Authors : SangGu Lee with
Jihoon Lee, GiSang Cheon, Soonhak Kwon, Changbum Cheon,
MeeKyoung Kim, Seki Kim, Jeong Hyeong Park, In Sung Hwang, Young Do Chai.
Calculus is the mathematical foundation for much of university mathematics, science, and engineering curriculum. For the mathematics student, it is a first exposure to rigorous mathematics. For the engineer, it is an introduction to the modeling and approximation techniques used throughout his engineering curriculum. And for future scientist, it is the mathematical language that will be used to express many of the most important scientific concepts.
In the first semester, that’s for the beginners of calculus, we start with differential and integral calculus on functions of single variable and then study L'Hospital's theorem, concavity, convexity, inflection points, optimization problems, and ordinary differential equations as applications of differential and integral calculus accordingly. In the second semester of calculus, we cover vector calculus that includes parameter equations, polar coordinates, infinite sequences and infinite series, vectors and coordinate space which uses partial derivatives. Modeling and approximation in calculus should resemble the techniques and methods currently in use. Concepts, definitions, terminology, and interpretation in calculus should be as current as possible. This booklet has many problems to present calculus as the foundation of modern mathematics, science and engineering.
This booklet is a Lab Manual for Calculus with SageMath. Most of recent calculus textbooks are using Computer Algebra System(CAS) including a variety of visual tools in it. But its use was limited to students in most of cases. Therefore, in this book, we adapted a wonderful opensource program, SAGE, for our students. With the new learning environment of universities, students will take a full advantage of 21C state of arts technology to learn calculus more easily and get better prepared for future job market. We can use the SageMath well on popular web browsers such as Firefox or Chrome (https://www.google.com/chrome).
More content and related materials will be added to be viewed on the web. When you see CAS or Web mark in the book, which means you will be able to find relevant informations by clicking http://math1.skku.ac.kr/ address. That will save lots of your work.
Finally, the booklet also combines technology, reform, and tradition in a way to offer a wider view to students. Most importantly, we appreciate all members of the Mathematics department at Sungkyunkwan University who supported our effort to make a small history.
Authors : SangGu Lee (성균관대학교 이상구) with
Jihoon Lee, GiSang Cheon, Soonhak Kwon, Changbum Cheon,
MeeKyoung Kim, Seki Kim, Jeong Hyeong Park, In Sung Hwang, Young Do Chai.
2012. 1. 1.
1. Calculus (미분적분학 1, 2)
Calculus 미분적분학 http://matrix.skku.ac.kr/CalBook/
Single Variable Calculus: http://matrix.skku.ac.kr/CalBook/part1/part1.html
Multivariate Calculus: http://matrix.skku.ac.kr/CalBook/part2/part2.html
2. Calculus 실습실 http://sagecell.sagemath.org/ http://sage.skku.edu/
3. Internet resources:
SageReference http://matrix.skku.ac.kr/2009Sage/SageReference.html
http://matrix.skku.ac.kr/2012LAwithSage/interact/
(Tool) http://matrix.skku.ac.kr/2012Sage/index.html
Sage Tutorial : http://www.youtube.com/watch?v=0SQpiNe2LU8
http://www.youtube.com/watch?v=GJcym7gMKrg&feature=results_main&playnext=1&list=PL9168C6B83FE306CE
2011How to use Sage 1 http://matrix.skku.ac.kr/2011Album/Sagemath02.html
2011How to use Sage 2 http://matrix.skku.ac.kr/2010Album/MathtalkSage.html
2011How to use Sage 3 http://matrix.skku.ac.kr/2011Album/Sagemath01.html
2011Visualization of LA with Sage http://www.youtube.com/watch?v=e5MDq8wNWmw
William Stein demos sage math http://www.youtube.com/watch?v=kIQZU_uZGlc
2011Mobile Sage http://matrix.skku.ac.kr/2011sage/sage/clawithsage.html
2011Mobile Math with Sage http://matrix.skku.ac.kr/2011Sage/2011MobileMath/MobileMath.html
2011Linear Algebra with Sage http://matrix.skku.ac.kr/2012sage/sagela/
2011Matrix Theory(Web version) http://matrix.skku.ac.kr/2011sage/sage/skkula.html
2011Mobile LA with Sage http://matrix.skku.ac.kr/2011Sage/2011MobileMath/MobileLA.html
2011How to do LU with Sage http://matrix.skku.ac.kr/2011Album/SKKUmatrixSage1.html
Sage Interact / ODE and Mandelbrot http://www.youtube.com/watch?v=_258y4kMYyQ
Sage Multivariable Calculus (1 of 2) by Travis http://www.youtube.com/watch?v=rqACCzGYOm8
Sage Multivariable Calculus (2 of 2) by Travis http://www.youtube.com/watch?v=SwgFWKK0oCg
http://bkmath.skku.ac.kr/bk21/index.html
http://matrix.skku.ac.kr/sglee/
1. Introduction and Use of SageMath^{1)}
Mathematical tools have long held an important place in classrooms. With the innovation of information and communication technologies, many tools have appeared and been adapted for educational purposes. SageMath is a popular mathematical software which was released in 2005. This software has efficient features which utilize the internet and can handle most mathematical problems, including linear algebra, algebra, combinatorics, numerical mathematics and calculus. In this book, we will introduce this powerful software and discuss how it can be used in classes.
