572
C H A P T E R 14
C A L C U L U S O F VE C T O R - VA L U E D F U N C T I O N S
SOLUTION
(ET CHAPTER 13)
Keplers Third Law states that the period T of the orbit is given by: T2 = 4 2 GM a3
or 2 3/2 T= a GM If a is increased four-fold the period becomes
S E C T I O N 14.4
Curvature
(ET Section 13.4)
511
25. In the notation of Exercise 23, assume that a b. Show that b/a 2 (t ) a /b2 for all t .
SOLUTION
In Exercise 23 we showed that the curvature of the ellipse r(t ) = a cos t , b sin t is the following f
324
C H A P T E R 13
V E C T O R G E O M E T RY
(ET CHAPTER 12)
We use the formula for the midpoint of a segment to nd the coordinates of the points P and Q . This gives P= Q= 1+0 0+1 1+1 , , 2 2 2 1+0 1+1 0+1 , , 2 2 2 = = 11 , ,1 22 1 1 , 1, 2 2
Substit
32A Stovall
Quiz 9
Name:
Sections A/C (Thu)
TA: Melissa or Duncan
Continues on back!
1. Find all critical points of
f (x, y ) = sin x sin y.
Use the second derivative test to classify the critical point at ( , ).
2
2
2. Find three positive numbers whose s
32A Stovall
Quiz 1
Name:
Sections A/C (Tue)
TA: Melissa or Duncan
Continues on back
1. (2 pts) Compute the rst two derivatives of f (x) = esin x .
2. (2 pts) Compute
1
0
x 1 x2 dx.
3. (2 pts) Write inequalities to describe the solid cylinder that lies on
32A Stovall
Midterm 2
Name:
November 9
Section: Tu/Th
Duncan/Melissa
I certify that the work appearing on this exam is completely my
own:
Signature:
There are 5 problems and a total of 8 pages. Please make sure that
you have all pages.
Please show your
32A Stovall
Midterm 1
Name:
October 17
Section: Tu/Th
Duncan/Melissa
I certify that the work appearing on this exam is completely my
own:
Signature:
There are 5 problems and a total of 8 pages. Please make sure that
you have all pages.
Please show your
S E C T I O N 14.2
y
Calculus of Vector-Valued Functions
(ET Section 13.2)
469
r'(1) = 2, 1
t=1
t=0 t = 1
x
17. Sketch the curve r1 (t ) = t , t 2 together with its tangent vector at t = 1. Then do the same for r2 (t ) = t 3 , t 6 .
SOLUTION
Note that r1
478
C H A P T E R 14
C A L C U L U S O F VE C T O R - VA L U E D F U N C T I O N S
(ET CHAPTER 13)
In Exercises 4653, nd the general solution r(t ) of the differential equation and the solution with the given initial condition. 46. dr = 1 2t , 4t , r(0) =
492
C H A P T E R 14
C A L C U L U S O F VE C T O R - VA L U E D F U N C T I O N S
(ET CHAPTER 13)
r (s ) = 4 sin (s ), 0, 4 cos (s ) Substituting in (1) we get: r1 (s ) = (s ) 4 sin (s ), 0, 4 cos (s ) = 4 (s ) sin (s ), 0, cos (s ) Hence, r1 (s ) = 4| (
556
C H A P T E R 14
C A L C U L U S O F VE C T O R - VA L U E D F U N C T I O N S
(ET CHAPTER 13)
v0 2 = For g = 32 ft/s2 we get:
2 v0 =
gd 2 gd 2 sec2 = 2 (d tan h ) 2 cos2 (d tan h )
16d 2 sec2 d tan h
v0 =
4d sec d tan h
.
25. At a certain moment, a
S E C T I O N 14.5
Motion in Three-Space
(ET Section 13.5)
565
Substituting (1), (3) and (6) into (5) gives: aN N = 2, 0 4t 4 t2 + 4 t 2t , 4 = 2, 0 2 2t , 4 t +4 (7)
8 2t 2 4t 4t 1 8, 4t = 2 2 , =2 , =2 t + 4 t2 + 4 t + 4 t2 + 4 t +4 Since aN = av , aN i
S E C T I O N 14.5
Motion in Three-Space
(ET Section 13.5)
565
Substituting (1), (3) and (6) into (5) gives: aN N = 2, 0 4t 4 t2 + 4 t 2t , 4 = 2, 0 2 2t , 4 t +4 (7)
8 2t 2 4t 4t 1 8, 4t = 2 2 , =2 , =2 t + 4 t2 + 4 t + 4 t2 + 4 t +4 Since aN = av , aN i
S E C T I O N 14.5
Motion in Three-Space
(ET Section 13.5)
543
We now compute T(0): T(0) = Finally we nd N = B T: 1 1 1 1 1 N(0) = 2, 0, 1 0, 1, 0 = (2i + k) j = (2i j + k j) = (2k i) = 1, 0, 2 5 5 5 5 5 (b) Differentiating r(t ) = t 2 , t 1 , t gives r (
Chapter Review Exercises
593
F
7 2
= 4e2(7/2)6 + 4e82(7/2) = 4e + 4e = 8e > 0
The Second Derivative Test implies that F (t ), hence v(t ) as well, have a minimum at t = 7 . The minimum speed is: 2 v 7 2 = 1 + e2(7/2)6 + e82(7/2) = 1 + 2e
25. Calculate the
32A Stovall
Quiz 1
Name:
Sections B/D (Thu)
TA: Melissa or Duncan
Continues on back!
