Question

find r(t), r(t_0), and r(t_0) for the given value of t_0. r(t)=(4 + t)i + t^3j, t_0 = 1 r(t) = 5i + j r(t) = r(t_0) =

Explanation:

Step1: Differentiate \(r(t)\)

Given \(r(t)=(4 + t)\mathbf{i}+t^{3}\mathbf{j}\), using the power - rule for differentiation. The derivative of \(4 + t\) with respect to \(t\) is \(1\), and the derivative of \(t^{3}\) with respect to \(t\) is \(3t^{2}\). So \(r^{\prime}(t)=\mathbf{i}+3t^{2}\mathbf{j}\).

Step2: Evaluate \(r(t)\) at \(t_0 = 1\)

Substitute \(t = 1\) into \(r(t)\): \(r(1)=(4 + 1)\mathbf{i}+1^{3}\mathbf{j}=5\mathbf{i}+\mathbf{j}\).

Step3: Evaluate \(r^{\prime}(t)\) at \(t_0 = 1\)

Substitute \(t = 1\) into \(r^{\prime}(t)\): \(r^{\prime}(1)=\mathbf{i}+3\times1^{2}\mathbf{j}=\mathbf{i}+3\mathbf{j}\).

Answer:

\(r(t)=(4 + t)\mathbf{i}+t^{3}\mathbf{j}\), \(r(1)=5\mathbf{i}+\mathbf{j}\), \(r^{\prime}(1)=\mathbf{i}+3\mathbf{j}\)