calcii_complete_assignments.pdf
TRANSCRIPT
-
7/28/2019 CalcII_Complete_Assignments.pdf
1/34
CALCULUS IIAssignment Problems
Paul Dawkins
-
7/28/2019 CalcII_Complete_Assignments.pdf
2/34
Calculus II
Table of Contents
Preface ........................................................................................................................................... iiiOutline ........................................................................................................................................... iiiIntegration Techniques ................................................................................................................. 5
Introduction ................................................................................................................................................ 5Integration by Parts .................................................................................................................................... 5Integrals Involving Trig Functions ............................................................................................................. 6Trig Substitutions ....................................................................................................................................... 8Partial Fractions ........................................................................................................................................11Integrals Involving Roots ..........................................................................................................................12Integrals Involving Quadratics ..................................................................................................................13Integration Strategy ...................................................................................................................................13Improper Integrals .....................................................................................................................................14Comparison Test for Improper Integrals ...................................................................................................15Approximating Definite Integrals .............................................................................................................16
Applications of Integrals ............................................................................................................. 17Introduction ...............................................................................................................................................17Arc Length ................................................................................................................................................17Surface Area ..............................................................................................................................................18Center of Mass ..........................................................................................................................................20Hydrostatic Pressure and Force .................................................................................................................20Probability .................................................................................................................................................23
Parametric Equations and Polar Coordinates .......................................................................... 25Introduction ...............................................................................................................................................25Parametric Equations and Curves .............................................................................................................25Tangents with Parametric Equations .........................................................................................................25Area with Parametric Equations ......... ........... ........... .......... ........... .......... ........... .......... ........... .......... ........25Arc Length with Parametric Equations .....................................................................................................26Surface Area with Parametric Equations...................................................................................................26Polar Coordinates ......................................................................................................................................26Tangents with Polar Coordinates ..............................................................................................................26Area with Polar Coordinates .....................................................................................................................26
Arc Length with Polar Coordinates .......... .......... ........... .......... ........... .......... ........... .......... ........... ........... ..26Surface Area with Polar Coordinates ........................................................................................................26Arc Length and Surface Area Revisited .......... .......... ........... .......... ........... .......... ........... .......... ........... ......26
Sequences and Series ................................................................................................................... 27Introduction ...............................................................................................................................................27Sequences ..................................................................................................................................................27More on Sequences ...................................................................................................................................28Series The Basics ...................................................................................................................................28Series Convergence/Divergence ............................................................................................................28Series Special Series ..............................................................................................................................28Integral Test ..............................................................................................................................................28Comparison Test / Limit Comparison Test ...............................................................................................28Alternating Series Test ..............................................................................................................................28Absolute Convergence ..............................................................................................................................28Ratio Test ..................................................................................................................................................29Root Test ...................................................................................................................................................29Strategy for Series .....................................................................................................................................29Estimating the Value of a Series ...............................................................................................................29Power Series ..............................................................................................................................................29Power Series and Functions ......................................................................................................................29Taylor Series .............................................................................................................................................29Applications of Series ...............................................................................................................................29
2007 Paul Dawkins i http://tutorial.math.lamar.edu/terms.aspx
-
7/28/2019 CalcII_Complete_Assignments.pdf
3/34
Calculus II
Binomial Series .........................................................................................................................................30Vectors .......................................................................................................................................... 30
Introduction ...............................................................................................................................................30Vectors The Basics ........... .......... ........... .......... ........... .......... ........... .......... ........... .......... ........... ........... ..30Vector Arithmetic .....................................................................................................................................30Dot Product ...............................................................................................................................................30Cross Product ............................................................................................................................................31
Three Dimensional Space............................................................................................................ 31
Introduction ...............................................................................................................................................31The 3-D Coordinate System ......................................................................................................................31Equations of Lines ....................................................................................................................................31Equations of Planes ...................................................................................................................................32Quadric Surfaces .......................................................................................................................................32Functions of Several Variables .................................................................................................................32Vector Functions .......................................................................................................................................32Calculus with Vector Functions ................................................................................................................32Tangent, Normal and Binormal Vectors ...................................................................................................32Arc Length with Vector Functions ............................................................................................................32Curvature...................................................................................................................................................33Velocity and Acceleration .........................................................................................................................33Cylindrical Coordinates ............................................................................................................................33Spherical Coordinates ...............................................................................................................................33
2007 Paul Dawkins ii http://tutorial.math.lamar.edu/terms.aspx
-
7/28/2019 CalcII_Complete_Assignments.pdf
4/34
Calculus II
Preface
Here are a set of problems for my Calculus I notes. These problems do not have any solutionsavailable on this site. These are intended mostly for instructors who might want a set of problems
to assign for turning in. I try to put up both practice problems (with solutions available) and theseproblems at the same time so that both will be available to anyone who wishes to use them.
