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Chapter 10: Problem 40

Find the curvature and radius of curvature of the plane curve at the given value of \(x\). $$ y=2 x+\frac{4}{x}, \quad x=1 $$

Short Answer

Expert verified
The curvature of the plane curve at \( x = 1 \) is \( \kappa = \frac{8}{3 \sqrt{5}} \) and the radius of curvature is \( R = \frac{3 \sqrt{5}}{8} \)

Step by step solution

01

Calculate the First Derivative

We need to calculate the first derivative of the function. The function is \( y = 2x + \frac{4}{x} \), which can be written as \( y = 2x + 4x^{-1} \). The derivative \( y' \) is then calculated as follows: \( y' = \frac{d}{dx}(2x) + \frac{d}{dx}(4x^{-1}) = 2 - 4x^{-2} \) at \( x = 1 \), it becomes \( y'(1) = 2 - 4 = -2 \).
02

Calculate the Second Derivative

The next step is to calculate the second derivative of the function. The derivative \( y'' \) is calculated from \( y' \) as follows: \( y'' = \frac{d}{dx}(2) - \frac{d}{dx}(4x^{-2}) = 0 + 8x^{-3} \) at \( x = 1 \), it becomes \( y''(1) = 8 \).
03

Calculate the Curvature

Now that we have the first and second derivatives, we can calculate the curvature using the formula \( \kappa = |y''| / (1 + (y')^2)^{3/2} \). Plug in \( y'(1) = -2 \) and \( y''(1) = 8 \) into the formula, we get \( \kappa = |8| / (1 + (-2)^2)^{3/2} = 8 / 3 \sqrt{5} \).
04

Find the Radius of Curvature

Finally, we can find the radius of curvature using the formula \( R = 1 / | \kappa | \). Substitute \( \kappa = 8 / 3 \sqrt{5} \) in the equation, we get \( R = 3 \sqrt{5} / 8 \).

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Most popular questions from this chapter

Consider a particle moving on a circular path of radius \(b\) described by $$ \begin{aligned} &\mathbf{r}(t)=b \cos \omega t \mathbf{i}+b \sin \omega t \mathbf{j}\\\ &\text { where } \omega=d \theta / d t \text { is the constant angular velocity. } \end{aligned} $$ Find the acceleration vector and show that its direction is always toward the center of the circle.

Use the given acceleration function to find the velocity and position vectors. Then find the position at time \(t=2\) $$ \begin{array}{l} \mathbf{a}(t)=\mathbf{i}+\mathbf{j}+\mathbf{k} \\ \mathbf{v}(0)=\mathbf{0}, \quad \mathbf{r}(0)=\mathbf{0} \end{array} $$

True or False? Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ \text { The acceleration of an object is the derivative of the speed. } $$

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

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. If \(\mathbf{r}(t) \cdot \mathbf{r}(t)\) is a constant, then \(\mathbf{r}(t) \cdot \mathbf{r}^{\prime}(t)=0\)
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