Question 1

Given the parabola parameterized by $$r(t) = (t, t^2$)$$, which of the following best describes the behavior of its curvature $$\kappa as$$ $$|t| 	o \infty$$?

Accepted Answer

The curvature $$\kappa$$ approaches zero.

Question 2

At which point on the parabola $$r(t) = (t, t^2$)$$ does the curvature attain its maximum value?

Accepted Answer

At $$t = 0$$

Question 3

Which of the following statements about the curvature of the parabola $$r(t) = (t, t^2$) is $$correct?

Accepted Answer

The curvature is maximum at $$t = 0$$ and decreases as $$|t|$$ increases.

Question 4

What does the flattening of the parabola as $$|t| 	o \infty$$ indicate about its curvature?

Accepted Answer

The curvature approaches zero as the curve flattens.

Question 5

If the curvature $$\kappa of a $$curve is large at a point, what does this imply about the curve at that point?

Accepted Answer

The curve changes direction rapidly at that point.

Question 6

For the parabola $$r(t) = (t, t^2$)$$, what is the geometric significance of the point where the curvature is maximum?

Accepted Answer

It is the vertex of the parabola.

Question 7

Which formula correctly gives the curvature $$\kappa$$ for a plane curve $$r(t) = (x(t), y(t))$$?

Accepted Answer

$$\kappa = \frac{|x'(t) y''(t) - y'(t) x''(t)|}{(x'(t)^2$ + y'(t)^2$)^{3/2}$$}$

Question 8

Suppose you are designing a roller coaster and want the ride to feel smooth at the ends. What should the curvature of the track do as you move away from the center?

Accepted Answer

The curvature should decrease to zero.

Question 9

If a curve has zero curvature at a point, what does this imply about the curve at that point?

Accepted Answer

The curve is a straight line at that point.

Question 10

For the parabola $$r(t) = (t, t^2$)$$, what happens to the radius of curvature as $$|t|$$ increases?

Accepted Answer

It increases.

Calc 8: Arc Length and Curvature of Vector Functions in Calculus

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