Chapter 10: Problem 41
In Exercises 41 and \(42,\) use the definition of the derivative to find \(\mathbf{r}^{\prime}(t)\). $$ \mathbf{r}(t)=(3 t+2) \mathbf{i}+\left(1-t^{2}\right) \mathbf{j} $$
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Use the model for projectile motion, assuming there is no air resistance. Determine the maximum height and range of a projectile fired at a height of 3 feet above the ground with an initial velocity of 900 feet per second and at an angle of \(45^{\circ}\) above the horizontal.
Use the given acceleration function to find the velocity and position vectors. Then find the position at time \(t=2\) $$ \begin{array}{l} \mathbf{a}(t)=2 \mathbf{i}+3 \mathbf{k} \\ \mathbf{v}(0)=4 \mathbf{j}, \quad \mathbf{r}(0)=\mathbf{0} \end{array} $$
The position vector \(r\) describes the path of an object moving in space. Find the velocity, speed, and acceleration of the object. $$ \mathbf{r}(t)=t \mathbf{i}+t \mathbf{j}+\sqrt{9-t^{2}} \mathbf{k} $$
Find \(\mathbf{r}(t)\) for the given conditions. $$ \mathbf{r}^{\prime}(t)=t e^{-t^{2}} \mathbf{i}-e^{-t} \mathbf{j}+\mathbf{k}, \quad \mathbf{r}(0)=\frac{1}{2} \mathbf{i}-\mathbf{j}+\mathbf{k} $$
Prove the property. In each case, assume that \(\mathbf{r}, \mathbf{u},\) and \(\mathbf{v}\) are differentiable vector-valued functions of \(t,\) \(f\) is a differentiable real-valued function of \(t,\) and \(c\) is a scalar. $$ D_{t}[\mathbf{r}(t) \times \mathbf{u}(t)]=\mathbf{r}(t) \times \mathbf{u}^{\prime}(t)+\mathbf{r}^{\prime}(t) \times \mathbf{u}(t) $$
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