r(t) = 〈4t², 9t〉

r(t) = 〈e^t, t²〉

r(t) = 〈3 + 4t, 3 − 5t, 9t〉

r(t) = 〈1 + 2t, t², 3 − t²〉

r(t) = 〈cos πt, sin πt, t〉

r(t) = 〈e^t, e^−t, t²〉

r(t) = 〈1, e^t, t〉

r(t) = 〈4 cos t, t, 4 sin t〉

r(t) = 〈4t + 1, 4t − 3, 2t〉

r(t) = 〈t⁻¹, 1, t〉

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

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.

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

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

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

Find the point of maximum curvature on y = e^x.

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

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.

In the notation of Exercise 25, assume that a ≥ b. Show that b/a² ≤ κ(t) ≤ a/b² for all t.

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

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

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).

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

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.

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

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?

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

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

Find the unit normal to the clothoid (Exercise 35) at t = π^1/3.

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).

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

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

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).

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 −S ≤ y ≤ S, 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.

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

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

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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)?

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

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

f(θ) = 2 cos θ

f(θ) = θ

f(θ) = e^θ

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

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).

Show that

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

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).

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).