Ranking Vector Function Acceleration Magnitudes

Question

For each of the vector–valued functions above (with time $$t$$ in an appropriate interval), the acceleration vector is found by differentiating each component twice. Rank the functions in order of increasing magnitude of the acceleration vector at the specified evaluation point: use $$t = 1$$ for Functions 1, 3, and 4; and $$t = \frac{\pi}{6}$$ for Function 2. (Recall that for a vector function $$\vec{r}(t) = \langle x(t), y(t) \rangle$$, the acceleration magnitude is given by $$|\vec{a}(t)| = \sqrt{(x''(t))^2+(y''(t))^2}.$$)
 
| Function      | $$\vec{r}(t)$$                               |
|---------------|----------------------------------------------|
| Function 1    | $$\langle t, t^2 \rangle$$                    |
| Function 2    | $$\langle \cos(t), \sin(t) \rangle$$          |
| Function 3    | $$\langle e^t, e^t \rangle$$                   |
| Function 4    | $$\langle t^2, \ln(t+1) \rangle$$             |

Accepted Answer

Function 2, Function 1, Function 4, Function 3

Answer

Function 2, Function 4, Function 1, Function 3

Answer

Function 1, Function 2, Function 4, Function 3

Answer

Function 1, Function 4, Function 2, Function 3

Function	$$\vec{r}(t)$$
Function 1	$$\langle t, t^2 \rangle$$
Function 2	$$\langle \cos(t), \sin(t) \rangle$$
Function 3	$$\langle e^t, e^t \rangle$$
Function 4	$$\langle t^2, \ln(t+1) \rangle$$

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