The given vector function is:

$\mathbf{r}(t) = \sin(4t) \, \mathbf{i} + \ln(t) \, \mathbf{j} + 2te^{-9t} \, \mathbf{k}$

To find the second derivative, $\mathbf{r}''(t)$, we need to first find the first derivative $\mathbf{r}'(t)$ and then differentiate again to obtain $\mathbf{r}''(t)$.

Step 1: First derivative $\mathbf{r}'(t)$

Differentiate each component of $\mathbf{r}(t)$ with respect to $t$:

1. $\sin(4t) \, \mathbf{i}$ $\frac{d}{dt} [\sin(4t)] = 4\cos(4t) \, \mathbf{i}$

2. $\ln(t) \, \mathbf{j}$ $\frac{d}{dt} [\ln(t)] = \frac{1}{t} \, \mathbf{j}$

3. $2te^{-9t} \, \mathbf{k}$ Using the product rule: [ \frac{d}{dt} [2te^{-9t}] = 2 \cdot \frac{d}{dt}[t] \cdot e^{-9t} + 2t \cdot \frac{d}{dt}[e^{-9t}] ] $= 2e^{-9t} + 2t(-9e^{-9t}) = 2e^{-9t} - 18te^{-9t} \, \mathbf{k}$

Thus, the first derivative is: $\mathbf{r}'(t) = 4\cos(4t) \, \mathbf{i} + \frac{1}{t} \, \mathbf{j} + (2e^{-9t} - 18te^{-9t}) \, \mathbf{k}$

Step 2: Second derivative $\mathbf{r}''(t)$

Now differentiate each component of $\mathbf{r}'(t)$ again:

1. $4\cos(4t) \, \mathbf{i}$ $\frac{d}{dt} [4\cos(4t)] = -16\sin(4t) \, \mathbf{i}$

2. $\frac{1}{t} \, \mathbf{j}$ $\frac{d}{dt} \left[\frac{1}{t}\right] = -\frac{1}{t^2} \, \mathbf{j}$

3. $2e^{-9t} - 18te^{-9t} \, \mathbf{k}$ Using the product rule on $-18te^{-9t}$: $\frac{d}{dt} [2e^{-9t}] = -18e^{-9t}$ $\frac{d}{dt} [-18te^{-9t}] = -18e^{-9t} + 162te^{-9t}$ So, the derivative of $2e^{-9t} - 18te^{-9t}$ is: $(-18e^{-9t}) + (-18e^{-9t} + 162te^{-9t}) = -36e^{-9t} + 162te^{-9t} \, \mathbf{k}$

Thus, the second derivative is: $\mathbf{r}''(t) = -16\sin(4t) \, \mathbf{i} - \frac{1}{t^2} \, \mathbf{j} + (-36e^{-9t} + 162te^{-9t}) \, \mathbf{k}$

Final Answer:

$\mathbf{r}''(t) = -16\sin(4t) \, \mathbf{i} - \frac{1}{t^2} \, \mathbf{j} + (-36e^{-9t} + 162te^{-9t}) \, \mathbf{k}$

Let me know if you need further clarification, or if you'd like more details on any step!

Here are 5 questions that build on this information:

1. How do we apply the product rule when differentiating vector functions?
2. What is the significance of the second derivative in the context of a position vector?
3. How does the exponential decay in the $\mathbf{k}$-component affect the behavior of the function as $t$ increases?
4. What is the geometrical interpretation of $\mathbf{r}(t)$ and $\mathbf{r}''(t)$?
5. How would the calculation change if $\mathbf{r}(t)$ were defined in terms of $e^{4t}$ instead of $\ln(t)$?

Tip: When differentiating vector functions, treat each component separately, applying the rules for differentiation as you would for scalar functions.