SageMath is a mathematical CAS tool and is based on Internet Web environment. This tool was introduced in April 2008 at University of Washington, USA. It is free and has a powerful capability that can be compared with expensive commercial softwares such as Mathematica, Maple or Matlab, but can do more than that without requiring separate installations of the program. It is more like WebMathematica, but has some better features. When you connect to any Web browser, you can really solve almost all calculus problems in the book by using resources that we are offering. You can easily find preexisting command to modify for your own problems.
Figure 1 We can use others existing codes as well in Sage
Korean SageMath model was developed and relevant experiments were done by BK 21 Mathematical Modelling HRD division at Sungkyunkwan University. We have built Korean servers that you can use as you see below. (Instructions will be given in your first class)
http://www.sagenb.org/ (Sign in)
http://math1.skku.ac.kr (ID: skku, Password: math)
http://math2.skku.ac.kr (Make your own accounts)
http://math3.skku.ac.kr (Mobile Server, No need for login)
Figure 2 SageMath Community (Left) http://www.sagemath.org,
SageMath Korean Version (Right) http://math1.skku.ac.kr
Use Chrome or Internet Explorer to make a connection to the one of the above servers. Then register your ID and password to start with.
1. Using the Internet to connect to http://math1.skku.ac.kr 2. ID: skku, Password: math 3. Click "New Worksheet" button in the upper left corner 4. Change a name of worksheet. 5. It is ready to use.
Figure 3 SageMath Worksheet 
2. The development of Korean Version of SageMath
You now have Quick Reference (simple manual) of SageMath in English and Korean. Those Quick References can be downloaded from the Web site address in the below.
http://wiki.sagemath.org/quickref
http://matrix.skku.ac.kr/2010Album/SageQReferenceSKKU.pdf
Sage Quick Reference: Basic Math and Calculus
Peter Jipsen, version 1.1 (Basic Math) Latest Version at http://wiki.sagemath.org/quickref
William Stein (Calculus) Sage Version 3.4 http://wiki.sagemath.org/quickref
GNU Free Document License, extend for your own use
Translated to Korean language by SangGu Lee and Jae Hwa Lee (Sungkyunkwan University)
Korean Version at http://matrix.skku.ac.kr/2010Album/SageQReferenceSKKU.pdf
Notebook (and Command line)
Evaluate cell: <shiftenter>
com <tab> tries to complete command
command?<tab> shows documentation
command??<tab> shows source
a.<tab>i shows all methods for object a (more: dir(a))
search_doc('string or regexp') shows links to docs
search_src('string or regexp') shows links to source
lprint() toggle LaTeX output mode
version() print version of Sage
Insert cell: click on blue line between cells
Delete cell: delete content then backspace
Numerical types
Integers: ℤ=ZZ e.g. 2 1 0 1 10^100
Rationals: ℚ=QQ e.g. 1/2 1/1000 314/100 42
Decimals: ℝRR e.g. .5 0.001 3.14 42.
Complex: ℂCC e.g. 1+i 2.53*i
Builtin constants and functions
Constants: =pi =e =I=i =oo
=oo=infinity NaN=NaN log(2)=log2
=golden_ratio =euler_gamma
Builtin functions: sin cos tan sec csc cot sinh cosh tanh sech csch coth log ln exp
=a*b =a/b =a^b =sqrt(x) =x^(1/n) =abs(x) =log(x,b)
Symbolic variables: e.g. t, u, v, y, z = var('t u v y z')
Define functions: e.g. f(x)=x^2 or f=lambda x: x^2 or def f(x): return x^2
Operations and equations
Relations: : f == g, : f != g, : f <= g, : f >= g, : f < g, : f > g
Solve : solve(f(x)==g(x),x)
Solve : solve([f(x,y)==0, g(x,y)==0], x,y)
Exact roots: (x^3+2*x+1).roots(x)
Real roots: (x^3+2*x+1).roots(x, ring=RR)
Complex roots: (x^3+2*x+1).roots(x, ring=CC)
: sum([f(i) for i in [k..n]])
: prod([f(i) for i in [k..n]])
Defining symbolic expressions
Create symbolic variables: var("t u theta") or var("t, u, theta") ...