1. (2 pts) Mark each item as true or false.
(a) ln(ab) = ln a + ln b
(b) (ln a)(ln b) = ln(a + b)
b
(c) (ea )b = ea
(d) (ea )b = ea+b
2. (2 pts) Find an equation of the s
32A Stovall
Quiz 2
Name:
Sections A/C (Tue)
TA: Melissa or Duncan
Continues on back
1. (2pts) What is the angle between the vector
x-axis?
3i + j and the positive
2. (2pts) Find the scalar and vector projections of b = i + 2j + 3k onto
a = i j.
3. (2pts)
32A Stovall
Quiz 9
Name:
Sections A/C (Tue)
TA: Melissa or Duncan
Continues on back!
1. Find and classify all critical points of
f (x, y ) = ey (y 2 x2 ).
2. Find three positive numbers whose sum is 100 and whose product is a
maximum.
32A Stovall
Quiz 8
Name:
Sections A/C (Thu)
TA: Melissa or Duncan
Continues on back!
1. Use the chain rule (do not use direct substitution) to nd
of t when
z = x2 + y 2 + xy, x = sin t, y = et .
dz
dt
as a function
2. Use the chain rule (do not use direct
32A Stovall
Quiz 8
Name:
Sections A/C (Tue)
TA: Melissa or Duncan
Continues on back!
1. Use the chain rule (do not use direct substitution) to nd
t if
z = 1 + x2 + y 2 , x = ln t, y = cos t.
dz
dt
in terms of
2. Let f be a dierentiable function of two var
32A Stovall
Quiz 7
Name:
Sections B/D (Thu)
TA: Melissa or Duncan
Continues on back!
1. (3pts) Find the dierential of the function
z = e2x cos(2t).
2. (3pts) Find an equation for the tangent plane to the surface
z = x sin(x + y )
at the point (1, 1, 0).
32A Stovall
Quiz 7
Name:
Sections A/C (Tue)
TA: Melissa or Duncan
Continues on back!
1. (3pts) Find all second partial derivatives of f (x, y ) = x3 y 5 + 2x4 y .
2. (3pts) Find the dierential of the function P = xyz and use it to solve
the following prob
32A Stovall
Quiz 6
Sections B/D (Thu)
Name:
TA: Melissa or Duncan
Continues on back!
1. (3pts) Use implicit dierentiation to nd
ez = xyz.
z
x
and
z
y
when
2. (3pts) Find grst when g (r, s, t) = er sin(st).
32A Stovall
Quiz 6
Sections A/C (Tue)
Name:
TA: Melissa or Duncan
Continues on back!
1. (3pts) Give two continuous vector functions r1 and r2 , 1 t 1 with
r1 (0) = r2 (0) = 0, 0, 0 so that
lim f (r1 (t) = f (r2 (t),
t0
where
f (x, y, z ) =
x2
xy + yz
.
+
32A Stovall
Quiz 5
Name:
Sections B/D (Thu)
TA: Melissa or Duncan
Continues on back!
1. (3pts) Reduce the equation to one of the standard forms, classify the
surface, and sketch it.
x2 y 2 + z 2 4x 2y 2z + 4 = 0.
2. (3pts) Give two continuous vector funct
32A Stovall
Quiz 5
Name:
Sections A/C (Tue)
TA: Melissa or Duncan
Continues on back!
1. (3pts) Reduce the equation to one of the standard forms and classify the
surface
4x2 + y 2 + z 2 4y 2z + 1 = 0.
2. (3pts) Find and sketch the domain of
f (x, y ) =
y+
32A Stovall
Quiz 4
Sections B/D (Thu)
Name:
TA: Melissa or Duncan
Continues on back!
1. (3pts) Find the acute angle of intersection between the curves
r1 (t) = cos t, sin t, t ,
at 1, 0, 0 .
r2 (t) = 1, t, t2
2. (3pts) Find an equation for the osculating
32A Stovall
Quiz 4
Name:
Sections A/C (Tue)
TA: Melissa or Duncan
Continues on back!
1. (3pts) Find the area of the triangle with vertices P (0, 1, 1), Q(1, 0, 1), and
R(1, 1, 0).
2. (3pts) Find the unit tangent and unit normal vectors to the curve parame
32A Stovall
Name:
Quiz 3
Sections B/D (Thu)
TA: Melissa or Duncan
Continues on back!
1. (3pts) Compute limt1 (e3t i + 1 + t2 j + sin(t) k).
ln t
2. (3pts) Let r(t) = t, t2 , t3 . Find the parametric equations for the tangent
line to the curve sketched out
32A Stovall
Quiz 3
Name:
Sections A/C (Tue)
TA: Melissa or Duncan
Continues on back!
1. (3pts) Find
dy
dx
and
d2 y
dx2
when x = 2t3 and y = 1 + 4t t2 .
3
2. (3pts) Compute limt tet , 2tt3+t1 , 1 sin t .
t