## Related questions with answers

r(t) = ti + 3tj + t²k, u(t) = 4ti + t²j + t³k Use the properties of the derivative to find the following. (a)r′(t) (b)d/dt [3r(t) - u(t)] (c)d/dt (5t)u(t) (d)d/dt [r(t)·u(t)] (e)d/dt [r(t)×u(t)] (f)d/dt r(2t)

Solutions

VerifiedIn this exercise, we need to use properties of the derivative to find the values of the derivative of combinations of the given vector-valued functions:

$\mathbf{r}(t)=f_1(t)\mathbf{i}+g_1(t)\mathbf{j}+h_1(t)\mathbf{k},$

where $f_1(t)=t,\,g_1(t)=3t,\,h_1(t)=t^2$, and

$\mathbf{u}(t)=f_2(t)\mathbf{i}+g_2(t)\mathbf{j}+h_2(t)\mathbf{k},$

where $f_2(t)=4t,\,g_2(t)=t^2,\,h_2(t)=t^3.$

a)Given

$r(t) = ti + 3tj + t^{2}k$ and $u(t) = 4ti + t^{2}j + t^{3}k$

$\implies r'(t) = i + 3j + 2tk$ and $u'(t) = 4i + 2tj + 3t^{2}k$

So, $r'(t) = i + 3j + 2tk$

If $r(t) = f(t)i + g(t)j + h(t)k$

then $r'(t) = f'(t)i + g'(t)j + h'(t)k$

where f,g,h are differentiable functions of t.

## Create a free account to view solutions

## Create a free account to view solutions

## Recommended textbook solutions

#### Thomas' Calculus

14th Edition•ISBN: 9780134438986 (11 more)Christopher E Heil, Joel R. Hass, Maurice D. Weir#### Calculus: Early Transcendentals

8th Edition•ISBN: 9781285741550 (6 more)James Stewart#### Calculus: Early Transcendentals

9th Edition•ISBN: 9781337613927 (1 more)Daniel K. Clegg, James Stewart, Saleem Watson## More related questions

1/4

1/7