Symbolic functions
Symbolic function (can integrate, dierentiate, etc.): f(a, b, theta) = a + b*theta^2
Also, a "formal" function of theta: f = function('f', theta)
Piecewise symbolic functions: Piecewise([[(0, pi/2), sin(1/x)], [(pi/2, pi), x^2+1]])
Python functions
Defining:
def f(a, b, theta=1):
c = a + b*theta^2
return c
Inline functions:
f = lambda a, b, theta = 1: a + b*theta^2
Factorization
Factored form: (x^3y^3).factor()
List of (factor, exponent) pairs: (x^3y^3).factor_list()
Limits
: limit(f(x), x=a)
Derivatives
: diff(f(x),x) or f.diff(x)
: diff(f(x,y),x)
e.g. diff(x*y + sin(x^2) + e^(x), x)
Integrals
: integral(f(x),x) and f.integrate(x)
e.g. integral(x*cos(x^2), x)
: integral(f(x),x,a,b)
e.g. integral(x*cos(x^2), x, 0, sqrt(pi))
numerical_integral(f(x),a,b)[0]
e.g. numerical_integral(x*cos(x^2),0,1)[0]
assume(...): use if integration asks a question
e.g. assume(x>0)
Taylor and partial fraction expansion ...
Numerical roots and optimization ...
Multivariable calculus
Gradient: f.gradient() or f.gradient(vars)
e.g. (x^2+y^2).gradient([x,y])
Hessian: f.hessian()
e.g. (x^2+y^2).hessian()
Jacobian matrix: jacobian(f, vars)
e.g. jacobian(x^2  2*x*y,(x,y))
2D graphics
Line: line([(),...,()], options)
Polygon: polygon([(),...,()], options)
Circle: circle((), , options)
Functions: plot(f(), (x, , ), options)
Parametric functions: parametric plot((f(),g()), (t, , ), options)
Polar functions: polar_plot(f(), (t, , ), options)
Animate: animate(list of graphics objects, options).show(delay=20)
3D graphics
Line: line3d([(),...,()], options)
Sphere: sphere((), , options)
Tetrahedron: tetrahedron((), size, options)
Functions: plot3d(f(), (x, ), (y, ), options)
add option plot_points=[]
Parametric functions: parametric_plot3d((f(), g(), h()), (t, ), options) or
parametric_plot3d((f(), g(), h(), (u, ), (v, ), options)
Linear algebra
: vector([1,2])
: matrix([[1,2], [3,4]])
: det(matrix([[1,2], [3,4]]))
: A*v
: A^1
: A.transpose()
Other methods: nrows(), ncols(), nullity(), rank(), trace(), etc.
Sage Problem: Exercise 22 in Section 9.3 (Polar Coordinates)
★ Sketch the curve with the given polar equation.
Published at http://math1.skku.ac.kr/home/pub/78 / http://math1.skku.ac.kr (ID/PW: skku/math)
Sage Problem: Exercise 16 in Section 11.6 (Cylinders and Quadric Surfaces)
★ Sketch the region bounded by the surfaces and for .
Published at http://math1.skku.ac.kr/home/pub/80 / http://math1.skku.ac.kr (ID/PW: skku/math)
All of the above Sage commands and many more in the published section of each Sage server (e.g. http://math1.skku.ac.kr/pub/) can be copied and pasted, so you can modify. Sungkyunkwan University have made more than 3,000 Sage commands. We hope all of you can take full advantage of SageMath in learning and teaching Calculus.
