Chapter 12 - Sequences and Series - 12.3 Taylor Polynomials at 0 - 12.3 Exercises - Page 632: 35

$f(x)=4x^{3}+6x^{2}+6x+3$

Let $f(x)=ax^{3}+bx^{2}+cx+d$ $f'(x)=3ax^{2}+2bx + c$ $f''(x)=6ax+2b$ $f'''(x)=6a$ The desired polynomial hence by the hypothesis: $f(0)=3 \rightarrow d=3$ $f'(0)=6 \rightarrow c=6$ $f''(0)=12 \rightarrow b=6$ $f'''(0)=24 \rightarrow a=4$ The polynomial of degree 3 is $f(x)=4x^{3}+6x^{2}+6x+3$