Question

How to solve this with steps? its Calculus-3

Answer 1

`\mathbf{v}(\pi/4)=-3\mathbf{i}+3\mathbf{j}+3 \mathbf{k}`

Explanation:

Instantaneous Velocity

Given r(t) as the vector function of the position for time t, the instantaneous velocity is computed as:

`\mathbf{v}=\frac{d\mathbf{r}}{dt}=\mathbf{r}'(t)`

We are given:

`\mathbf{r}=\sin^2(3t)\mathbf{i}+3t\mathbf{j}-\cos^2(3t)\mathbf{k}`

Thus:

`\mathbf{v}(t)=\mathbf{r}'(t)=[\sin^2(3t)\mathbf{i}+3t\mathbf{j}-\cos^2(3t)\mathbf{k}]'`

Computing the derivative:

`\mathbf{v}(t)=2\sin(3t)\cos(3t)(3)\mathbf{i}+3\mathbf{j}+2\cos(3t)\sin(3t)(3)\mathbf{k}`

`\mathbf{v}(t)=6\sin(3t)\cos(3t)\mathbf{i}+3\mathbf{j}+6\cos(3t)\sin(3t)\mathbf{k}`

Evaluating for t=π/4:

`\mathbf{v}(\pi/4)=6\sin(3\pi/4)\cos(3\pi/4)\mathbf{i}+3\mathbf{j}+6\cos(3\pi/4)\sin(3\pi/4) \mathbf{k}`

`\mathbf{v}(\pi/4)=6(√(2)/2)(-√(2)/2)\mathbf{i}+3\mathbf{j}+6(-√(2)/2))(√(2)/2)) \mathbf{k}`

`\mathbf{v}(\pi/4)=-3\mathbf{i}+3\mathbf{j}-3 \mathbf{k}`