Contents
Calculus
Chapter 1. Functions 1
Chapter 2. Limits and Continuity 11
Chapter 3. Theory of Differentiation 21
Chapter 4. Applications of Differentiation 39
Chapter 5. Integrals 63
Chapter 6. Applications of Integration 81
Chapter 7. Techniques of Integration 101
Chapter 8. Further Applications of Integration 123
Chapter 9. Parametric Equations and Polar Coordinates 133
Chapter 10. Infinite Sequences and infinite Series 163
Chapter 11. Vectors and the Geometry of Space 183
Chapter 12. Partial Derivatives and Local Maxima and Minima 211
Chapter 13. Vector Functions 213
Chapter 14. Vector Calculus 233
History of Calculus
1.Celsius and Fahrenheit. If the temperature is degrees Celsius, then the temperature is also degrees Fahrenheit, where
(a) Find , , and .
(b)Suppose the outside temperature is degrees Celsius. What is the temperature in degrees Fahrenheit?
(c)What temperature is the same in both degrees Fahrenheit and in degrees Celsius?
(a)
(b)
(c) Let . Then .
That is .
Hence .
Therefore the temperature that the same in both degrees Fahrenheit and in degrees Celsius is .
2. Brain Weight Problem. The weight of a human’s brain is directly proportional to his or her body weight .
(a) It is known that a person who weights has a brain that weight . Find an equation of variation expressing as a function of .
(b) Express the variation constant as a percent and interpret the resulting equation.
(c) What is the weight of the brand of a person who weighs ?
(a).
Since , .
.
(b).
That is .
Hence the variation constant is and brain weight is of body weight.
(c) Since , .
3. Muscle Weight. The weight of the muscles in a human is directly proportional to his or her body weight .
(a) It is known that a person who weighs has of muscles. Find an equation of variation expressing as a function of .
(b) Express the variation constant as a percent and interpret the resulting equation.
(a) Since , .
So, we know that .
Hence .
(b) .
That is .
Hence the variation constant is and muscles weight is of body weight.
4. Estimating Heights. An anthropologist can use certain linear functions to estimate the height of a male or female, given the length of certain bones. The humerus is the bone from the elbow to the shoulder. Let be the length of the humerus, in centimeters. Then the heights, in centimeters, of a male with a humerus of length is given by
The height, in centimeters, of a female with a humerus of length is given by
A humerus was uncovered in a ruins.
(a) If we assume it was from a male, how tall was he?
(b) If we assume it was from a female, how tall was she?
(a)
(b)
5. Urban Population. The population of a town is . After a growth of , its new population is .
(a) Assuming that is directly proportional to , find an equation of variation.
(b) Find when .
(c) Find when .
(a)
(b)
(c) .
That is .
6. Median Age of Women at first Marriage. In general, our society is marrying at a later age. The median age of women at first marriage can be approximated by the linear function
,
where is the median age of women at frist marriage years after . Thus, is the median age of women at first marriage in the year , is the median age in , and so on.
(a) Find , , , and .
(b) What will be the median age of women at first marriage in ?
(c) Graph .
(a)
(b)
(c)
Symmetry
15. PiecewiseDefined Functions :
Graph the following functions.
1.
2.
3.
4.
5. Find a formula for given graphed.
610. Graph the functions in Exercises 610. What symmetric, if any, do the graphs have? Specify the intervals over which the functions is increasing and the intervals where it is decreasing.
6.
symmetric with respect to (w.r.t.) the origin.
decreasing on
7.
symmetric w.r.t. the origin.
decreasing on
increasing on
8.
symmetric with respect to the axis.
decreasing on
increasing on
9.
symmetric w.r.t the origin.
increasing on
10.
it has no symmetry.
decreasing on
1116.Say whether the functions is even, odd, or neither. Give reasons for your answer.
11.
even function
12.
even function
13.
odd function
14.
even function
15.
neither odd or even function
16.
neither odd or even function
Common Functions
1. If , find and .
2. Find the domain of the function.
(a)
(b)
(c)
(d)
(e)
(a)
(b)
(c)
(d)
(e)
3. Find the domain and sketch the graph of the function.
(a)
(b)
(c)
(a)
(b)
(c)
4. Determine whether is even, odd or neither. If is even or odd, use symmetry to sketch its graph.
(a)
(b)
(c)
(d)
(e)
(f)
(a) even
(b) odd
(c) neither
(d) even
(e) odd
(f) neither
5. Draw the original and given functions graphs together.
5.
Translation, stretching and
rotation of functions
16. Exercises 16 tell how many units and in what directions the graphs of the given equations are to be shifted. Give an equation for the shifted graph. Then sketch the original and shifted graphs together labeling each graph with its equation.
1. . Down , left .
2. . Left , down .
3. . Left .
4. . Up .
5. . Up , right .
6. . Down , right .
