Question

Compute \(\mathbf{r}^{\prime \prime}(t)\) and \(\mathbf{r}^{\prime \prime \prime}(t)\) for the following functions. $$\mathbf{r}(t)=\left\langle t^{2}+1, t+1,1\right\rangle$$

Short answer

Short Answer: The second derivative of the vector-valued function \(\mathbf{r}(t)\) is \(\mathbf{r}^{\prime\prime}(t) = \langle 2, 0, 0 \rangle\), and the third derivative is \(\mathbf{r}^{\prime\prime\prime}(t) = \langle 0, 0, 0 \rangle\).

Step-by-step solution

Compute the first derivative

Differentiate each component of \(\mathbf{r}(t)\) with respect to time \(t\) to obtain the first derivative, \(\mathbf{r}^{\prime}(t)\). $$\frac{d}{dt}(\langle t^2+1,t+1,1 \rangle)=\langle \frac{d(t^2+1)}{dt}, \frac{d(t+1)}{dt}, \frac{d(1)}{dt}\rangle$$ Applying the derivative rules: $$\mathbf{r}^{\prime}(t) =\langle 2t, 1, 0\rangle$$

Compute the second derivative

Differentiate each component of \(\mathbf{r}^{\prime}(t)\) with respect to time \(t\) to obtain the second derivative, \(\mathbf{r}^{\prime\prime}(t)\). $$\frac{d^2}{dt^2}(\langle 2t, 1, 0 \rangle)=\langle \frac{d(2t)}{dt}, \frac{d(1)}{dt}, \frac{d(0)}{dt} \rangle$$ Applying the derivative rules: $$\mathbf{r}^{\prime\prime}(t) =\langle 2, 0, 0 \rangle$$

Compute the third derivative

Differentiate each component of \(\mathbf{r}^{\prime\prime}(t)\) with respect to time \(t\) to obtain the third derivative, \(\mathbf{r}^{\prime\prime\prime}(t)\). $$\frac{d^3}{dt^3}(\langle 2, 0, 0 \rangle) = \langle \frac{d(2)}{dt}, \frac{d(0)}{dt}, \frac{d(0)}{dt}\rangle$$ Applying the derivative rules: $$\mathbf{r}^{\prime\prime\prime}(t) =\langle 0, 0, 0 \rangle$$ In conclusion, the second derivative of the vector-valued function \(\mathbf{r}(t)\) is \(\mathbf{r}^{\prime\prime}(t)=\langle 2, 0, 0 \rangle\), and the third derivative is \(\mathbf{r}^{\prime\prime\prime}(t)=\langle 0, 0, 0 \rangle\).