Preview Activity 7.2.1.
Let’s consider the initial value problem
\begin{equation*}
\frac{dy}{dt} = t - 2, \ \ y(0) = 1\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 initial value to find the equation of the tangent line at \(t=0\text{.}\) Sketch this tangent line over the interval \(-0.25 \leq t \leq 0.25\) on the axes provided below.
(b)
Also shown in the figure in (a) are the tangent lines to the solution \(y(t)\) at the points \(t=1, 2\text{,}\) and \(3\) (we will see how to find these later). Use the graph to measure the slope of each tangent line and verify that each agrees with the value specified by the differential equation.
(c)
Using the tangent lines from (a) and (b) as a guide, sketch a graph of the solution \(y(t)\) over the interval \(0\leq t\leq 3\) so that the lines are tangent to the graph of \(y(t)\text{.}\)
(d)
Use the Fundamental Theorem of Calculus to find \(y(t)\text{,}\) the solution to original initial value problem,
\begin{equation*}
\frac{dy}{dt} = t - 2, \ \ y(0) = 1\text{.}
\end{equation*}
(e)
Graph the solution you found in (d) on the axes provided in (a), and compare it to the sketch you made using the tangent lines.
