Question

Compute \(\mathbf{r}^{\prime \prime}(t)\) and \(\mathbf{r}^{\prime \prime \prime}(t)\) for the following functions. $$\mathbf{r}(t)=\left\langle e^{4 t}, 2 e^{-4 t}+1,2 e^{-t}\right\rangle$$

Short answer

Question: Find the acceleration and jerk vectors of the given function: $$\mathbf{r}(t)=\left\langle e^{4 t}, 2 e^{-4 t}+1,2 e^{-t}\right\rangle$$ Answer: The acceleration (second-order derivative) and jerk (third-order derivative) vectors are as follows: Acceleration: $$\mathbf{r}^{\prime\prime} (t) = \left\langle 16e^{4t}, 32e^{-4t}, 2e^{-t} \right\rangle$$ Jerk: $$\mathbf{r}^{\prime\prime\prime} (t) = \left\langle 64e^{4t}, -128e^{-4t}, 2e^{-t} \right\rangle$$

Step-by-step solution

Compute the first-order derivative (velocity) for each component function.

To find the velocity vector, take the first derivative for each of the component functions: $$\frac{d}{dt} e^{4t}, \frac{d}{dt}(2e^{-4t} + 1), \frac{d}{dt}(2e^{-t})$$ Compute the derivatives: $$4 e^{4t}, -8e^{-4t}, -2e^{-t}$$ Now that we have the first-order derivative of each component, let's write it in a vector form: $$\mathbf{r}^\prime (t) = \left\langle 4 e^{4t}, -8e^{-4t}, -2e^{-t} \right\rangle$$

Compute the second-order derivative (acceleration) for each component function.

To find the acceleration vector, take the second derivative for each of the component functions: $$\frac{d^2}{dt^2} 4e^{4t}, \frac{d^2}{dt^2}(-8e^{-4t}), \frac{d^2}{dt^2}(-2e^{-t})$$ Compute the derivatives: $$16e^{4t}, 32e^{-4t}, 2e^{-t}$$ Now that we have the second-order derivative of each component, let's write it in a vector form: $$\mathbf{r}^{\prime\prime} (t) = \left\langle 16e^{4t}, 32e^{-4t}, 2e^{-t} \right\rangle$$

Compute the third-order derivative (jerk) for each component function.

To find the jerk vector, take the third derivative for each of the component functions: $$\frac{d^3}{dt^3} 16e^{4t}, \frac{d^3}{dt^3}(32e^{-4t}), \frac{d^3}{dt^3}(2e^{-t})$$ Compute the derivatives: $$64e^{4t}, -128e^{-4t}, 2e^{-t}$$ Now that we have the third-order derivative of each component, let's write it in a vector form: $$\mathbf{r}^{\prime\prime\prime} (t) = \left\langle 64e^{4t}, -128e^{-4t}, 2e^{-t} \right\rangle$$ Now we have both the acceleration and jerk vectors, let's write them out together: Acceleration (second-order derivative): $$\mathbf{r}^{\prime\prime} (t) = \left\langle 16e^{4t}, 32e^{-4t}, 2e^{-t} \right\rangle$$ Jerk (third-order derivative): $$\mathbf{r}^{\prime\prime\prime} (t) = \left\langle 64e^{4t}, -128e^{-4t}, 2e^{-t} \right\rangle$$