Worked example: Maclaurin polynomial | Series | AP Calculus BC | Khan Academy

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17 Dec 201604:43

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TLDRIn this instructional video, the concept of finding the second-degree Maclaurin polynomial for the function f(x) = 1/√(x+1) is explored. The instructor guides viewers through the process of deriving the polynomial by evaluating the function and its first and second derivatives at zero. The final result is a polynomial, p(x) = 1 - 1/2x + 3/8x², which serves as an approximation for f(x) when x is near zero, providing a clear and concise explanation of the Maclaurin polynomial's application.

Takeaways

📚 The script is a tutorial on finding the second degree Maclaurin polynomial for a given function.
🔍 The function in question is f(x) = 1 / √(x + 1), and the goal is to determine its Maclaurin polynomial.
📘 A Maclaurin polynomial is defined as a Taylor polynomial centered at zero, which is a polynomial approximation of a function near zero.
🔑 The form of the second degree Maclaurin polynomial is given by p(x) = f(0) + f'(0)x + f''(0)/2 * x^2.
📈 The first step is to evaluate the function and its derivatives at zero to find the necessary coefficients for the polynomial.
🧩 f(0) is calculated as 1 / √1, which simplifies to 1, since the principal root of 1 is positive.
🔄 The first derivative, f'(x), is found using the chain rule and power rule, resulting in f'(0) = -1/2.
🔄🔄 The second derivative, f''(x), is also calculated using the chain rule and power rule, yielding f''(0) = 3/4.
🔢 The coefficients for the Maclaurin polynomial are then substituted into the formula to obtain the polynomial: p(x) = 1 - 1/2x + 3/8x^2.
📝 The final result is the second degree Maclaurin polynomial for the function f(x), which can be used to approximate the function's value near zero.

Q & A

What is a Maclaurin polynomial?
-A Maclaurin polynomial is a Taylor polynomial centered at zero. It is used to approximate a function near zero by expanding it in a power series.
What is the form of the second degree Maclaurin polynomial for a function f(x)?
-The second degree Maclaurin polynomial for a function f(x) is given by p(x) = f(0) + f'(0)x + (f''(0)/2)x^2.
What is the function f(x) in the given script?
-The function f(x) is defined as f(x) = 1/√(x + 1), which can also be written as (x + 1)^(-1/2).
How do you find the Maclaurin polynomial for a function?
-To find the Maclaurin polynomial, you need to evaluate the function and its derivatives at x = 0 and then plug these values into the polynomial form.
What is the value of f(0) for the given function?
-The value of f(0) is 1, as f(0) = 1/√(0 + 1) = 1.
How is the first derivative of f(x) calculated?
-The first derivative of f(x), denoted as f'(x), is calculated using the chain rule and power rule. It is f'(x) = -1/2 * (x + 1)^(-3/2).
What is the value of f'(0) for the given function?
-The value of f'(0) is -1/2, as f'(0) = -1/2 * (0 + 1)^(-3/2) = -1/2.
How is the second derivative of f(x) calculated?
-The second derivative of f(x), denoted as f''(x), is calculated by differentiating f'(x) again, using the power rule and chain rule.
What is the value of f''(0) for the given function?
-The value of f''(0) is 3/4, as f''(0) = 3/4 * (0 + 1)^(-5/2) = 3/4.
What is the final form of the second degree Maclaurin polynomial for the given function f(x)?
-The final form of the second degree Maclaurin polynomial for the given function is p(x) = 1 - (1/2)x + (3/8)x^2.

Outlines

00:00

🔢 Understanding Maclaurin Polynomials

In this section, the instructor introduces the concept of the Maclaurin polynomial. A Maclaurin polynomial is a special case of a Taylor polynomial centered at zero. The main objective is to find the second-degree Maclaurin polynomial for a given function, f(x) = 1/(√(x+1)). The instructor highlights that the polynomial will be in the form: P(x) = f(0) + f'(0)x + (f''(0)/2)x², emphasizing that for this exercise, we only need terms up to the second degree.

📊 Calculating the Function at Zero

The focus shifts to calculating f(0), the first term of the Maclaurin polynomial. The instructor substitutes x = 0 into the function f(x) = 1/(√(x+1)), simplifying it to f(0) = 1/(√(1)), which equals 1. This step establishes the first component of the polynomial, showing that the constant term f(0) is simply 1.

✏️ Deriving the First Derivative

Here, the instructor derives the first derivative, f'(x), using the chain rule and power rule. By expressing the function as (x+1)^(-1/2), the derivative becomes f'(x) = -1/2 * (x+1)^(-3/2). Evaluating this at x = 0 gives f'(0) = -1/2, indicating that the coefficient of x in the polynomial is -1/2. This calculation shows the role of the derivative in shaping the polynomial.

