13.2 Videos Guide

13.2a

• Derivative of a vector function

  • $\frac{d\mathbf{r}}{dt}=\mathbf{r}'(t)=\left\langle f'(t),\ g'(t),\ h'(t)\right\rangle =f'(t)\mathbf{i}+g'(t)\mathbf{j}+h'(t)\mathbf{k}$

• Product rules with vector functions

• The unit tangent vector: $\mathbf{T}(t)=\frac{\mathbf{r}'(t)}{\left|\mathbf{r}'(t)\right|}$

Exercises:

13.2b

• Sketch the plane curve with the given vector equation.
  (b) Find $\mathbf{r}'(t)$.
  (c) Sketch the position vector $\mathbf{r}(t)$ and the tangent vector $\mathbf{r}'(t)$ for the given value of $t$.

$\mathbf{r}(t)=e^{t}\mathbf{i}+2t\mathbf{j}$, $t=0$

• Find the derivative of the vector function $\mathbf{r}(t)=\sin^{2}{at}\mathbf{i}+te^{bt}\mathbf{\ j}+\cos^{2}{ct}\mathbf{k}$.

13.2c

• Find the unit tangent vector $\mathbf{T}(t)$ at the point with the given value of the parameter $t$.
  $\mathbf{r}(t)=\left\langle \tan^{-1}t,\ 2e^{2t},\ 8te^{t}\right\rangle$, $t=0$

• Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point.
  $x=ln(t+1)$, $y=t\cos{2t}$, $z=2^{t}$, $(0,\ 0,\ 1)$

13.2d

• Proof that if $\left|\mathbf{r}(t)\right|=c$, a constant, then $\mathbf{r}'(t)$ is orthogonal to $\mathbf{r}(t)$ for all $t$

13.2e

• Integrals of vector functions

  • Indefinite: $\int^ {}_{}{\mathbf{r}(t)\ dt}=\int^ {}_{}{f(t)\mathbf{i}}+\int^ {}_{}{g(t)\mathbf{j}}+\int^ {}_{}{h(t)\mathbf{k}}+\mathbf{C}$

  • Definite: $\int^{b}_{a}{\mathbf{r}(t)\ dt}=\int^{b}_{a}{f(t)\mathbf{i}\ dt}+\int^{b}_{a}{g(t)\mathbf{j}\ dt}+\int^{b}_{a}{h(t)\mathbf{k}\ dt}$

Exercise:

• Evaluate the integral.
  $$\int^{4}_{1}{\left(2t^{\frac{3}{2}}\mathbf{\ i}+(t+1)\sqrt{t}\mathbf{\ k}\right)\ dt}$$