Let $y = f ( x )$ be the particular solution to the differential equation $\frac { d y } { d x } = ( 3 - x ) y ^ { 2 }$ with initial condition $f ( 1 ) = - 1$. A. Find $f ^ { \prime \prime } ( 1 )$, the value of $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ at the point $( 1 , - 1 )$. Show the work that leads to your answer. B. Write the second-degree Taylor polynomial for $f$ about $x = 1$. C. The second-degree Taylor polynomial for $f$ about $x = 1$ is used to approximate $f ( 1.1 )$. Given that $\left| f ^ { \prime \prime \prime } ( x ) \right| \leq 60$ for all $x$ in the interval $1 \leq x \leq 1.1$, use the Lagrange error bound to show that this approximation differs from $f ( 1.1 )$ by at most 0.01. D. Use Euler's method, starting at $x = 1$ with two steps of equal size, to approximate $f ( 1.4 )$. Show the work that leads to your answer.
Let $y = f ( x )$ be the particular solution to the differential equation $\frac { d y } { d x } = ( 3 - x ) y ^ { 2 }$ with initial condition $f ( 1 ) = - 1$.
A. Find $f ^ { \prime \prime } ( 1 )$, the value of $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ at the point $( 1 , - 1 )$. Show the work that leads to your answer.
B. Write the second-degree Taylor polynomial for $f$ about $x = 1$.
C. The second-degree Taylor polynomial for $f$ about $x = 1$ is used to approximate $f ( 1.1 )$. Given that $\left| f ^ { \prime \prime \prime } ( x ) \right| \leq 60$ for all $x$ in the interval $1 \leq x \leq 1.1$, use the Lagrange error bound to show that this approximation differs from $f ( 1.1 )$ by at most 0.01.
D. Use Euler's method, starting at $x = 1$ with two steps of equal size, to approximate $f ( 1.4 )$. Show the work that leads to your answer.