations, analysis, and understanding the behavior of functions around a given point.

  • How does the Taylor series for cosine of x relate to the series for sine of x squared?

    -The Taylor series for cosine of x provides the coefficients for the even-powered terms in the Taylor series for sine of x squared plus cosine of x. This is because the cosine function contributes terms with even powers of x when expanded in a Taylor series.

  • What is the third non-zero term in the Taylor series for the function f(x) = sine of x squared plus cosine of x?

    -The third non-zero term in the Taylor series for the function f(x) = sine of x squared plus cosine of x is x^4/4!, which comes from the Taylor series of cosine x.

  • What is the fourth non-zero term in the Taylor series for the function f(x) = sine of x squared plus cosine of x?

    -The fourth non-zero term in the Taylor series for the function f(x) = sine of x squared plus cosine of x is -121x^6/6!, which results from combining the terms of sine of x squared and cosine of x.

  • How does the process of finding the Taylor series for a function involve derivatives?

    -The process of finding the Taylor series for a function involves taking derivatives of the function and evaluating them at the point around which the series is centered. Each derivative provides the coefficient for a term in the series.

  • What is the role of factorial in the Taylor series?

    -Factorials are used in the denominators of the terms in the Taylor series to control the growth of the terms as the power of x increases. They are a mathematical way of ensuring that higher powers of x have smaller coefficients.

Outlines
00:00
📚 Derivation of Taylor Series for Cosine Function

This paragraph discusses the process of deriving the first four non-zero terms of the Taylor series for the cosine function about x equals 0. The explanation begins by setting up the Taylor series formula with x centered at 0. It then proceeds to calculate the derivatives of the cosine function to find the coefficients for the series. The first derivative at 0 is found to be 0 (since sine of 0 is 0), the second derivative is -1 (negative cosine of 0 which is 1), the third derivative is 0 (since sine of 0 is still 0), and the fourth derivative returns to the function value at 0, which is 1. The paragraph then constructs the Taylor series for cosine of x using these derivatives, highlighting the first four non-zero terms: 1, 0, -x^2/2!, and 1x^4/4!. The explanation is clear and methodical, ensuring a solid understanding of the Taylor series approximation for the cosine function.

05:05
📈 Approximation of Function f(x) using Taylor Series

The second paragraph focuses on approximating a given function f(x) = sin(x^2) + cos(x) using the Taylor series derived in the previous section and the sine squared series from part A. It begins by identifying the lowest degree terms from both the cosine and sine squared series to construct the approximation of f(x). The paragraph then systematically combines like terms, taking into account the coefficients and the corresponding powers of x, to determine the first four non-zero terms of the Taylor series for f(x). The result is an approximation that includes a constant term (1), a term with x^2 (-1/2 x^2), a fourth-degree term (x^4/4!), and a sixth-degree term (-121x^6/6!). The explanation is detailed, walking through each step of the approximation process and providing a clear understanding of how the Taylor series is applied to the function f(x).

Mindmap
Keywords
💡Taylor series
The Taylor series is a mathematical representation that approximates a function as an infinite sum of terms calculated from the values of its derivatives at a single point. In the video, the Taylor series is used to expand the functions cosine of x and sine of x squared about x equals 0, allowing for approximations of these functions without complex calculations.
💡Derivatives
Derivatives are mathematical operations that find the rate of change or slope of a function at a particular point. In the context of the video, derivatives are used to determine the coefficients for the Taylor series expansion of the cosine function, with each derivative evaluated at x=0 contributing to a term in the series.
💡Cosine function
The cosine function is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of its adjacent side length to its hypotenuse. In the video, the cosine function is approximated using its Taylor series, which is a polynomial representation that closely mirrors the behavior of the function near x=0.
💡Sine function
The sine function is another fundamental trigonometric function that relates the angle of a right triangle to the ratio of its opposite side length to its hypotenuse. In the video, the sine function is squared and then expanded using a Taylor series, which is used in combination with the cosine series to approximate the function sine of x squared plus cosine of x.
💡Trigonometric functions
Trigonometric functions are mathematical functions that relate angles to the ratios of the sides of a right triangle. In the video, both cosine and sine functions are discussed, and they are used to construct a Taylor series for the function sine of x squared plus cosine of x.
💡Series expansion
A series expansion is a way to represent a function as the sum of a sequence of simpler functions, often polynomials. In mathematics, this is a common technique for approximating complex functions based on their local behavior. The video focuses on the Taylor series expansion, which is a specific type of series expansion centered around a point.
💡Approximation
Approximation in mathematics refers to the process of finding a simpler function that is close in value to a more complex function, often for practical or computational purposes. In the video, the Taylor series is used to approximate trigonometric functions, allowing for easier computation and analysis.
💡Polynomial
A polynomial is an algebraic expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. In the video, polynomials are used as the building blocks for the Taylor series, which approximates functions as the sum of polynomial terms.
💡Factorial
The factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. Factorials are used in mathematics, particularly in combinatorics and the study of permutations and combinations. In the video, factorials appear in the denominators of terms in the Taylor series, indicating the number of possible permutations or combinations of objects in a set.
💡x equals 0
The point x equals 0, or simply x=0, is the central point around which the Taylor series expansions in the video are centered. It is the value at which all functions and their derivatives are evaluated to generate the terms of the series.
💡Non-zero terms
Non-zero terms are those terms in a mathematical series that do not equal zero. In the context of the Taylor series, non-zero terms are the coefficients and variable terms that contribute to the overall sum and shape of the series.
Highlights

Introduction to finding the first four non-zero terms of the Taylor series for cosine of x about x equals 0.

Identification of h(x) as cosine of x and calculation of h(0) equals 1.

First derivative of h(x) leading to h prime of 0 equals 0.

Second derivative showing a cycle with h prime prime of 0 equals negative 1.

Observation of the cyclic nature of the derivatives of cosine, simplifying further calculations.

Derivation of the first four non-zero terms of the cosine Taylor series: 1, -x^2/2!, +x^4/4!, -x^6/6!.

Transition to combining the derived series with the series for sine of x squared from part A.

Formulation of the function f(x) = sine of x squared plus cosine of x.

Analysis to find the first four non-zero terms of the Taylor series for f(x).

Simplification strategy by combining like terms from both series.

Identification of terms of different degrees and their contributions to the series.

Detailed breakdown of the addition and simplification process for the terms of the series.

Explanation of the method for combining terms of the sixth degree from both series.

Final expression of the first four non-zero terms for f(x) as a Taylor series approximation.

Conclusion with the simplified Taylor series for f(x), showcasing the applied mathematical techniques.

Transcripts

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