Question

2. (7 pts.) Let $f(x) = \ln(1+x)$. i. Calculate the first, second, third, and fourth derivatives of $f$. ii. Using these derivatives, obtain the Taylor polynomials of degree $n$ of $f$ at $b = 0$, for $n$ equals to zero, one, two, and three; i.e., $P_{0,b}(x)$, $P_{1,b}(x)$, $P_{2,b}(x)$, $P_{3,b}(x)$, with $b = 0$. iii. Use the direct subtraction $R_{n,b}(x) = \ln(1+x) - P_{n,b}(x)$, to evaluate the errors when approximating $f(0.5)$ using $P_{n,b}(0.5)$ where $b = 0$ and $n =$ 0, 1, 2, 3. 2. iv. Calculate $\frac{|R_{n+1,b}(x)|}{|R_{n,b}(x)|}$ for $n =$ 0, 1, 2, where $b = 0$ and $x = 0.5$ using the results you get from point iii. v. Write the residual formula $R_{n,b}(x)$ using only the $n + 1$ derivative of $f$ for $n = 2, 3$. vi. Using the formula you get above, give an estimate of $\frac{|R_{n,b}(0.25)|}{|R_{n,b}(0.5)|}$ where $b = 0$ and $n = 2$ or $n = 3$. As this is just an estimate, you should not do this task by using direct subtraction as in iii.

          2. (7 pts.) Let $f(x) = \ln(1+x)$. 
i. Calculate the first, second, third, and
fourth derivatives of $f$.
ii. Using these derivatives, obtain the
Taylor polynomials of degree $n$ of $f$
at $b = 0$, for $n$ equals to zero, one,
two, and three; i.e.,
$P_{0,b}(x)$, $P_{1,b}(x)$, $P_{2,b}(x)$, $P_{3,b}(x)$, 
with $b = 0$.
iii. Use the direct subtraction
$R_{n,b}(x) = \ln(1+x) - P_{n,b}(x)$, 
to evaluate the errors when
approximating $f(0.5)$ using
$P_{n,b}(0.5)$ where $b = 0$ and $n =$
0, 1, 2, 3.
2. iv. Calculate $\frac{|R_{n+1,b}(x)|}{|R_{n,b}(x)|}$ for $n =$
0, 1, 2, where $b = 0$ and $x = 0.5$
using the results you get from point
iii.
v. Write the residual formula $R_{n,b}(x)$
using only the $n + 1$ derivative of $f$
for $n = 2, 3$.
vi. Using the formula you get above, give
an estimate of $\frac{|R_{n,b}(0.25)|}{|R_{n,b}(0.5)|}$ where
$b = 0$ and $n = 2$ or $n = 3$. As this
is just an estimate, you should not do
this task by using direct subtraction as
in iii.

2. (7 pts.) Let f(x) = ln(1+x).
i. Calculate the first, second, third, and
fourth derivatives of f.
ii. Using these derivatives, obtain the
Taylor polynomials of degree n of f
at b = 0, for n equals to zero, one,
two, and three; i.e.,
P0,b(x), P1,b(x), P2,b(x), P3,b(x),
with b = 0.
iii. Use the direct subtraction
Rn,b(x) = ln(1+x) - Pn,b(x),
to evaluate the errors when
approximating f(0.5) using
Pn,b(0.5) where b = 0 and n =
0, 1, 2, 3.
2. iv. Calculate (|Rn+1,b(x)|)/(|Rn,b(x)|) for n =
0, 1, 2, where b = 0 and x = 0.5
using the results you get from point
iii.
v. Write the residual formula Rn,b(x)
using only the n + 1 derivative of f
for n = 2, 3.
vi. Using the formula you get above, give
an estimate of (|Rn,b(0.25)|)/(|Rn,b(0.5)|) where
b = 0 and n = 2 or n = 3. As this
is just an estimate, you should not do
this task by using direct subtraction as
in iii.

Added by Emilio C.

Question

Please give Ace some feedback