Exercises

In Exercises 1–6, calculate r′(t) and T(t), and evaluate T(1).

Question 13.177

r(t) = 〈4t2, 9t

Question 13.178

r(t) = 〈et, t2

Question 13.179

r(t) = 〈3 + 4t, 3 − 5t, 9t

Question 13.180

r(t) = 〈1 + 2t, t2, 3 − t2

Question 13.181

r(t) = 〈cos πt, sin πt, t

Question 13.182

r(t) = 〈et, et, t2

In Exercises 7–10, use Eq. (3) to calculate the curvature function κ(t).

Question 13.183

r(t) = 〈1, et, t

Question 13.184

r(t) = 〈4 cos t, t, 4 sin t

Question 13.185

r(t) = 〈4t + 1, 4t − 3, 2t

Question 13.186

r(t) = 〈t−1, 1, t

In Exercises 11–14, use Eq. (3) to evaluate the curvature at the given point.

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Question 13.190

In Exercises 15–18, find the curvature of the plane curve at the point indicated.

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Question 13.195

Find the curvature of r(t) = 〈2 sin t, cos 3t, t〉 at and (Figure 16).

The curve r(t) = 〈2 sin t, cos 3t, t〉.

Question 13.196

Find the curvature function κ(x) for y = sin x. Use a computer algebra system to plot κ(x) for 0 ≤ x ≤ 2π. Prove that the curvature takes its maximum at and . Hint: As a shortcut to finding the max, observe that the maximum of the numerator and the minimum of the denominator of κ(x) occur at the same points.

Question 13.197

Show that the tractrix r(t) = 〈t − tanh t, sech t〉 has the curvature function κ(t) = sech t.

Question 13.198

Show that curvature at an inflection point of a plane curve y = f(x) is zero.

Question 13.199

Find the value of α such that the curvature of y = e αx at x = 0 is as large as possible.

Question 13.200

Find the point of maximum curvature on y = ex.

Question 13.201

Show that the curvature function of the parametrization r(t) = 〈a cos t, b sin t〉 of the ellipse is

Question 13.202

Use a sketch to predict where the points of minimal and maximal curvature occur on an ellipse. Then use Eq. (9) to confirm or refute your prediction.

Question 13.203

In the notation of Exercise 25, assume that ab. Show that b/a2κ(t) ≤ a/b2 for all t.

Question 13.204

Use Eq. (3) to prove that for a plane curve r(t) = 〈x(t), y(t)〉,

In Exercises 29–32, use Eq. (10) to compute the curvature atthe given point.

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Question 13.209

Let for the Bernoulli spiral r(t) = 〈et cos 4t, et sin 4t〉 (see Exercise 29 in Section 13.3). Show that the radius of curvature is proportional to s(t).

Question 13.210

The Cornu spiral is the plane curve r(t) = 〈x(t), y(t)〉, where

Verify that κ(t) = |t|. Since the curvature increases linearly, the Cornu spiral is used in highway design to create transitions between straight and curved road segments (Figure 17).

Cornu spiral.

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Question 13.211

Plot and compute the curvature κ(t) of the clothoid r(t) = 〈x(t), y(t)〉, where

Question 13.212

Find the unit normal vector N(θ) to r(θ) = R 〈cos θ, sin θ〉, the circle of radius R. Does N(θ) point inside or outside the circle? Draw N(θ) at with R = 4.

Question 13.213

Find the unit normal vector N(t) to r(t) = 〈4, sin 2t, cos 2t〉.

Question 13.214

Stetch the graph of r(t) = 〈t, t3〉. Since r′(t) = 〈1, 3t2〉, the unit normal N(t) points in one of the two directions ±〈−3t2, 1〉. Which sign is correct at t = 1? Which is correct at t = −1?

Question 13.215

Find the normal vectors to r(t) = 〈t, cos t〉 at and .

Question 13.216

Find the unit normal to the Cornu spiral (Exercise 34) at .

Question 13.217

Find the unit normal to the clothoid (Exercise 35) at t = π1/3.

