Find r'(t)·r''(t) of r(t) = <2e²t, 4e⁻²t, te²t>

[chickyroger]

Homework Statement

If r(t) = $$\left\langle\$$2e^{2t},4e^{-2t},te^{2t} $$\right\rangle$$, find r'(t) $$\cdot$$ r''(t).

Homework Equations

r'(t) = $$\left\langle$$4e^{2t},-8e^{-2t},2te^{2t}+e^{2t} $$\right\rangle$$
r''(t) = $$\left\langle$$8e^{2t},16e^{2t},4te^{2t}+4e^{2t} $$\right\rangle$$

The Attempt at a Solution

r'(t) $$\cdot$$ r''(t) = 32e^{2t} - 128 + 8t^2e^{2t} + 8te^{2t} + 4te^{2t} + 4e^{2t}
r'(t) $$\cdot$$ r''(t) = 36e^{4t}+8t^2e^{2t}+12te^{2t}-128

I cannot see why my answer is wrong. Please help!

 

[vela]

Fixed your post to make it readable:

Homework Statement

If $\mathbf{r}(t) = \langle 2e^{2t}, 4e^{-2t}, te^{2t} \rangle$, find $\mathbf{r}'(t)\cdot\mathbf{r}''(t)$.

Homework Equations

$$\mathbf{r}'(t) = \langle 4e^{2t},-8e^{-2t},2te^{2t}+e^{2t} \rangle$$
$$\mathbf{r}''(t) = \langle 8e^{2t},16e^{2t},4te^{2t}+4e^{2t} \rangle$$

The Attempt at a Solution

$$\mathbf{r}'(t) \cdot \mathbf{r}''(t) = 32e^{2t} - 128 + 8t^2e^{2t} + 8te^{2t} + 4te^{2t} + 4e^{2t}$$
$$\mathbf{r}'(t) \cdot \mathbf{r}''(t) = 36e^{4t}+8t^2e^{2t}+12te^{2t}-128$$

I cannot see why my answer is wrong. Please help!

You didn't combine the exponentials together correctly.

 

[HallsofIvy]

Homework Statement

If r(t) = $$\left\langle\$$2e^{2t},4e^{-2t},te^{2t} $$\right\rangle$$, find r'(t) $$\cdot$$ r''(t).

Homework Equations

r'(t) = $$\left\langle$$4e^{2t},-8e^{-2t},2te^{2t}+e^{2t} $$\right\rangle$$

This is correct.
r''(t) = $$\left\langle$$8e^{2t},16e^{2t},4te^{2t}+4e^{2t} $$\right\rangle$$[/quote]
This is not but probably just a typo: the second component is $16e^{-2t}$.

The Attempt at a Solution

r'(t) $$\cdot$$ r''(t) = $$32e^{2t} - 128 + 8t^2e^{2t} + 8te^{2t} + 4te^{2t} + 4e^{2t}$$

And you have propogated the typo: the product in the second term is [itex](-8e^{-2t})(16e^{-2t})= -128e^{-2t}[/tex]<br /> <br /> <blockquote data-attributes="" data-quote="" data-source="" class="bbCodeBlock bbCodeBlock--expandable bbCodeBlock--quote js-expandWatch"> <div class="bbCodeBlock-content"> <div class="bbCodeBlock-expandContent js-expandContent "> <b>r'</b>(t) $$\cdot$$ <b>r''</b>(t) = 36e^{4t}+8t^2e^{2t}+12te^{2t}-128<br /> <br /> I cannot see why my answer is wrong. Please help! </div> </div> </blockquote>[/itex]