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Chapter 11: Problem 1

What is the curvature of a straight line?

Short Answer

Expert verified
Answer: The curvature of a straight line is zero, as there is no deviation from being a straight line.

Step by step solution

01

Understand Curvature

Curvature is a measure of how much a curve deviates from being a straight line. A curve with higher curvature changes its direction more rapidly, while one with lower curvature changes its direction more slowly.
02

Determine Curvature of a Curve

The curvature of a curve can be found using the following formula, given a function y=f(x): $$k(x) = \frac{|f''(x)|}{(1 + (f'(x))^2)^{\frac{3}{2}}}$$ Where: - \(k(x)\) is the curvature at any point x - f'(x) is the first derivative of the function, which represents the slope at point x - f''(x) is the second derivative of the function, which represents the rate of change of the slope at point x
03

Determine Function of a Straight Line

A straight line can be represented by a linear function of the form \(y = mx + b\), where m is the slope of the line and b is the y-intercept.
04

Calculate the Derivatives

For a straight line with the function \(y = mx + b\), calculate the first and second derivatives: - First Derivative: \(f'(x) = m\) - Second Derivative: \(f''(x) = 0\)
05

Calculate the Curvature of a Straight Line

Plug the derivatives of the straight line into the curvature formula: \begin{align*} k(x) &= \frac{|0|}{(1 + (m)^2)^{\frac{3}{2}}} \\ &= 0 \end{align*}
06

Conclusion

The curvature of a straight line is zero, as there is no deviation from being a straight line.

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Most popular questions from this chapter

Prove the following vector properties using components. Then make a sketch to illustrate the property geometrically. Suppose \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) are vectors in the \(x y\) -plane and a and \(c\) are scalars. $$a(c \mathbf{v})=(a c) \mathbf{v}$$

Let $$\mathbf{u}(t)=2 t^{3} \mathbf{i}+\left(t^{2}-1\right) \mathbf{j}-8 \mathbf{k} \text { and } \mathbf{v}(t)=e^{t} \mathbf{i}+2 e^{-t} \mathbf{j}-e^{2 t} \mathbf{k}$$ Compute the derivative of the following functions. $$\mathbf{v}(\sqrt{t})$$

Evaluate the following definite integrals. $$\int_{0}^{\ln 2}\left(e^{-t} \mathbf{i}+2 e^{2 t} \mathbf{j}-4 e^{t} \mathbf{k}\right) d t$$

Orthogonal lines Recall that two lines \(y=m x+b\) and \(y=n x+c\) are orthogonal provided \(m n=-1\) (the slopes are negative reciprocals of each other). Prove that the condition \(m n=-1\) is equivalent to the orthogonality condition \(\mathbf{u} \cdot \mathbf{v}=0,\) where \(\mathbf{u}\) points in the direction of one line and \(\mathbf{v}\) points in the direction of the other line..

A golfer launches a tee shot down a horizontal fairway; it follows a path given by \(\mathbf{r}(t)=\left\langle a t,(75-0.1 a) t,-5 t^{2}+80 t\right\rangle,\) where \(t \geq 0\) measures time in seconds and \(\mathbf{r}\) has units of feet. The \(y\) -axis points straight down the fairway and the \(z\) -axis points vertically upward. The parameter \(a\) is the slice factor that determines how much the shot deviates from a straight path down the fairway. a. With no slice \((a=0),\) sketch and describe the shot. How far does the ball travel horizontally (the distance between the point the ball leaves the ground and the point where it first strikes the ground)? b. With a slice \((a=0.2),\) sketch and describe the shot. How far does the ball travel horizontally? c. How far does the ball travel horizontally with \(a=2.5 ?\)
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