Question

Assignment5: Problem 11 Previous Problem Problem List Next Problem (1 point) Find the polynomial of degree 9 (centered at zero) that best approximates f(x) = ln(x + 4). Hint: First find a Taylor polynomial for g(x) = ln(x + 4), then use this to find the Taylor polynomial you want. f(x) 1/6 Now use this polynomial to approximate Luca In(x + 4) dx. -1/6 1/6 f(x) dx

Please answer both questions if possible. Thank you :)

Answer 1

Now, the taylor polynomial of degree g is f(x) 23 x6 + 29 4 2142 3:43. Again, s f(x) da [ 2 '2² 4 26 29 2.4L 3.43 Jdx -YG alo x 4 4.4 72 + 3.4310 2.427 lo 1/67 lid 2 2.427 21427 67 7.4767. =-0.000000032 % f(x) ~ 0.000000032 Again, we first find the taylor polynomial for goviste ga)= (1+2) g 0021 g'(x) = -√ (144) 1 g'lo=& = -4 Clea% 8 72 g'om) - tk 3. (1975), g'o- 9"00) - 1.2-13 g"(a) 118.15 C1+x) - 22 7.7.7 73

23 2173 Now, g(x) = 1 - 4x + x - 2 Replace all the a with up to get a series for Tirya -xo_8115x9 2.73 7/142² fou ñ 1-2 2 23 + 2 2 X o Now, sufera de a 5 1 1 - 2 - 2 + 1 - 2 - 3.153 dr. 260) 8.x? ay + 920766 [ " 8115 2.73.10 2- 7.4 727 =-19.28109
To find Taylor polynomial of f(x)=ln(1+x^3), We start by working out a Taylor series for g(x)=ln(1+x). I will expanding around 0 , so it will be a Maclaurin series.