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

question 167

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 $\frac { x ^ { 3 } } { 128 }$ .

---

Definitions:

Likely

Likely refers to a high probability or chance of something happening or being the case, often used to forecast outcomes based on current trends or evidence.

Evolution Of Money

The historical development and transformation of money from traditional bartering systems to modern digital currencies.

Physical Commodities

Raw materials or primary agricultural products that can be bought, sold, or traded, often through futures contracts.

Barter

The direct exchange of goods and services between parties without the use of money.