Answer
\[K = \frac{{12}}{{125}}\]
Work Step by Step
\[\begin{gathered}
{\mathbf{r}}\left( t \right) = 3t{\mathbf{i}} + 2{t^2}{\mathbf{j}},{\text{ }}P\left( { - 3,2} \right) \hfill \\
{\text{By Theorem 12}}{\text{.8 }} \hfill \\
{\text{If }}C{\text{ is a smooth curve given by }}{\mathbf{r}}\left( t \right),{\text{ then the curvature }}K{\text{ of}} \hfill \\
C{\text{ at }}t{\text{ is }}K = \frac{{\left\| {{\mathbf{r}}'\left( t \right) \times {\mathbf{r}}''\left( t \right)} \right\|}}{{{{\left\| {{\mathbf{r}}'\left( t \right)} \right\|}^3}}} \hfill \\
{\mathbf{r}}\left( t \right) = 3t{\mathbf{i}} + 2{t^2}{\mathbf{j}} \hfill \\
{\mathbf{r}}'\left( t \right) = 3{\mathbf{i}} + 4t{\mathbf{j}} \hfill \\
{\mathbf{r}}''\left( t \right) = 4{\mathbf{j}} \hfill \\
{\mathbf{r}}'\left( t \right) \times {\mathbf{r}}''\left( t \right) = \left| {\begin{array}{*{20}{c}}
{\mathbf{i}}&{\mathbf{j}}&{\mathbf{k}} \\
3&4&0 \\
0&4&0
\end{array}} \right| \hfill \\
{\mathbf{r}}'\left( t \right) \times {\mathbf{r}}''\left( t \right) = \left| {\begin{array}{*{20}{c}}
4&0 \\
4&0
\end{array}} \right|{\mathbf{i}} - \left| {\begin{array}{*{20}{c}}
3&0 \\
0&0
\end{array}} \right|{\mathbf{j}} + \left| {\begin{array}{*{20}{c}}
3&4 \\
0&4
\end{array}} \right|{\mathbf{k}} \hfill \\
{\mathbf{r}}'\left( t \right) \times {\mathbf{r}}''\left( t \right) = 12{\mathbf{k}} \hfill \\
\left\| {{\mathbf{r}}'\left( t \right) \times {\mathbf{r}}''\left( t \right)} \right\| = 12 \hfill \\
and \hfill \\
\left\| {{\mathbf{r}}'\left( t \right)} \right\| = \left\| {3{\mathbf{i}} + 4t{\mathbf{j}}} \right\| \hfill \\
\left\| {{\mathbf{r}}'\left( t \right)} \right\| = \sqrt {{{\left( 3 \right)}^2} + {{\left( 4 \right)}^2}} \hfill \\
\left\| {{\mathbf{r}}'\left( t \right)} \right\| = 5 \hfill \\
{\text{Therefore,}} \hfill \\
K = \frac{{\left\| {{\mathbf{r}}'\left( t \right) \times {\mathbf{r}}''\left( t \right)} \right\|}}{{{{\left\| {{\mathbf{r}}'\left( t \right)} \right\|}^3}}} = \frac{{12}}{{{{\left( 5 \right)}^3}}} \hfill \\
K = \frac{{12}}{{125}} \hfill \\
{\text{The curvature is constant, so at the point }}P\left( { - 3,2} \right),{\text{ the}} \hfill \\
{\text{curvature is }}\frac{{12}}{{125}} \hfill \\
\end{gathered} \]
