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7/21/2019 tut_2

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Math236 – Week 2 Tutorial Exercises

Some comments for question 6. The hypocycloid based on a small wheel of radius r rollinginside a circle of radius R creates a path in the plane given by

ϕ(θ) =

(R − r)cos θ + r cos

R − r

r

θ

, (R − r)sin θ − r sin

R − r

r

θ

This is derived on page 97 the book Trigonometric Delights by Eli Maor, which is available at

the link:http://press.princeton.edu/books/maor/chapter_7.pdf.The derivative of this path is

ϕ(θ) = (R − r)

− sin θ − sin

R − r

r

θ

, cos θ − cos

R − r

r

θ

The speed of the path is ϕ(θ) · ϕ(θ)

and

ϕ(θ) · ϕ(θ) = (R − r)2

2 + 2 sin(θ)sin

θ(R − r)

r

− 2 cos(θ)cos

θ(R − r)

r

= 4(R − r)2 sin2

Rθ

2r

.

This means that the speed is

ϕ(θ) = 2(R − r)

sin

Rθ

2r

.The cusps occur at the points corresponding to

Rθ

2r = mπ, for some integer m.

For example to get exactly three cusps, we would need them to occur at θ = 0, 2π/3 and 4π/3,

so having R three times r works.

0.4 0.2 0.2 0.4 0.6 0.8 1.0

0.5

0.5

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Ex. 1. Find φ(0), φ�(t) and φ�(0) in each of the following cases of paths in R2 or R3 :

(a) φ : [0 , 1] → R3 : φ(t) =

sin2πt , cos2πt , 2t− t2

;

(b) φ : [0 , 4π] → R3 : φ(t) =et, cos t , sin t

;

(c) φ : [0 , 4] → R2 : φ(t) =t2, t3 − 4t

;

(d) φ : [0 , 6π] → R3 : φ(t) =

sin2t , log(1 + t) , t

.

Ex. 2. Determine the velocity vector and the equation of the tangent line for each of the following paths,at the specified value of t.

(a) r(t) = (6t , 3t2, t3), at t = 0

(b) φ(t) = (sin 3t , cos3t , 2t3/2) , at t = 1

(c) φ(t) = (cos2 t , 3t− t3, t) , at t = 0

(d) φ(t) = (0 , 0 , t) at t = 1 .

Ex. 3. Find the path φ such that φ(0) = (0 ,−5 , 1) and φ�

(t) =t , et , t

2.

Ex. 4. Suppose a particle follows the path φ(t) =et, e−t, cos t

until it flies off on a tangent at t = 1 .

Where is it at t = 2 ?

Ex. 5. Suppose a particle following the path φ(t) =t2, t3 − 4t

flies off on a tangent at t = 2 .

Compute the position of the particle at t = 3 .

Ex. 6. Find an explicit form describing the path of a particle moving along a hypocycloid with (i) 5 cusps,(ii) 6 cusps, (iii) 3 cusps. (iv) What can you say about a hypocycloid with only 2 cusps?[In each case centre the hypocycloid at (0, 0) and let there be a cusp at the point (1, 0).]

Ex. 7. Suppose that z0 ∈ C is fixed. A polynomial P (z) is said to be divisible by z − z0 if there isanother polynomial Q(z) such that P (z) = (z − z0)Q(z) .

(a) Show that for every c ∈ C and k ∈ N , the polynomial c(zk − zk0

) is divisible by z − z0 .

(b) Consider the polynomial P (z) = a0 + a1z + a2z2 + . . . + anz

n, where a0, a1, a2, . . . , an ∈ Care arbitrary. Show that the polynomial P (z)− P (z0) is divisible by z − z0 .

(c) Deduce that P (z) is divisible by z − z0 if P (z0) = 0 .

(d) Suppose that a polynomial P (z) of degree n vanishes at n distinct values z1, z2, . . . , zn ∈ C ,so that P (z1) = P (z2) = . . . = P (zn) = 0 .Show that P (z) = c(z − z1)(z − z2) . . . (z − zn) , where c ∈ C is a constant.

(e) Suppose that a polynomial P (z) of degree at most n vanishes at more than n distinct values.

Show that P (z) = 0 identically.

Ex. 8. Suppose that α ∈ C is fixed and |α| < 1 . Show that |z| ≤ 1 if and only if z − α

1 − αz

≤ 1 .

Ex. 9. Suppose that z = x + iy, where x, y ∈ R .Express each of the following in terms of x and y:

(a) |z − 1|3 (b)z + 1

z − 1

(c)

z + i

−iz

.

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Ex. 10. Suppose that c ∈ R and α ∈ C with α �= 0 .

(a) Show that αz + αz + c = 0 is the equation of a straight line on the plane.

(b) What does the equation zz + αz + αz + c = 0 represent if |α|2 ≥ c?

Ex. 11. Suppose that z, w ∈ C. Show that |z + w|2 + |z − w|2 = 2|z|2 + |w|2

.

Ex. 12. Find all the roots of the equation (z8 − 1)(z3 + 8) = 0.

Ex. 13. For each of the following, compute all the values and plot them on the plane:

(a) (1 + i)−1/2 (b) (−4)3/4 (c) (1− i)3/8.

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