Activity 10.5.4

In general, calculating $\vN$ as a function of the parameter $t$ ends up being very difficult because there are multiple compositions of functions involved which means there is a nesting of chain rules involved in the definition. In this activity, we will examine how to go through the direct calculation of $\vN$ for a curve parameterized by $\vr(t) =\langle t,t^2,t^3\rangle$.

1. Calculate velocity $\vv(t)$ and $\text{speed}(t)$ for the parameterization $\vr(t) =\langle t,t^2,t^3\rangle$.

2. Now calculate $\vT(t)$ for the path given by $\vr(t) =\langle t,t^2,t^3\rangle$.

3. Calculate the first component of $\vT\, '=\frac{d\vT}{dt}$.

4. Calculate the second component of $\vT\, '$.

5. Calculate the third component of $\vT\, '$.

6. Put together your work for the three components of $\vT\, '$.

7. In the previous part, you likely decided that trying to simplify your formulas for $\frac{d\vT}{dt}$ was more complicated than you had space for on your paper, and note that we still would need to calculate $\vecmag{\frac{d\vT}{dt}}$. Instead of doing more intricate algebraic computations, describe how you would calculate $\vN$ if you had nice formulas for $\frac{d\vT}{dt}$ and $\vecmag{\frac{d\vT}{dt}}$.