Use of Tech Linear and quadratic approximation


a. Find the linear approximating polynomial for the following functions centered at the given point a.


b. Find the quadratic approximating polynomial for the following functions centered at a.


c Use the polynomials obtained in parts (a) and (b) to approximate the given quantity.


Find the Taylor polynomials p₁, p₂, and p₃ centered at a=1 for f(x)=x³.

Verified step by step guidance

1. Step 1: Understand the problem. We need to find the Taylor polynomials of degrees 1, 2, and 3 (denoted as \(p_1\), \(p_2\), and \(p_3\)) centered at \(a=1\) for the function \(f(x) = x^3\). Taylor polynomials approximate a function near a point using derivatives at that point.
2. Step 2: Recall the Taylor polynomial formula centered at \(a\): \[ P_n(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x - a)^n \] We will need to compute derivatives of \(f(x)\) up to the third order and evaluate them at \(x = 1\).
3. Step 3: Compute the derivatives of \(f(x) = x^3\): - First derivative: \(f'(x) = 3x^2\) - Second derivative: \(f''(x) = 6x\) - Third derivative: \(f^{(3)}(x) = 6\) Then evaluate each at \(x=1\): - \(f(1) = 1^3 = 1\) - \(f'(1) = 3(1)^2 = 3\) - \(f''(1) = 6(1) = 6\) - \(f^{(3)}(1) = 6\)
4. Step 4: Write the Taylor polynomials using the formula: - Linear polynomial (\(p_1\)): \[ p_1(x) = f(1) + f'(1)(x - 1) \] - Quadratic polynomial (\(p_2\)): \[ p_2(x) = p_1(x) + \frac{f''(1)}{2!}(x - 1)^2 \] - Cubic polynomial (\(p_3\)): \[ p_3(x) = p_2(x) + \frac{f^{(3)}(1)}{3!}(x - 1)^3 \]
5. Step 5: To approximate values near \(x=1\), substitute the desired \(x\) value into \(p_1(x)\) and \(p_2(x)\) for linear and quadratic approximations respectively, and into \(p_3(x)\) for the cubic approximation. This will give increasingly accurate estimates of \(f(x)\) near \(x=1\).

Key Concepts

Taylor Polynomials

Taylor polynomials approximate a function near a point by using derivatives at that point. The nth-degree Taylor polynomial uses derivatives up to order n to create a polynomial that closely matches the function's behavior near the center. For example, the first-degree polynomial is a linear approximation, while higher degrees provide better accuracy.

Linear and Quadratic Approximations

Linear approximation uses the first-degree Taylor polynomial to estimate function values near a point, relying on the function's value and first derivative. Quadratic approximation extends this by including the second derivative, forming a second-degree polynomial that captures curvature, improving the estimate's accuracy near the center.

Derivatives and Their Role in Approximation

Derivatives measure how a function changes and are essential for constructing Taylor polynomials. The value of the function and its derivatives at the center point determine the coefficients of the approximating polynomial, enabling the polynomial to mimic the function's slope, curvature, and higher-order behavior near that point.