Q: Which of the following best describes the curvature $$\kappa of a $$curve given by $$\mathbf{r}(t)$$?

$$\kappa = \dfrac{\left| \mathbf{r}'(t) \times \mathbf{r}''(t) \right|}{\left| \mathbf{r}'(t) \right|^3}$$

Question 1

Suppose a curve is given by the vector function $$\mathbf{r}(t) = \langle f_1$(t), f_2$(t), f_3$(t) \rangle$$ for $$a \leq t \leq b. $$What is the formula for the arc length $$L of $$the curve from $$t = a to$$ $$t = b$$?

Accepted Answer

$$L = \int_a^b \sqrt{[f_1$'(t)]^2 + [f_2$'(t)]^2 + [f_3$'(t)]^2} \, dt$$

Question 2

Given the vector function $$\mathbf{r}(t) = \cos t \, \mathbf{i} + \sin t \, \mathbf{j} + t \, \mathbf{k}$$, what is the length of the arc from $$t = 0 to$$ $$t = 2\pi$$?

Accepted Answer

$$2\pi \sqrt{2}$$

Question 3

Which of the following best describes the curvature $$\kappa of a $$curve given by $$\mathbf{r}(t)$$?

Accepted Answer

$$\kappa = \dfrac{\left| \mathbf{r}'(t) 	imes \mathbf{r}''(t) ight|}{\left| \mathbf{r}'(t) ight|^3}$$

Question 4

For a plane curve $$y = f(x)$$, what is the formula for curvature $$\kappa at a $$point $$x$$?

Accepted Answer

$$\kappa = \dfrac{f''(x)}{[1 + (f'(x))^2$]^{3/2}}$$

Question 5

A circular helix is described by $$\mathbf{r}(t) = \cos t \, \mathbf{i} + \sin t \, \mathbf{j} + t \, \mathbf{k}. $$What is the unit tangent vector $$\mathbf{T}(t) at $$any point $$t$$?

Accepted Answer

$$\mathbf{T}(t) = \dfrac{\langle -\sin t, \cos t, 1 \rangle}{\sqrt{2}}$$

Question 6

If a curve $$C is $$parameterized by arc length $$s$$, what is the value of $$\left| \dfrac{d\mathbf{r}}{ds} \right|$$?

Accepted Answer

$$1$$

Question 7

The radius of curvature $$\rho at a $$point on a curve is related to the curvature $$\kappa by $$which formula?

Accepted Answer

$$\rho = \dfrac{1}{\kappa}$$

Question 8

A car travels along a path described by $$y = x^2$. At the point (1$$, 1)$$, what is the curvature of the path?

Accepted Answer

$$\dfrac{2}{[1 + 4]^{3/2}}$$

Question 9

Which vector is always perpendicular to both the tangent and normal vectors of a curve?

Accepted Answer

The binormal vector $$\mathbf{B}(t)$$

Question 10

Suppose a roller coaster track is modeled by $$\mathbf{r}(t) = t^3$ \, \mathbf{i} + t^2$ \, \mathbf{j} + t \, \mathbf{k}. $$What is the curvature $$\kappa at$$ $$t = 1$$?

Accepted Answer

$$\kappa = \dfrac{\sqrt{29}}{14$$^{3/2}$$}$$

Question 11

Which of the following statements about arc length parameterization is TRUE?

Accepted Answer

Arc length parameterization is possible for any smooth curve.

Question 12

A medical imaging device traces a path in space given by $$\mathbf{r}(t) = e^t \, \mathbf{i} + e$$^{-t}$$ \, \mathbf{j} + t \, \mathbf{k}. $$What is the formula for the arc length from $$t = 0 to$$ $$t = 1$$?

Accepted Answer

$$\$$int_0$$^1 \sqrt{[e^t]^2 + [e$$^{-t}$$]^2 + 1} \, dt$$

Question 13

Which of the following is the correct expression for the unit tangent vector $$\mathbf{T}(t)$$ for a curve $$\mathbf{r}(t)$$?

Accepted Answer

$$\mathbf{T}(t) = \dfrac{\mathbf{r}'(t)}{\left| \mathbf{r}'(t) \right|}$$

Question 14

In the context of curvature, what does the principal unit normal vector $$\mathbf{N}(t)$$ indicate?

Accepted Answer

The direction in which the curve is turning at each point.

Question 15

A circle of radius $$a$$ has curvature $$\kappa at $$any point. What is $$\kappa$$?

Accepted Answer

$$\kappa = \dfrac{1}{a}$$

Question 16

A curve $$\mathbf{r}(t) is $$traversed exactly once as $$t$$ increases from $$a to$$ $$b. $$Which of the following is NOT a valid way to compute its arc length?

Accepted Answer

$$L = \int_a^b \mathbf{r}(t) \, dt$$

Question 17

A drone follows a path $$\mathbf{r}(t) = t \, \mathbf{i} + t^2$ \, \mathbf{j} + t^3$ \, \mathbf{k}. $$What is the magnitude of the velocity vector at $$t = 2$$?

Accepted Answer

$$\sqrt{1^2$ + (2 \cdot 2)^2$ + (3 \cdot 8)^2$}$$

Question 18

Which of the following best describes the osculating plane at a point $$P on a $$curve?

Accepted Answer

The plane containing the tangent and normal vectors at $$P.$$

Section 3.3 - Arc Length and Curvature

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