Solução_Calculo_Stewart_6e

F.
TX.10
CHAPTER 14 REVIEW ET CHAPTER 13 ¤ 157
39. The tangential component of a is the length of the projection of a onto T,sowesketch
the scalar projection of a in the tangential direction to the curve and estimate its length to
be 4.5 (using the fact that a has length 10 as a guide). Similarly, the normal component of
a is the length of the projection of a onto N,sowesketchthescalarprojectionofa in the
normal direction to the curve and estimate its length to be 9.0. Thus a T ≈ 4.5 cm/s 2 and
a N ≈ 9.0 cm/s 2 .
41. If the engines are turned off at time t, then the spacecraft will continue to travel in the direction of v(t), so we need a t such
that for some scalar s>0, r(t)+s v(t) =h6, 4, 9i. v(t) =r 0 (t) =i + 1 t j + 8t
(t 2 +1) k ⇒
2
r(t)+s v(t) = 3+t + s, 2+lnt + s t , 7 − 4
t 2 +1 + 8st
⇒ 3+t + s =6 ⇒ s =3− t,
(t 2 +1) 2
so 7 − 4 8(3 − t)t
+
t 2 +1 (t 2 +1) =9 ⇔ 24t − 12t2 − 4
=2 ⇔ t 4 +8t 2 − 12t +3=0.
2 (t 2 +1) 2
It is easily seen that t =1is a root of this polynomial. Also 2+ln1+ 3 − 1
1
=4,sot =1is the desired solution.
14 Review ET 13
1. A vector function is a function whose domain is a set of real numbers and whose range is a set of vectors. To find the derivative
or integral, we can differentiate or integrate each component of the vector function.
2. Thetipofthemovingvectorr(t) of a continuous vector function traces out a space curve.
3. The tangent vector to a smooth curve at a point P with position vector r(t) is the vector r 0 (t). The tangent line at P is the line
through P parallel to the tangent vector r 0 (t). The unit tangent vector is T(t) = r0 (t)
|r 0 (t)| .
4. (a) – (f ) See Theorem 14.2.3 [ET 13.2.3].
5. Use Formula 14.3.2 [ ET 13.3.2], or equivalently, 14.3.3 [ ET 13.3.3].
6. (a) The curvature of a curve is κ =
dT
ds where T is the unit tangent vector.
(b) κ(t) =
T0 (t)
r 0 (t)
(c) κ(t) = |r0 (t) × r 00 (t)|
|f 00 (x)|
|r 0 (t)| 3 (d) κ(x) =
[1 + (f 0 (x)) 2 ] 3/2
7. (a) The unit normal vector: N(t) = T0 (t)
. The binormal vector: B(t) =T(t) × N(t).
|T 0 (t)|
(b) See the discussion preceding Example 7 in Section 14.3 [ ET 13.3].