Outline
Here is a list of sections for which problems have been written.
Integration Techniques
Integration by PartsIntegrals Involving Trig Functions
Trig SubstitutionsPartial FractionsIntegrals Involving Roots
Integrals Involving QuadraticsUsing Integral Tables
Integration StrategyImproper Integrals
Comparison Test for Improper IntegralsApproximating Definite Integrals
Applications of IntegralsArc Length No problems written yet.
Surface Area No problems written yet.Center of Mass No problems written yet.Hydrostatic Pressure and Force No problems written yet.Probability No problems written yet.
Parametric Equations and Polar CoordinatesParametric Equations and Curves No problems written yet.Tangents with Parametric Equations No problems written yet.Area with Parametric Equations No problems written yet.Arc Length with Parametric Equations No problems written yet.
Surface Area with Parametric Equations No problems written yet.Polar Coordinates No problems written yet.Tangents with Polar Coordinates No problems written yet.Area with Polar Coordinates No problems written yet.Arc Length with Polar Coordinates No problems written yet.
Surface Area with Polar Coordinates No problems written yet.Arc Length and Surface Area Revisited No problems written yet.
2007 Paul Dawkins iii http://tutorial.math.lamar.edu/terms.aspx
-
7/28/2019 CalcII_Complete_Assignments.pdf
5/34
Calculus II
Sequences and Series
Sequences No problems written yet.More on Sequences No problems written yet.
Series The Basics No problems written yet.Series Convergence/Divergence No problems written yet.
Series Special Series No problems written yet.Integral Test No problems written yet.Comparison Test/Limit Comparison Test No problems written yet.Alternating Series Test No problems written yet.Absolute Convergence No problems written yet.Ratio Test No problems written yet.
Root Test No problems written yet.Strategy for Series No problems written yet.Estimating the Value of a Series No problems written yet.Power Series No problems written yet.
Power Series and Functions No problems written yet.Taylor Series No problems written yet.
Applications of Series No problems written yet.Binomial Series No problems written yet.
VectorsVectors The Basics No problems written yet.Vector Arithmetic No problems written yet.Dot Product No problems written yet.Cross Product No problems written yet.
Three Dimensional SpaceThe 3-D Coordinate System No problems written yet.Equations of Lines No problems written yet.
Equations of Planes No problems written yet.Quadric Surfaces No problems written yet.Functions of Several Variables No problems written yet.Vector Functions No problems written yet.Calculus with Vector Functions No problems written yet.Tangent, Normal and Binormal Vectors No problems written yet.
Arc Length with Vector Functions No problems written yet.Curvature No problems written yet.Velocity and Acceleration No problems written yet.Cylindrical Coordinates No problems written yet.Spherical Coordinates No problems written yet.
2007 Paul Dawkins iv http://tutorial.math.lamar.edu/terms.aspx
-
7/28/2019 CalcII_Complete_Assignments.pdf
6/34
Calculus II
Integration Techniques
Introduction
Here are a set of problems for which no solutions are available. The main intent of theseproblems is to have a set of problems available for any instructors who are looking for some extraproblems.
Note that some sections will have more problems than others and some will have more or less ofa variety of problems. Most sections should have a range of difficulty levels in the problemsalthough this will vary from section to section.
Here is a list of topics in this chapter that have problems written for them.
Integration by Parts
Integrals Involving Trig FunctionsTrig Substitutions
Partial FractionsIntegrals Involving RootsIntegrals Involving Quadratics
Using Integral TablesIntegration Strategy
Improper IntegralsComparison Test for Improper Integrals
Approximating Definite Integrals
Integration by Parts
Evaluate each of the following integrals.
