The First Three Derivatives Of $f ( x ) = ( x + 4 ) ^ { 3 / 2 }$

Essay

The first three derivatives of $f ( x ) = ( x + 4 ) ^ { 3 / 2 }$ are $f ^ { \prime } ( x ) = \frac { 3 ( x + 4 ) ^ { 1 / 2 } } { 2 }$ , $f ^ { \prime \prime } ( x ) = \frac { 3 } { 4 ( x + 4 ) ^ { 1 / 2 } }$ and $f ^ { \prime \prime } ( x ) = \frac { - 3 } { 8 ( x + 4 ) ^ { 3 / 2 } }$ .(a) Give the first four terms of the Taylor series associated with f at $a = - 3$ .(b) Give the second-order Taylor polynomial, $T _ { 2 } ( x )$ , associated with f at $a = 0$ .(c) Suppose that $x \geq 0$ and that $T _ { 2 } ( x )$ from part (b) is used to approximate $f ( x )$ . Prove that the error in this approximation does not exceed $\frac { x ^ { 3 } } { 128 }$ .