The image contains a vector calculus problem, which describes a curve by the vector equation $\mathbf{r}(t) = t\mathbf{i} + t^2\mathbf{j}$, where $t > 0$. There are three parts to the question:

1. Calculate $\mathbf{r}'(t)$, which is the derivative of the vector equation.
2. Calculate the gradient of the curve at the point where $t = 1$.
3. Determine a unit vector $\hat{s}$ parallel to the tangent to the curve where $t = 1$.

Let’s solve each part step by step:

1. Calculate $\mathbf{r}'(t)$

The vector equation is: $\mathbf{r}(t) = t\mathbf{i} + t^2\mathbf{j}$

To find $\mathbf{r}'(t)$, differentiate each component with respect to $t$: $\mathbf{r}'(t) = \frac{d}{dt}(t\mathbf{i}) + \frac{d}{dt}(t^2\mathbf{j})$ $\mathbf{r}'(t) = \mathbf{i} + 2t\mathbf{j}$

So, $\mathbf{r}'(t) = \mathbf{i} + 2t\mathbf{j}$.

2. Calculate the gradient at $t = 1$

The gradient of the curve is given by $\mathbf{r}'(t)$, so we substitute $t = 1$ into $\mathbf{r}'(t)$: $\mathbf{r}'(1) = \mathbf{i} + 2(1)\mathbf{j} = \mathbf{i} + 2\mathbf{j}$

Thus, the gradient at $t = 1$ is $\mathbf{i} + 2\mathbf{j}$.

3. Determine a unit vector $\hat{s}$ parallel to the tangent at $t = 1$

To find a unit vector, we need to normalize $\mathbf{r}'(1)$. The formula for the magnitude of a vector $\mathbf{v} = a\mathbf{i} + b\mathbf{j}$ is: $|\mathbf{v}| = \sqrt{a^2 + b^2}$

For $\mathbf{r}'(1) = \mathbf{i} + 2\mathbf{j}$, we have: $|\mathbf{r}'(1)| = \sqrt{1^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5}$

Now, the unit vector $\hat{s}$ is: $\hat{s} = \frac{\mathbf{r}'(1)}{|\mathbf{r}'(1)|} = \frac{\mathbf{i} + 2\mathbf{j}}{\sqrt{5}} = \frac{1}{\sqrt{5}}\mathbf{i} + \frac{2}{\sqrt{5}}\mathbf{j}$

Thus, the unit vector $\hat{s}$ parallel to the tangent to the curve at $t = 1$ is: $\hat{s} = \frac{1}{\sqrt{5}}\mathbf{i} + \frac{2}{\sqrt{5}}\mathbf{j}$

Would you like further details or clarification on any step? Here are five follow-up questions to consider:

1. What is the geometric interpretation of $\mathbf{r}'(t)$?
2. How would you calculate the tangent vector at any arbitrary point $t = t_0$?
3. Can you explain the significance of the unit vector $\hat{s}$?
4. How does the magnitude of $\mathbf{r}'(t)$ relate to the speed of motion along the curve?
5. What would change if the curve were described in three dimensions?

Tip: The unit vector always has a magnitude of 1, which makes it useful in defining direction without scaling effects.