1.7
8t
t dt e
2. ( ) ( )2
1
21 3 sinx x dx
3.2
2 4
1
ww dw
e
4. ( ) ( )3 2
1
2 ln 4x x dx
5. ( ) ( )6 3 cos 1 4z z dz+ +
2007 Paul Dawkins 5 http://tutorial.math.lamar.edu/terms.aspx
-
7/28/2019 CalcII_Complete_Assignments.pdf
7/34
Calculus II
6. ( )22 cos 9y y dy
7. ( ) ( )23 sinz z z dz+
8. ( )3 3
lnx x dx
9. ( )2 7 12 ww w dw e
10. ( )29 sec 2t t dt
11. ( )80
sin 4x x dx
e
12. ( )18 tan 2y dy
13. ( )6 cos 2t t dte
14. ( )13sin 10x dx
15. ( )3 sin 2z z dz +e
16.90
17 1
1
2x
x dx+
e
17. ( )11 69 cos 1t t dt
18.
7
41
xdx
x +
19. ( ) ( )4 125 sinx x dx+
20.5 1
2z
z dz
e
21. ( ) ( )3 55 2 cos 3w w w dw+
Integrals Involving Trig Functions
Evaluate each of the following integrals.
2007 Paul Dawkins 6 http://tutorial.math.lamar.edu/terms.aspx
-
7/28/2019 CalcII_Complete_Assignments.pdf
8/34
Calculus II
1. ( ) ( )5 2cos 2 sin 2t t dt
2. ( )3cos 12x dx
3. ( ) ( )2 4cos sinz z dz
4. ( ) ( )3
5 63 3
4 4sin cosw w dw
5. ( ) ( )11 30
cos 5 sin 5z z dz
6. ( )2sin 7x dx
7. ( ) ( )6 3 30
tan 8 sec 8x x dx
8. ( ) ( )8 51 12 2
sec tant t dt
9. ( ) ( )2 3sec 9 tan 9z z dz
10. ( ) ( )3
4
6 4sec 10 tan 10t t dt
11. ( ) ( )12 6tan 2 sec 2w w dw
12. ( ) ( )2 6cot 3 csc 3x x dx
13. ( ) ( )2
3
3
3 31 1
4 4csc cotw w dw
14. ( )4csc 6w dw
15. ( ) ( )12 5
csc cotx x dx
16. ( )cot x dx
17. ( )3cot x dx
2007 Paul Dawkins 7 http://tutorial.math.lamar.edu/terms.aspx
-
7/28/2019 CalcII_Complete_Assignments.pdf
9/34
Calculus II
18. ( )csc x dx
19. ( )3csc x dx
20. ( ) ( )4
2sin 8 cos 15x x dx
21. ( ) ( )cos 2 cos 24x x dx
22. ( ) ( )sin 13 sin 9z z dz
23.( )( )
5
3
cos 2
sin 2
tdt
t
24. ( )( )
3
2
sin 2
cos 2
xdx
x
25.( )
( )
6 1
2
8 1
2
sec
tan
zdz
z
26.( )( )
5
2
tan
sec
xdx
x
27. ( )( )
5
2
1 9 cos 8
sin 8
wdw
w+
28. ( )( ) ( )3 23 7cos sinx x dx+
29. ( ) ( )3 2sin 9 sec 9y y dy
30. ( ) ( )5 5tan cosz z dz
31. ( ) ( ) ( )3 3tan 2 sin 2 sec 2t t t dt
Trig Substitutions
For problems 1 15 use a trig substitution to eliminate the root.
2007 Paul Dawkins 8 http://tutorial.math.lamar.edu/terms.aspx
-
7/28/2019 CalcII_Complete_Assignments.pdf
10/34
Calculus II
1.2
64 1t +
2.2
4 49z
3.2
7 w
4. ( )7
2216 81x
5.2
6 9y+
6. ( )3
221 8z
7. ( )2
9 16 3 1x
8. ( )( )5
222
11 1t+ +
9. ( )2
144 8 3z +
10.2
4 24 43x x +
11.
( )
1122
2 24 36z z
+
12.2
4 10 5t t
13. ( )29 sin 4 1t
14.3
36 9z e
15. 16x +
For problems 16 42 use a trig substitution to evaluate the given integral.