🔍 Calculating the Second Derivative

The instructor explains the process of finding the second derivative, f''(x), building on the previous step. Using the power rule again, f''(x) is determined as (3/4) * (x+1)^(-5/2). When x = 0, this results in f''(0) = 3/4. The term (f''(0)/2)x² thus becomes (3/8)x², adding the quadratic component to the polynomial. This section emphasizes the importance of higher-order derivatives in obtaining a more accurate polynomial approximation.

📝 Formulating the Second-Degree Maclaurin Polynomial

The instructor combines the calculated values to construct the second-degree Maclaurin polynomial: P(x) = 1 - (1/2)x + (3/8)x². This polynomial serves as an approximation for the original function f(x) near x = 0. The instructor concludes by reiterating the usefulness of Maclaurin polynomials in approximating functions, especially for values close to the center point, zero.

Mindmap

Keywords

💡Maclaurin Polynomial

A Maclaurin Polynomial is a type of Taylor series expansion that is centered at zero. It is used to approximate functions around the point of zero. In the video, the instructor is focusing on finding the second degree Maclaurin polynomial for a given function, which is a specific case of the Taylor series. This concept is central to the video's theme of approximating functions using polynomials.

💡Taylor Polynomial

A Taylor Polynomial is a mathematical tool used to approximate a function by a polynomial of a certain degree. It is named after the mathematician Brook Taylor. The video script discusses the Maclaurin polynomial as a specific case of the Taylor polynomial, which is centered at zero. The concept is integral to the video's lesson on approximating functions using polynomials.

💡Second Degree

In the context of the video, 'second degree' refers to the second power in a polynomial, which is the highest power of the variable in the polynomial. The instructor is calculating the second degree Maclaurin polynomial, which includes terms up to x squared. This is a key aspect of the video's demonstration of polynomial approximation.

💡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 script discusses finding the Maclaurin polynomial for a specific function, f(x), which is defined as one over the square root of x plus one.

💡Derivative

A derivative in calculus represents the rate at which a function changes with respect to one of its variables. The video script involves calculating the first and second derivatives of the function f(x) to determine the coefficients for the Maclaurin polynomial. Derivatives are essential in understanding how the function behaves and in constructing the polynomial approximation.

💡Chain Rule

The Chain Rule is a fundamental theorem in calculus for finding the derivative of a composite function. In the video, the instructor uses the chain rule to find the first derivative of the function f(x), which is expressed as one over the square root of (x + 1), or equivalently, as (x + 1) to the power of -1/2.

💡Power Rule

The Power Rule is a basic principle in calculus for finding the derivative of a function that is a power of the variable. The video script uses the power rule to find the derivative of the function f(x) when expressed in the form of (x + 1) to a power, specifically to the power of -1/2.

💡Factorial

A factorial, denoted by an exclamation mark (e.g., n!), is the product of all positive integers less than or equal to a given number n. In the video, the instructor mentions 'two factorial' when discussing the second derivative, which is a way to express the division by 2 in the formula for the second degree term of the Maclaurin polynomial.

💡Zeroth Derivative

The zeroth derivative of a function is the function itself, as the derivative of a constant is zero. In the context of the video, the instructor evaluates the zeroth derivative of the function f(x) at zero, which is simply f(0), to find the constant term in the Maclaurin polynomial.

💡Approximation

Approximation in mathematics refers to the process of finding a value that is close to the actual value but easier to use. The video script is about approximating the function f(x) near zero using a second degree Maclaurin polynomial, which is a simpler polynomial that closely resembles the function's behavior around zero.

Highlights

Introduction to the problem of finding the second degree Maclaurin polynomial of a given function.

Definition of a Maclaurin polynomial as a Taylor polynomial centered at zero.

Explanation of the form of the second degree Maclaurin polynomial.

Identification of the need to find the function and its first and second derivatives at zero.

Evaluation of the function \(f(x) = \frac{1}{\sqrt{x+1}}\) at zero, resulting in \(f(0) = 1\).

Derivation of the first derivative \(f'(x)\) using the chain rule and power rule.

Evaluation of the first derivative \(f'(0)\) resulting in \(f'(0) = -\frac{1}{2}\).

Derivation of the second derivative \(f''(x)\) using the chain rule and power rule.

Evaluation of the second derivative \(f''(0)\) resulting in \(f''(0) = \frac{3}{4}\).

Calculation of the second derivative at zero divided by two, yielding \(f''(0)/2 = \frac{3}{8}\).

Construction of the second degree Maclaurin polynomial using the evaluated derivatives.

Final expression of the second degree Maclaurin polynomial \(p(x) = 1 - \frac{1}{2}x + \frac{3}{8}x^2\).

Discussion on the use of the Maclaurin polynomial for approximations of the function near zero.

Emphasis on the importance of understanding the derivatives in constructing the polynomial.

Highlight of the practical applications of the Maclaurin polynomial in approximating functions.

Encouragement for viewers to pause and attempt the problem before watching the solution.

Clarification that only the second degree term is needed for this problem.

Transcripts

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