344 CHAPTER 16. SPACE CURVES

4. Let r (t) =(

4+ t2,√

t2 +1t3, t3)

describe the position of an object in R3 as a func-tion of t where t is measured in seconds and r (t) is measured in meters. Is thevelocity of this object ever equal to zero? If so, find the value of t at which thisoccurs and the point in R3 at which the velocity is zero.

5. Let r (t) =(sin2t, t2,2t +1

)for t ∈ [0,4]. Find a tangent line to the curve parame-

terized by r at the point r (2).

6. Let r (t)=(t,sin t2, t +1

)for t ∈ [0,5]. Find a tangent line to the curve parameterized

by r at the point r (2).

7. Let r (t) =(sin t, t2,cos

(t2))

for t ∈ [0,5]. Find a tangent line to the curve parame-terized by r at the point r (2).

8. Let r (t) =(sin t,cos

(t2), t +1

)for t ∈ [0,5]. Find the velocity when t = 3.

9. Let r (t) =(sin t, t2, t +1

)for t ∈ [0,5]. Find the velocity when t = 3.

10. Let r (t) =(t, ln

(t2 +1

), t +1

)for t ∈ [0,5]. Find the velocity when t = 3.

11. Suppose an object has position r (t) ∈ R3 where r is differentiable and suppose alsothat |r (t)|= c where c is a constant.

(a) Show first that this condition does not require r (t) to be a constant. Hint: Youcan do this either mathematically or by giving a physical example.

(b) Show that you can conclude that r′ (t) ·r (t) = 0. That is, the velocity is alwaysperpendicular to the displacement.

12. Prove 16.4 from the component description of the cross product.

13. Prove 16.4 from the formula (f ×g)i = ε i jk f jgk.

14. Prove 16.4 directly from the definition of the derivative without considering compo-nents.

15. A Bezier curve in Rp is a vector valued function of the form

y (t) =n

∑k=0

(nk

)xk (1− t)n−k tk

where here the(n

k

)are the binomial coefficients and xk are n+1 points in Rn. Show

that y (0) = x0, y (1) = xn, and find y′ (0) and y′ (1). Recall that(n

0

)=(n

n

)= 1 and( n

n−1

)=(n

1

)= n. Curves of this sort are important in various computer programs.

16. Suppose r (t), s(t), and p(t) are three differentiable functions of t which have valuesin R3. Find a formula for (r (t)×s(t) ·p(t))′.

17. If F ′ (t) = f (t) for all t ∈ (a,b) and F is continuous on [a,b], show that∫ b

a f (t) dt =F (b)−F (a).

18. If r′ (t) = 0 for all t ∈ (a,b), show that there exists a constant vector c such thatr (t) = c for all t ∈ (a,b).