16.5 2
3 16x x dx
17. ( )523 2
25 81t t dt +
2007 Paul Dawkins 9 http://tutorial.math.lamar.edu/terms.aspx
-
7/28/2019 CalcII_Complete_Assignments.pdf
11/34
Calculus II
18.
14 3
2
0 1 9
wdw
w
19.
( )32
5
29 25
zdz
z
20.1
3 2
3
49 4y y dy
21.
5
2 21
5
4
dxx x +
22.
2
2
3 4tdt
t
23.
5
28 1
wdw
w +
34.
2
3
15xdx
x
35.
( )
6 2
2
3 6 5
dx
x x x +
36.
( ) ( )322 2
1
1 2 4 34
dz
z z z+ +
37.( )
2
6
4 16 19
2
y ydy
y
+
38.( )
12 3
2
9
4
8 7
tdt
t t
+
39.6
2
0
5 10 6x x dx+ +
40.7 49x x dx
2007 Paul Dawkins 10 http://tutorial.math.lamar.edu/terms.aspx
-
7/28/2019 CalcII_Complete_Assignments.pdf
12/34
Calculus II
41.
12
64 1
t
tdt
e
e
42.
( ) ( ) ( )
3 2sin cos 16 cosz z z dz+
Partial Fractions
Evaluate each of the following integrals.
1.2
9
12dz
z z
2. 2
7
14 40
x
dxx x+ +
3.
4
20
8 1
2 15 8
ydy
y y
4.( )( )( )
29
1 3 5 4
wdw
w w w
+ +
5.
8
3 2
1
12
2 63
dz
z z z
6.( )( ) ( )
27 2
4 2 3 2 1
x xdx
x x x
+
+ +
7.( )( )
2
4 10
2 1
xdx
x x
+
8.
2
4 3
1
24
6dt
t t
9.( ) ( )
2 2
10 2
1 3
zdz
z z
+
+
10.( ) ( )
2
2
8
7 16
w wdw
w w
+
+
2007 Paul Dawkins 11 http://tutorial.math.lamar.edu/terms.aspx
-
7/28/2019 CalcII_Complete_Assignments.pdf
13/34
Calculus II
11.( )( )2
6 7
2 1 4 1
ydy
y y
+ +
12.( )( )
3 2
2 2
8 5 72 10
2 9t t t dt
t t
+
+ +
13.( )( )
3 2
2 2
16 6 12 21
9 4 3
w w wdw
w w
+ + +
+ +
14.
( )
4 3
22
5 20 16
4
x x xdx
x x
+ + +
+
15.2
2
6
2 21
zdz
z z
+
16.
3
2
4
30
x xdx
x x
17.( )( )
3
2
8
3 1
tdt
t t
+
18.
6 5 4 3 2
4 2
6 3 10 9 12 27
3x x x x x x dx
x x + + +
Integrals Involving Roots
Evaluate each of the following integrals.
1.5
4 3dz
z
2.1
4 3dx
x x+
3.4
7 1dt
t t + +
2007 Paul Dawkins 12 http://tutorial.math.lamar.edu/terms.aspx
-
7/28/2019 CalcII_Complete_Assignments.pdf
14/34
Calculus II
4.2 3 2 1 9
zdz
z z+ +
5.2 6 3 5
wdw
w w
6. ( )cos x dx
Integrals Involving Quadratics
Evaluate each of the following integrals.
1.2
1
10 18
xdx
x x
+
+ +
2.2
15
3 4 4dy
y y +
3.2
12 9
1 4 4
zdz
z z
4.
( )2
2
3 7
12 40
tdt
t t
+ +
5.
( )2
2
11 4
3 6
wdw
w w
+
+
6.
( )722
3
2 10 4
dx
x x+ +
Integration Strategy
Problems have not yet been written for this section.
I was finding it very difficult to come up with a good mix of new problems and decided mytime was better spent writing problems for later sections rather than trying to come up with asufficient number of problems for what is essentially a review section. I intend to come back at a
later date when I have more time to devote to this section and add problems then.
2007 Paul Dawkins 13 http://tutorial.math.lamar.edu/terms.aspx
-
7/28/2019 CalcII_Complete_Assignments.pdf
15/34
Calculus II
Improper Integrals
Determine if each of the following integrals converge or diverge. If the integral converges
determine its value.
1.2
4
2 4 6x x dx
+
2.
