Chapter 14

How do you find the indefinite integral of $\mathbf{r}(t)=⟨f(t),g(t),h(t)⟩?$

An object moves on a trajectory given by $$\mathbf{r}(t)=⟨10\cos 2t,10\sin 2t⟩,\text{ for }0\le t\le \pi$$ How far does it travel?

$$\text{ How do you evaluate }lim\mathbf{r}(t),\text{ where }\mathbf{r}(t)=⟨f(t),g(t),h(t)⟩?$$

Use calculus to find the length of the line segment $\mathbf{r}(t)=⟨t,-8t,4t⟩,$ for $0\le t\le 2.$ Verify your answer without using calculus.

Explain how to decompose the acceleration vector of a moving object into its tangential and normal components.

How do you evaluate ${\int }_a^b\mathbf{r}(t)dt?$

Given the velocity of an object and its initial position, how do you find the position of the object, for $t\ge 0?$

How do you determine whether $\mathbf{r}(t)=f(t)\mathbf{i}+g(t)\mathbf{j}+h(t)\mathbf{k}$ is continuous at $t=a?$

The velocity of a moving object, for $t\ge 0,$ is ${\mathbf{r}}^{\prime }(t)=⟨60,96-32t⟩\mathrm{ft}/\mathrm{s}$ a. When is the vertical component of velocity of the object equal to $0?$ b. Find $r(t)$ if $r(0)=(0,3)$.

Explain what it means for a curve to be parameterized by its are length.