Question 13.218

Method for Computing N Let υ(t) = ∥r′(t)∥. Show that

Hint: N is the unit vector in the direction T′(t). Differentiate T(t) = r′(t)/υ(t) to show that υ(t)r″(t) − υ′(t)r′(t) is a positive multiple of T′(t).

In Exercises 43–48, use Eq. (11) to find N at the point indicated.

Question 13.219

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Question 13.225

Let f(x) = x2. Show that the center of the osculating circle at is given by .

Question 13.226

Use Eq. (8) to find the center of curvature r(t) = 〈t2, t3〉 at t = 1.

In Exercises 51–58, find a parametrization of the osculating circle at the point indicated.

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Question 13.235

Figure 18 shows the graph of the half-ellipse , where r and p are positive constants. Show that the radius of curvature at the origin is equal to r. Hint: One way of proceeding is to write the ellipse in the form of Exercise 25 and apply Eq. (9).

The curve and the osculating circle at the origin.

Question 13.236

In a recent study of laser eye surgery by Gatinel, Hoang-Xuan, and Azar, a vertical cross section of the cornea is modeled by the half-ellipse of Exercise 59. Show that the half-ellipse can be written in the form x = f(y), where . During surgery, tissue is removed to a depth t(y) at height y for −SyS, where t(y) is given by Munnerlyn’s equation (for some R > r):

After surgery, the cross section of the cornea has the shape x = f(y) + t(y) (Figure 19). Show that after surgery, the radius of curvature at the point P (where y = 0) is R.

Contour of cornea before and after surgery.

Question 13.237

The angle of inclination at a point P on a plane curve is the angle θ between the unit tangent vector T and the x-axis (Figure 20). Assume that r(s) is a arc length parametrization, and let θ = θ(s) be the angle of inclination at r(s). Prove that

The curvature at P is the quantity |/ds|.

Hint: Observe that T(s) = 〈cos θ(s), sin θ(s)〉.

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Question 13.238

A particle moves along the path y = x3 with unit speed. How fast is the tangent turning (i.e., how fast is the angle of inclination changing) when the particle passes through the point (2, 8)?

Question 13.239

Let θ(x) be the angle of inclination at a point on the graph y = f(x) (see Exercise 61).

  • Use the relation f′(x) = tan θ to prove that .

  • Use the arc length integral to show that .

  • Now give a proof of Eq. (5) using Eq. (12).

Question 13.240

Use the parametrization r(θ) = 〈f(θ) cos θ, f(θ) sin θ〉 to show that a curve r = f(θ) in polar coordinates has curvature

In Exercises 65–67, use Eq. (13) to find the curvature of the curve given in polar form.

Question 13.241

f(θ) = 2 cos θ

Question 13.242

f(θ) = θ

Question 13.243

f(θ) = eθ

Question 13.244

Use Eq. (13) to find the curvature of the general Bernoulli spiral r = aebθ in polar form (a and b are constants).

Question 13.245

Show that both r′(t) and r″(t) lie in the osculating plane for a vector function r(t). Hint: Differentiate r′(t) = υ(t)T(t).

Question 13.246

Show that

is an arc length parametrization of the osculating circle at r(t0).

Question 13.247

Two vector-valued functions r1(s) and r2(s) are said to agree to order 2 at s0 if

Let r(s) be an arc length parametrization of a path , and let P be the terminal point of r(0). Let γ(s) be the arc length parametrization of the osculating circle given in Exercise 70. Show that r(s) and γ(s) agree to order 2 at s = 0 (in fact, the osculating circle is the unique circle that approximates to order 2 at P).

Question 13.248

Let r(t) = 〈x(t), y(t), z(t)〉 be a path with curvature κ(t) and define the scaled path r1(t) = 〈λx(t), λy(t), λz(t)〉, where λ ≠ 0 is a constant. Prove that curvature varies inversely with the scale factor. That is, prove that the curvature κ1(t) of r1(t) is κ1(t) = λ−1κ(t). This explains why the curvature of a circle of radius R is proportional to 1/R (in fact, it is equal to 1/R). Hint: Use Eq. (3).