5
0
1
4 20dw
w
3.
2
61
3
4 2dz
z
4.0
2 3xx dx+
e
5.2 3
0
xx dx
+ e
6.2
2
1
1dx
x
+
7.
3
2
0
1
4dz
z z
8.
1
21
xdx
x +
9.
2
2
1
1
2 3dy
y y
10. ( )0
cos w dw
11.( )
2
10
1
5 2
dzz
12.
3
41
zdz
z
+
2007 Paul Dawkins 14 http://tutorial.math.lamar.edu/terms.aspx
-
7/28/2019 CalcII_Complete_Assignments.pdf
16/34
Calculus II
13.
4
1
1
6 2dy
y
14.
5
3
1
1
2
dw
w
15.
11
2
2
x
dxx
e
16.3
2 xx dx
e
17.
( )
32
1
ydy
y
+
18.
33
2
0 9
wdw
w
19.
1
2
3
1
2dw
w w +
20.
1
2
0
x
dxx
e
21.
( )2
0
1
ln
dzz z
22.0
1
1dw
w
Comparison Test for Improper Integrals
Use the Comparison Test to determine if the following integrals converge or diverge.
1.5
4
1
2dz
z
2007 Paul Dawkins 15 http://tutorial.math.lamar.edu/terms.aspx
-
7/28/2019 CalcII_Complete_Assignments.pdf
17/34
Calculus II
2.6
0 2
wdw
w
+
3.
( )4
2
1
2 3
dw
w
+
4.
2
12
4 2
7
y ydy
y
+
5.( )2
1
lndx
x
Hint : Sketch the graph of y x= and ( )lny x= on the same axis system.
6.
( )2
32
4sinz z
dzz
7.( )
( )
23
2
20
2 sin
cos
x xdx
x x
+
8.3
0 1
zzdz
z
+
e
Approximating Definite Integrals
For each of the following integrals use the given value ofn to approximate the value of thedefinite integral using
(a) the Midpoint Rule,(b) the Trapezoid Rule, and(c) Simpsons Rule.
Use at least 6 decimal places of accuracy for your work.
1. ( )4
2
2
sin 2x dx
+ using 6n =
2.4
3 4
0
6x dx+ using 6n =
3.( )5 cos
1
xdx e using 8n =
2007 Paul Dawkins 16 http://tutorial.math.lamar.edu/terms.aspx
-
7/28/2019 CalcII_Complete_Assignments.pdf
18/34
Calculus II
4.( )
5
3
1
1 lndx
x
using 6n =
5. ( ) ( )1
2
3
sin cosx x dx using 8n =
Applications of Integrals
Introduction
Here are a set of problems for which no solutions are available. The main intent of theseproblems is to have a set of problems available for any instructors who are looking for some extra
problems.
Note that some sections will have more problems than others and some will have more or less ofa variety of problems. Most sections should have a range of difficulty levels in the problemsalthough this will vary from section to section.
Here is a list of topics in this chapter that have problems written for them.
Arc Length No problems written yet.
Surface Area No problems written yet.Center of Mass No problems written yet.Hydrostatic Pressure and Force No problems written yet.
Probability No problems written yet.
Arc Length
1. Set up, but do not evaluate, an integral for the length of 14 9y x= , 22 31y using,
(a)
2
1dy
ds dxdx
= +
(b)
2
1
dx
ds dydy
= +
2. Set up, but do not evaluate, an integral for the length of2y
x = e , 1 0y using,
(a)
2
1dy
ds dxdx
= +
2007 Paul Dawkins 17 http://tutorial.math.lamar.edu/terms.aspx
-
7/28/2019 CalcII_Complete_Assignments.pdf
19/34
Calculus II
(b)
2
1dx
ds dydy
= +
3. Set up, but do not evaluate, an integral for the length of ( )tan 2y x= ,3
0 x using,
(a)
2
1dy
ds dxdx
= +
(b)
2
1dx
ds dydy
= +
4. Set up, but do not evaluate, an integral for the length of
2
29 1
16
xy+ = .
5. For 6 1x y= + , 2 8y (a) Use an integral to find the length of the curve.
(b) Verify your answer from part (a) geometrically.
6. Determine the length of 43
2y x= + , 0 9x .
