[4 pts] State a formula using limits for the tangent vector to a parametrized curve at a point on the curve. Draw a sketch that explains this formula.

The sketch above illustrates the concept of a tangent vector to a parametrized curve. It shows the curve, points \(P_0 = \mathbf{r}(t_0)\) and \(P_1 = \mathbf{r}(t_0 + \Delta t)\), a secant vector between these points, and the tangent vector at \(P_0\), which represents the limiting direction as \(\Delta t\) approaches zero.

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Here are 5 related questions to expand your understanding:
1. How is the tangent vector useful in describing the velocity of an object moving along a parametrized curve?
2. Can you find the tangent vector for a given parametrized curve in a specific example, such as \(\mathbf{r}(t) = (t^2, t^3)\)?
3. How does the concept of the unit tangent vector differ from the general tangent vector?
4. What is the geometric meaning of the magnitude of the tangent vector?
5. How would the tangent vector be affected if the curve is reparametrized?

**Tip:** The direction of the tangent vector provides crucial insight into how fast and in what direction a particle is moving along a curve at a given instant.

State a formula using limits for the tangent vector to a parametrized curve at a point on the curve. Draw a sketch that explains this formula.

Learn how to find the tangent vector to a parametrized curve using limits. This calculus problem explains the key concepts of parametrized curves, limits, and the formula for a tangent vector, with an illustrative sketch. Suitable for college-level calculus students.

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