716. Graph the functions in Exercises 716.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
Limits of functions
17. Find the following limits or explain why the limit does not exist.
1.
2.
does not exist.
3.
http://math2.skku.ac.kr/home/pub/33
You can draw to use Sage. Then diverge to at .
+Infinity
4.
5.
6.
7.
http://math2.skku.ac.kr/home/pub/34
0
8. The sign function, denoted by , is defined by the following formulas
Find the following limits or explain why the limit does not exist.
(a)
(b) does not exist.
(
)
(c)
(d)
9. Consider the function .
(a) Find and .
(b) Find the asymptotes of ; vertical, horizontal, vertical and oblique.(For the oblique asymptote, please find the straight line which is closer and closer to as )
(c) Sketch the graph.
(a) ,
(b) : vertical asymptote
: oblique asymptote
(c)
10. Draw the graph of a function with all of the following properties:
(a) its domain is
(b) there is a vertical asymptote at
(c)
(d)
(e) does not exist.
(f) does not exist.
(g)
11. Let .
(a) Find or explain why it does not exist.
(b) Find and such that for all .
(c) Use squeeze Theorem to find .
(a)
(b) Since , we have
.
so,
(c) We know that and
, so by the Squeeze Theorem,
.
12. Use squeeze Theorem to find
.
13. Use squeeze Theorem to find .
(※We can use Squeeze Theorem around 0.)
14. Let . Find all positive integer such that
ⅰ) ;
ⅱ) ;
ⅲ) ;
15. Find all the asymptotes (vertical, horizontal, and oblique) of the function .
,
so ,,
and .
Thus, and are vertical asymptotes.
,
Thus, is oblique asymptotes.
16. Find the limit
17. Consider .
(a) Find all the vertical asymptotes for .
(b) If we restrict our domain to , then show that there exists an inverse function defined on .
(c) If the above inverse function is , then find all the horizontal asymptotes.
(a) , so
.
.
Thus, .
18.Find such t hat w henever .
19. Use the argument to prove that .
Let be a given positive number. Here &. Claim is to find a number s.t. whenever . With easy computation, we may choose to get the desired result.
20. Use the argument to prove that if .
Let be a given positive number. Here &. Claim is to find a number s.t. whenever . With easy computation, we may choose to get the desired result.
2126. Prove the statements using the argument.
21.
Let be a given positive number. Here &. Claim is to find a number s.t. whenever . With easy computation, we may choose to get the desired result.
22.
Let be a given positive number. Here
.
Claim is to find a number s.t. whenever . With easy computation, we may choose to get the desired result.
23.
Let be a given positive number. Here &. Claim is to find a number s.t. whenever . With easy computation, we may choose to get the desired result.
24.
Let be a given positive number. Here &. Claim is to find a number s.t. whenever . With easy computation, we may choose to get the desired result.
25.
Given any (large) number to find such
that whenever . Since
both and are positive,
whenever . Taking the square root of
both side and recalling that , we
get whenever .
So for any , choose .
Now if , then , that is,
.
Thus whenever .
Therefore .
26.
Let be a given positive number. Here &. Claim is to find a number
s.t. whenever . With easy computation, we may choose to get the desired result.
27. Use the argument to prove that
28. If and , where is a real number. Show that
(a)
(b) if
(a)
(b)
Continuity
1. If and are continuous functions with and , find .
2. If and are continuous functions with and , find .
Since is continuous, .
3. Show that the function is discontinuous at .
http://math1.skku.ac.kr/home/skku/328/
+Infinity
47. Determine the points of discontinuity of . At which of these numbers is continuous from the right, from the left or neither? Sketch the graph of .
4.
http://math2.skku.ac.kr/home/pub/35
5.
We see that exists for all a except . Notice that the right and left limits are different .
6.
We see that exists for all a except . Notice that the right and left limits are different and we see that exists for all a except . Notice that the right and left limits are different .
7.
We see that exists for all a except . Notice that the right and left limits are different and we see that exists for all a except . Notice that the right and left limits are different .
810. For what values of the constant is the function continuous on ?
8.
Thus, for to be continuous on .
9.
10.
Since is not continuous at , solution is .
1113. Show that the following functions has the removable discontinuity at . Also find a function that agrees with for and is continuous on ℝ.
11.
for . The discontinuity is removable and agrees with for and is continuous on ℝ.
12.
for . The discontinuity is removable and agrees with for and is continuous on ℝ.
13.
for . The discontinuity is
removable and agrees with
for and is continuous on ℝ.