7. Determine the length of ( )3
28 3y x= + ,
3 32 211 27y .
8. Determine the length of ( )3
210 2x y= , 1 2y .
9. Determine the length of ( )2
2 1x y= + , 2 5y .
10. Determine the length of ( )2
3 2y x= + , 1 4x .
Surface Area
1. Set up, but do not evaluate, an integral for the surface area of the object obtained by rotating
7 2y x= + , 5 0y about the x-axis using,
(a)
2
1dy
ds dxdx
= +
(b)
2
1dx
ds dydy
= +
2007 Paul Dawkins 18 http://tutorial.math.lamar.edu/terms.aspx
-
7/28/2019 CalcII_Complete_Assignments.pdf
20/34
Calculus II
2. Set up, but do not evaluate, an integral for the surface area of the object obtained by rotating5
1 2y x= + , 0 1x about the x-axis using,
(a)
2
1dy
ds dxdx
= +
(b)2
1dx
ds dydy
= +
3. Set up, but do not evaluate, an integral for the surface area of the object obtained by rotating2yx = e , 1 0y about the y-axis using,
(a)
2
1dy
ds dxdx
= +
(b)
2
1dx
ds dydy
= +
4. Set up, but do not evaluate, an integral for the surface area of the object obtained by rotating
( )12cosy x= , 0 x about(a) thex-axis
(b) the y-axis.
5. Set up, but do not evaluate, an integral for the surface area of the object obtained by rotating
3 7x y= + , 0 1y about(a) thex-axis
(b) the y-axis.
6. Find the surface area of the object obtained by rotating 14
6 2y x= + , 522 2
y about thex-axis.
7. Find the surface area of the object obtained by rotating 4y x= , 1 6x about the y-axis.
8. Find the surface area of the object obtained by rotating 22 5x y= + , 1 2x about the y-axis.
9. Find the surface area of the object obtained by rotating 21x y= , 0 3y about thex-axis.
10. Find the surface area of the object obtained by rotating2y
x = e , 1 0y about the y-axis.
11. Find for the surface area of the object obtained by rotating ( )12
cosy x= , 0 x about
the x-axis.
2007 Paul Dawkins 19 http://tutorial.math.lamar.edu/terms.aspx
-
7/28/2019 CalcII_Complete_Assignments.pdf
21/34
Calculus II
Center of Mass
Find the center of mass for each of the following regions.
1. The region bounded by3
y x= , 2x = and the x-axis.
2. The triangle with vertices (-2, -2), (4, -2) and (4,4).
3. The region bounded by ( )2
2y x= and 4y = .
4. The region bounded by ( )cosy x= and the x-axis between2 2
x .
5. The region bounded by2
y x= and 6y x= .
6. The region bounded by 2xy = e and the x-axis between 1 1x .
7. The region bounded by2xy = e and ( )cosy x= between 1 1
2 2x .
Hydrostatic Pressure and Force
Find the hydrostatic force on the following plates submerged in water as shown in each image. Ineach case consider the top of the blue box to be the surface of the water in which the plate issubmerged. Note as well that the dimensions in many of the images will not be perfectly to scalein order to better fit the plate in the image. The lengths given in each image are in meters.
1.
2.
2007 Paul Dawkins 20 http://tutorial.math.lamar.edu/terms.aspx
-
7/28/2019 CalcII_Complete_Assignments.pdf
22/34
Calculus II
3.
4.
5.
2007 Paul Dawkins 21 http://tutorial.math.lamar.edu/terms.aspx
-
7/28/2019 CalcII_Complete_Assignments.pdf
23/34
Calculus II
6.
7.
2007 Paul Dawkins 22 http://tutorial.math.lamar.edu/terms.aspx
-
7/28/2019 CalcII_Complete_Assignments.pdf
24/34
Calculus II
Probability
1. Let,
( )( )2
32 if 0 2
4
0 otherwise
x x xf x
=
(a) Show that ( )f x is a probability density function.(b) Find ( )0.25P X .(c) Find ( )1.4P X .(d) Find ( )0.1 1.2P X .(e) Determine the mean value ofX.
2. Let,
( ) ( )( )24
if 1 6ln 3 4
0 otherwise
xx xf x
+=
(a) Show that ( )f x is a probability density function.(b) Find ( )1P X .(c) Find ( )5P X .(d) Find ( )1 5P X .(e) Determine the mean value ofX.
