Preview Activity 7.3.1.
Consider the initial value problem
\begin{equation*}
\frac{dy}{dt} = \frac12 (y + 1), \ y(0) = 0\text{.}
\end{equation*}
(a)
Use the differential equation to find the slope of the tangent line to the solution \(y(t)\) at \(t=0\text{.}\) Then use the given initial value to find the equation of the tangent line at \(t=0\text{.}\)
(b)
Sketch the tangent line you found in (a) on the axes provided on the interval \(0\leq t\leq 2\) and use the tangent line to approximate \(y(2)\text{,}\) the value of the solution at \(t=2\text{.}\)
(c)
Assuming that your approximation for \(y(2)\) from (b) is the actual value of \(y(2)\text{,}\) use the differential equation to find the slope of the tangent line to \(y(t)\) at \(t=2\) (i.e., at the point \((2,y(2))\)). Then, write the equation of the tangent line at \(t=2\text{.}\)
(d)
Add a sketch of this tangent line from (c) on the interval \(2\leq t\leq 4\) to your plot in (b); use this new tangent line to approximate \(y(4)\text{,}\) the value of the solution at \(t=4\text{.}\)
(e)
Repeat the same step to find an approximation for \(y(6)\text{.}\)
