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Section 13.2: Derivatives and Integrals of Vector Functions

Exercise 2

  1. Make a large sketch of the curve described by the vector function \(\mathbf{r}(t) = \langle t^2, t \rangle\), \(0 \le t \le 2\), and draw the vectors \(\mathbf{r}(1)\), \(\mathbf{r}(1.1)\), and \(\mathbf{r}(1.1) - \mathbf{r}(1)\).
  2. Draw the vector \(\mathbf{r}'(1)\) starting at (1, 1), and compare it with the vector \[ \frac{\mathbf{r}(1.1) - \mathbf{r}(1)}{0.1} \] Explain why these vectors are so close to each other in length and direction.