3. Let,
( )1
1 sin if 0 1010 2
0 otherwise
x xf x
+ =
(a) Show that ( )f x is a probability density function.(b) Find ( )3P X .(c) Find ( )5P X .(d) Find ( )2.5 7P X .(e) Determine the mean value ofX.
2007 Paul Dawkins 23 http://tutorial.math.lamar.edu/terms.aspx
-
7/28/2019 CalcII_Complete_Assignments.pdf
25/34
Calculus II
4. The probability density function of the life span of a battery is given by the function below,where tis in years.
( )0.8
1.25 if 0
0 if 0
tt
f t
t
=
-
7/28/2019 CalcII_Complete_Assignments.pdf
26/34
Calculus II
Parametric Equations and Polar Coordinates
Introduction
Here are a set of problems for which no solutions are available. The main intent of theseproblems is to have a set of problems available for any instructors who are looking for some extra
problems.
Note that some sections will have more problems than others and some will have more or less ofa variety of problems. Most sections should have a range of difficulty levels in the problemsalthough this will vary from section to section.
Here is a list of topics in this chapter that have problems written for them.
Parametric Equations and Curves No problems written yet.Tangents with Parametric Equations No problems written yet.
Area with Parametric Equations No problems written yet.Arc Length with Parametric Equations No problems written yet.
Surface Area with Parametric Equations No problems written yet.Polar Coordinates No problems written yet.Tangents with Polar Coordinates No problems written yet.
Area with Polar Coordinates No problems written yet.Arc Length with Polar Coordinates No problems written yet.
Surface Area with Polar Coordinates No problems written yet.Arc Length and Surface Area Revisited No problems written yet.
Parametric Equations and Curves
Problems have not yet been written for this section.
Tangents with Parametric Equations
Problems have not yet been written for this section.
Area with Parametric Equations
Problems have not yet been written for this section.
2007 Paul Dawkins 25 http://tutorial.math.lamar.edu/terms.aspx
-
7/28/2019 CalcII_Complete_Assignments.pdf
27/34
Calculus II
Arc Length with Parametric Equations
Problems have not yet been written for this section.
Surface Area with Parametric Equations
Problems have not yet been written for this section.
Polar Coordinates
Problems have not yet been written for this section.
Tangents with Polar Coordinates
Problems have not yet been written for this section.
Area with Polar Coordinates
Problems have not yet been written for this section.
Arc Length with Polar Coordinates
Problems have not yet been written for this section.
Surface Area with Polar Coordinates
Problems have not yet been written for this section.
Arc Length and Surface Area Revisited
2007 Paul Dawkins 26 http://tutorial.math.lamar.edu/terms.aspx
-
7/28/2019 CalcII_Complete_Assignments.pdf
28/34
Calculus II
Problems have not yet been written for this section and probably wont be to be honest since thisis just a summary section.
Sequences and Series
Introduction
Here are a set of problems for which no solutions are available. The main intent of theseproblems is to have a set of problems available for any instructors who are looking for some extraproblems.
Note that some sections will have more problems than others and some will have more or less of
a variety of problems. Most sections should have a range of difficulty levels in the problemsalthough this will vary from section to section.
Here is a list of topics in this chapter that have problems written for them.
Sequences No problems written yet.More on Sequences No problems written yet.Series The Basics No problems written yet.
Series Convergence/Divergence No problems written yet.Series Special Series No problems written yet.
Integral Test No problems written yet.Comparison Test/Limit Comparison Test No problems written yet.
Alternating Series Test No problems written yet.Absolute Convergence No problems written yet.Ratio Test No problems written yet.
Root Test No problems written yet.Strategy for Series No problems written yet.
Estimating the Value of a Series No problems written yet.Power Series No problems written yet.Power Series and Functions No problems written yet.Taylor Series No problems written yet.Applications of Series No problems written yet.
Binomial Series No problems written yet.
Sequences
Problems have not yet been written for this section.
2007 Paul Dawkins 27 http://tutorial.math.lamar.edu/terms.aspx
-
7/28/2019 CalcII_Complete_Assignments.pdf
29/34
Calculus II
More on Sequences
Problems have not yet been written for this section.
Series The Basics
Problems have not yet been written for this section.
