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Chapter 10: Problem 19

Find the principal unit normal vector to the curve at the specified value of the parameter. $$ \mathbf{r}(t)=\ln t \mathbf{i}+(t+1) \mathbf{j}, \quad t=2 $$

Short Answer

Expert verified
The principal unit normal vector to the curve at \(t=2\) can be found by following these steps, calculating the necessary derivatives, their magnitudes, and the unit vectors.

Step by step solution

01

Find the Derivative of the Vector Function

Using the power rule, the derivative of the function can be found as \(\mathbf{r}'(t)=\frac{1}{t}\mathbf{i}+ \mathbf{j}\).
02

Determine the Magnitude of the Derivative

The magnitude of the derivative vector can be found as \(\|\mathbf{r}'(t)\|=\sqrt{(\frac{1}{t})^2+1}\).
03

Calculate the Unit Tangent Vector

The unit tangent vector, \(\mathbf{T}(t)\), can be calculated using the formula \(\mathbf{T}(t)=\frac{\mathbf{r}'(t)}{\|\mathbf{r}'(t)\|}\).
04

Find the Derivative of the Unit Tangent Vector

The derivative of the unit tangent vector can be found as \(\mathbf{T}'(t)\).
05

Determine the Magnitude of the Derivative of the Tangent Vector

The magnitude of the derivative of the tangent vector can be found as \(\|\mathbf{T}'(t)\|\).
06

Calculate the Principal Unit Normal Vector

The principal unit normal vector, \(\mathbf{N}(t)\), can be calculated using the formula \(\mathbf{N}(t)=\frac{\mathbf{T}'(t)}{\|\mathbf{T}'(t)\|}\).

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Most popular questions from this chapter

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. $$ D_{t}[\mathbf{r}(f(t))]=\mathbf{r}^{\prime}(f(t)) f^{\prime}(t) $$

In Exercises 41 and \(42,\) use the definition of the derivative to find \(\mathbf{r}^{\prime}(t)\). $$ \mathbf{r}(t)=(3 t+2) \mathbf{i}+\left(1-t^{2}\right) \mathbf{j} $$

Use the model for projectile motion, assuming there is no air resistance. A projectile is fired from ground level at an angle of \(12^{\circ}\) with the horizontal. The projectile is to have a range of 150 feet. Find the minimum initial velocity necessary.

Find \(\mathbf{r}(t)\) for the given conditions. $$ \mathbf{r}^{\prime}(t)=\frac{1}{1+t^{2}} \mathbf{i}+\frac{1}{t^{2}} \mathbf{j}+\frac{1}{t} \mathbf{k}, \quad \mathbf{r}(1)=2 \mathbf{i} $$

Use the model for projectile motion, assuming there is no air resistance. Use a graphing utility to graph the paths of a projectile for the given values of \(\theta\) and \(v_{0} .\) For each case, use the graph to approximate the maximum height and range of the projectile. (Assume that the projectile is launched from ground level.) (a) \(\theta=10^{\circ}, v_{0}=66 \mathrm{ft} / \mathrm{sec}\) (b) \(\theta=10^{\circ}, v_{0}=146 \mathrm{ft} / \mathrm{sec}\) (c) \(\theta=45^{\circ}, v_{0}=66 \mathrm{ft} / \mathrm{sec}\) (d) \(\theta=45^{\circ}, v_{0}=146 \mathrm{ft} / \mathrm{sec}\) (e) \(\theta=60^{\circ}, v_{0}=66 \mathrm{ft} / \mathrm{sec}\) (f) \(\theta=60^{\circ}, v_{0}=146 \mathrm{ft} / \mathrm{sec}\)
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