Consider the logistic differential equation $\frac { d y } { d t } = \frac { y } { 8 } ( 6 - y )$. Let $y = f ( t )$ be the particular solution to the differential equation with $f ( 0 ) = 8$. (a) A slope field for this differential equation is given below. Sketch possible solution curves through the points $( 3,2 )$ and $( 0,8 )$. (b) Use Euler's method, starting at $t = 0$ with two steps of equal size, to approximate $f ( 1 )$. (c) Write the second-degree Taylor polynomial for $f$ about $t = 0$, and use it to approximate $f ( 1 )$. (d) What is the range of $f$ for $t \geq 0$ ?
Consider the logistic differential equation $\frac { d y } { d t } = \frac { y } { 8 } ( 6 - y )$. Let $y = f ( t )$ be the particular solution to the differential equation with $f ( 0 ) = 8$.
(a) A slope field for this differential equation is given below. Sketch possible solution curves through the points $( 3,2 )$ and $( 0,8 )$.
(b) Use Euler's method, starting at $t = 0$ with two steps of equal size, to approximate $f ( 1 )$.
(c) Write the second-degree Taylor polynomial for $f$ about $t = 0$, and use it to approximate $f ( 1 )$.
(d) What is the range of $f$ for $t \geq 0$ ?