Let's work through the problem step by step.

Given:
$r(t) = \langle t^2, t^3 \rangle$ with $t = 1$.

(a) Sketch the plane curve with the given vector equation.

The parametric equation $r(t) = \langle t^2, t^3 \rangle$ represents a curve in the plane, where the $x$-coordinate is given by $t^2$ and the $y$-coordinate by $t^3$. To sketch this curve, it's helpful to plug in various values of $t$ to see how $x = t^2$ and $y = t^3$ behave. For example:

• At $t = 0$, $x = 0$, $y = 0$
• At $t = 1$, $x = 1$, $y = 1$
• At $t = -1$, $x = 1$, $y = -1$
• At $t = 2$, $x = 4$, $y = 8$

The curve passes through the origin and extends into the first and fourth quadrants.

(b) Find $r'(t)$ and evaluate it at $t = 1$.

To find the derivative of $r(t)$, we differentiate each component with respect to $t$: $r'(t) = \langle \frac{d}{dt}(t^2), \frac{d}{dt}(t^3) \rangle = \langle 2t, 3t^2 \rangle$ Now, evaluate $r'(t)$ at $t = 1$: $r'(1) = \langle 2(1), 3(1)^2 \rangle = \langle 2, 3 \rangle$

(c) Sketch the position vector $r(1)$ and the tangent vector $r'(1)$.

The position vector $r(1)$ is the vector from the origin to the point $\langle 1, 1 \rangle$, and the tangent vector at $t = 1$ is $\langle 2, 3 \rangle$. These vectors can be sketched at the point $(1, 1)$, with the tangent vector indicating the direction in which the curve is moving at that point.

If you'd like, I can create a sketch of the curve and the vectors to visualize this further.

Do you have any specific questions, or would you like more details? Here are some related questions for deeper exploration:

1. What does the derivative $r'(t)$ represent geometrically?
2. How would the curve change if the exponents of $t$ were different?
3. What is the significance of the tangent vector in the context of motion along a curve?
4. How would you find the length of the curve from $t = 0$ to $t = 1$?
5. What happens to the curve as $t$ approaches large positive or negative values?

Tip: When analyzing vector functions, it's useful to think of each component as a separate function of time and examine how they work together to describe the curve's path.