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 polynomial p₃ centered at a=e for f(x)=ln x.

Verified step by step guidance

1. Recall that the Taylor polynomial of degree 3 for a function \( f(x) \) centered at \( a \) is given by: \[ p_3(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f'''(a)}{3!}(x - a)^3 \] This polynomial approximates \( f(x) \) near \( x = a \).
2. Identify the function and center: here, \( f(x) = \ln x \) and \( a = e \). We need to compute \( f(a) \), \( f'(a) \), \( f''(a) \), and \( f'''(a) \).
3. Calculate the derivatives of \( f(x) = \ln x \): - First derivative: \( f'(x) = \frac{1}{x} \) - Second derivative: \( f''(x) = -\frac{1}{x^2} \) - Third derivative: \( f'''(x) = \frac{2}{x^3} \)
4. Evaluate each derivative at \( x = a = e \): - \( f(e) = \ln e \) - \( f'(e) = \frac{1}{e} \) - \( f''(e) = -\frac{1}{e^2} \) - \( f'''(e) = \frac{2}{e^3} \)
5. Substitute these values into the Taylor polynomial formula: \[ p_3(x) = f(e) + f'(e)(x - e) + \frac{f''(e)}{2}(x - e)^2 + \frac{f'''(e)}{6}(x - e)^3 \] This gives the cubic approximation of \( \ln x \) near \( x = e \).

Key Concepts

Taylor Polynomials

Taylor polynomials approximate a function near a point 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 third-degree Taylor polynomial p₃ at a point a approximates f(x) using f(a), f'(a), f''(a), and f'''(a).

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, providing a more accurate estimate with a second-degree polynomial. These approximations simplify complex functions for easier calculation.

Derivatives of Logarithmic Functions

Understanding the derivatives of f(x) = ln(x) is essential for constructing Taylor polynomials. The first derivative is 1/x, the second is -1/x², and the third is 2/x³. Evaluating these at the center a = e allows building the polynomial terms needed for accurate approximations near that point.