Question 1

Which of the following best describes the curvature $$\kappa of a $$straight line in the plane?

Accepted Answer

$$\kappa is $$zero everywhere along the line.

Question 2

Given a circle of radius $$a$$, what is the curvature $$\kappa at $$any point on the circle?

Accepted Answer

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

Question 3

For a smooth curve parameterized by $$\vec{r}(t)$$, which formula correctly computes the curvature $$\kappa$$?

Accepted Answer

$$\kappa = \dfrac{1}{v} \left| \dfrac{d\vec{T}}{dt} \right|$$

Question 4

The principal normal vector $$\vec{N} at a $$point on a curve is defined as:

Accepted Answer

$$\vec{N} = \dfrac{\dfrac{d\vec{T}}{dt}}{\left| \dfrac{d\vec{T}}{dt} \right|}$$

Question 5

At which points on an ellipse is the curvature maximal?

Accepted Answer

At the endpoints of the major axis.

Question 6

Suppose a car is driving along a plane curve. In which direction does the principal normal vector $$\vec{N}$$ point?

Accepted Answer

Toward the concave side of the curve.

Question 7

For the parabola $$y = x^2$, what is the radius of curvature at the origin?

Accepted Answer

$$\dfrac{1}{2}$$

Question 8

Which of the following statements about the osculating circle at a point $$P on a $$plane curve is TRUE?

Accepted Answer

It has the same tangent and curvature as the curve at $$P.$$

Question 9

Given the helix $$\vec{r}(t) = (a \cos t, a \sin t, bt)$$ with $$a \u003e 0, b \u003e 0$$, what is the curvature $$\kappa$$?

Accepted Answer

$$\kappa = \dfrac{a}{a^2$ + b^2$}$$

Question 10

A vertex of a plane curve is defined as:

Accepted Answer

A point where the curvature has a local maximum or minimum.

Question 11

If $$\vec{T} is $$the unit tangent vector and $$s is $$the arc length, what does $$\left| \dfrac{d\vec{T}}{ds} \right|$$ represent?

Accepted Answer

The curvature of the curve.

Question 12

For a curve $$\vec{r}(t)$$, the speed $$v is $$given by:

Accepted Answer

$$v = \left| \dfrac{d\vec{r}}{dt} \right|$$

Question 13

Which of the following is NOT a property of the osculating circle at a point $$P on a $$curve?

Accepted Answer

Its radius is always equal to the arc length at $$P.$$

Question 14

Suppose $$\vec{r}(t) is a $$smooth curve in space. Which formula gives the principal unit normal vector $$\vec{N}$$?

Accepted Answer

$$\vec{N} = \dfrac{\dfrac{d\vec{T}}{dt}}{\left| \dfrac{d\vec{T}}{dt} \right|}$$

Question 15

If the curvature $$\kappa at a $$point on a curve is very large, what does this indicate about the curve at that point?

Accepted Answer

The curve is turning sharply.

Question 16

For the parabola $$y = x^2$, what are the coordinates of the center of curvature at the origin?

Accepted Answer

$$(0, \frac{1}{2})$$

Question 17

Which of the following best describes the direction of the principal normal vector $$\vec{N}$$ for the helix $$\vec{r}(t) = (a \cos t, a \sin t, bt)$$?

Accepted Answer

$$\vec{N}$$ points parallel to the $$xy-$$plane toward the $$z-$$axis.

Question 18

Which of the following is a correct interpretation of the Four-Vertex Theorem for an ellipse?

Accepted Answer

An ellipse has exactly four points where the curvature is locally maximal or minimal.

Curvature and Normal Vectors of a Curve

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