Series Convergence/Divergence
Problems have not yet been written for this section.
Series Special Series
Problems have not yet been written for this section.
Integral Test
Problems have not yet been written for this section.
Comparison Test / Limit Comparison Test
Problems have not yet been written for this section.
Alternating Series Test
Problems have not yet been written for this section.
Absolute Convergence
Problems have not yet been written for this section.
2007 Paul Dawkins 28 http://tutorial.math.lamar.edu/terms.aspx
-
7/28/2019 CalcII_Complete_Assignments.pdf
30/34
Calculus II
Ratio Test
Problems have not yet been written for this section.
Root Test
Problems have not yet been written for this section.
Strategy for Series
Problems have not yet been written for this section.
Estimating the Value of a Series
Problems have not yet been written for this section.
Power Series
Problems have not yet been written for this section.
Power Series and Functions
Problems have not yet been written for this section.
Taylor Series
Problems have not yet been written for this section.
Applications of Series
2007 Paul Dawkins 29 http://tutorial.math.lamar.edu/terms.aspx
-
7/28/2019 CalcII_Complete_Assignments.pdf
31/34
Calculus II
Problems have not yet been written for this section.
Binomial Series
Problems have not yet been written for this section.
Vectors
Introduction
Here are a set of problems for which no solutions are available. The main intent of these
problems is to have a set of problems available for any instructors who are looking for some extraproblems.
Note that some sections will have more problems than others and some will have more or less ofa variety of problems. Most sections should have a range of difficulty levels in the problemsalthough this will vary from section to section.
Here is a list of topics in this chapter that have problems written for them.
Vectors The Basics No problems written yet.
Vector Arithmetic No problems written yet.Dot Product No problems written yet.
Cross Product No problems written yet.
Vectors The Basics
Problems have not yet been written for this section.
Vector Arithmetic
Problems have not yet been written for this section.
Dot Product
Problems have not yet been written for this section.
2007 Paul Dawkins 30 http://tutorial.math.lamar.edu/terms.aspx
-
7/28/2019 CalcII_Complete_Assignments.pdf
32/34
Calculus II
Cross Product
Problems have not yet been written for this section.
Three Dimensional Space
Introduction
Here are a set of problems for which no solutions are available. The main intent of theseproblems is to have a set of problems available for any instructors who are looking for some extra
problems.
Note that some sections will have more problems than others and some will have more or less ofa variety of problems. Most sections should have a range of difficulty levels in the problemsalthough this will vary from section to section.
Here is a list of topics in this chapter that have problems written for them.
The 3-D Coordinate System No problems written yet.Equations of Lines No problems written yet.
Equations of Planes No problems written yet.Quadric Surfaces No problems written yet.
Functions of Several Variables No problems written yet.Vector Functions No problems written yet.Calculus with Vector Functions No problems written yet.
Tangent, Normal and Binormal Vectors No problems written yet.Arc Length with Vector Functions No problems written yet.
Curvature No problems written yet.Velocity and Acceleration No problems written yet.
Cylindrical Coordinates No problems written yet.Spherical Coordinates No problems written yet
The 3-D Coordinate System
Problems have not yet been written for this section.
Equations of Lines
2007 Paul Dawkins 31 http://tutorial.math.lamar.edu/terms.aspx
-
7/28/2019 CalcII_Complete_Assignments.pdf
33/34
Calculus II
Problems have not yet been written for this section.
Equations of Planes
Problems have not yet been written for this section.
Quadric Surfaces
Problems have not yet been written for this section.
Functions of Several Variables
Problems have not yet been written for this section.
Vector Functions
Problems have not yet been written for this section.
Calculus with Vector Functions
Problems have not yet been written for this section.
Tangent, Normal and Binormal Vectors
Problems have not yet been written for this section.
Arc Length with Vector Functions
Problems have not yet been written for this section.
2007 Paul Dawkins 32 http://tutorial.math.lamar.edu/terms.aspx
-
7/28/2019 CalcII_Complete_Assignments.pdf
34/34
Calculus II
Curvature
Problems have not yet been written for this section.
Velocity and Acceleration
Problems have not yet been written for this section.
Cylindrical Coordinates
Problems have not yet been written for this section.
Spherical Coordinates
Problems have not yet been written for this section.