Answered: f(x) = Vx 4; approximate with a V3

Algebra: Structure And Method, Book 1

(REV)00th Edition

ISBN:9780395977224

Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Chapter8: Introduction To Functions

Section8.8: Linear And Quadratic Functions

Problem 4ST

See similar textbooks

Related questions

Question

Approximating powers Compute the coefficients for the Taylor series for the following functions about the given point a, and then use the first four terms of the series to approximate the given number.

Expert Solution

Step 1

Consider the provided function,

$f (x) = \frac{1}{\sqrt{x}} with a = 4; approximation \frac{1}{\sqrt{3}}$ .

Compute the coefficients for the Taylor series for the following functions about the given point a, and then use the first four terms of the series to approximate the given number.

The Taylor series as a form of $\sum_{k = 0}^{\infty} c_{k} {(x - a)}^{k}, where c_{k} = \frac{f^{k} (a)}{k!}, k = 0, 1, 2 . . . .$ .

Step 2

We evaluate the derivative of the function at point $a = 4$ ,

$\begin{array}{rcl} f (x) & = & \frac{1}{\sqrt{x}} \Rightarrow f (4) = \frac{1}{2} \\ f' (x) & = & - \frac{1}{2 x^{\frac{3}{2}}} \Rightarrow f' (4) = - \frac{1}{16} \\ f'' (x) & = & \frac{3}{4 x^{\frac{5}{2}}} \Rightarrow f'' (4) = \frac{3}{128} \\ f''' (x) & = & - \frac{15}{8 x^{\frac{7}{2}}} \Rightarrow f''' (4) = - \frac{15}{1024} \end{array}$

Step by step

Solved in 3 steps

Knowledge Booster

$Power Series$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

Recommended textbooks for you

Algebra: Structure And Method, Book 1

Algebra

ISBN:

9780395977224

Author:

Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole

Publisher:

McDougal Littell