14. Let . Is removable discontinuous. ?
Since
is not removable discontinuous.
15. If , show that there is a number such that .
is continuous on the interval , and . Since , there is a number in such that by the Intermediate Value Theorem.
16. Prove using Intermediate Value Theorem that there is a positive number such that .
Let . is continuous on the interval , and . Since , there is a number in such that by the Intermediate Value Theorem.
1419. Prove that there is a root of the given equation in the specified interval by using the Intermediate Value Theorem.
14.
15. ,
True
True
16.
1
1
17. ,
True
True
18.
is continuous on the interval , and . Since , there is a number in such that by the Intermediate Value Theorem. Thus, there is root of the equation in the interval .
19.
is continuous on the interval , and . Since , there is a number in such that by the Intermediate Value Theorem. Thus, there is root of the equation in the interval .
2023. Show that each of the following equation has at least one real root.
20.
http://math2.skku.ac.kr/home/pub/37
We can draw and at the same time. There are real roots at or .
0.37055809596982464
1.3649584337330951
21.
Let . Then and . So by the Intermediate Value Theorem. There is a number in such that . This implies that .
22.
Let . Then and . So by the Intermediate Value Theorem. There is a number in such that . This implies that .
23.
Let . Then and , and is continuous So by the Intermediate Value Theorem. there is a number in such that . This implies that .
2425. Find the values of for which is continuous.
24.
The value of don’t exist.
25.
is continuous at .
Derivatives of Polynomials,
Exponential Functions,
Trigonometric Functions,
The Product Rule
15. Find the derivative where is
1.
2.
3.
4.
5.
6. Find where .
15/14*x^(1/14) + 5*e^x
7. Find the equation of the tangent line to the curve at .
. So the slope of the tangent line is 20.
( passes through ).
20*x48
8. The normal line to a curve at a point is the line that passes through and os perpendicular to the tangent line to at . Find an equation of the normal line to the curve at the point .
So the slope of the tangent line is . ()
Then the slope of the normal line is .
Thus, .
The normal line is .
x+2
9. where is the function differentiable? Give a formula for .
,
then .
So is continuous on .
,
then is not differentiable at .
is differentiable on .
10. Let . Find the values of and that make differentiable everywhere.
To be differentiable at , , so .
And also have to be continuous at .
, since , .
11. Evaluate .
Method (1) Let , and . Then by the definition of a derivative,
.
Method (2) Note that
.
So
2020
1213. Differentiate the following functions.
12.
13.
1417. Find
14.
15.
16.
17.
It is appear that the denominator terms is , and numerator is .
18. Find the th derivative of .
19. Prove if , then satisfies the identity .
20. If satisfies the identity for all and and , then show that satisfies for all .
Thus, is or .
Since and , is for all .
Now,
Since , then
.
Therefore, .
2123. Find the following derivatives.
21.
22.
23.
2425. Find the following limit.
24.
(by L’Hospital)
or
25.
(by L’Hospital)
or
26. Show that the curve has no tangent line with slope .
Since is always positive, there is no such that .
So has no tangent line with slope 0.
2728. Find and of the followings.
27.
28.
29. Given , show that it satisfies the following identity.
Using this identity, find .
Therefore,
.
Therefore, for is even,
for is odd.
30. Given , show that .
Since so is continuous on .
And
3133. Find of the following expressions.
31.
32.
33.
34. If , where and are three times differentiable, find expressions for and .
35. Given , find at the point .
,
36. A stone is thrown into a pond, creating wave whose radius increases at the rate of meter per second. In square meter per second, how fast is the area of the circular ripple increasing seconds after the stone hits the water?
radius, time
37. A particle moves along a straight line with equation of motion .
(a) When is the particle moving forward?
(b) when is the acceleration zero?
(c) when is the particle speeding up? Slowing down?
(a)
(b)
(c) acceleration zero is .
In , the particle is speeding up, in , the particle is slowing down.
38. A particle moves in a straight line with equation of motion , where is measured in second and in meters.
(a) What is the position of the particle at and ?
(b) Find the velocity of the particle at time .
(c) When is the particle moving forward?
(d) Find the total distance traveled by particle on the time interval .
(e) Find the acceleration of the particle at time .
(a)
(b)
(c) When ,
(d)
(e)
39. The population of the bacteria colony after hours is . Find the growth rate when .
40. A cost function is given by
.
(a) Find the marginal cost function.
(b) Find .
(a)
(b)
41. If a stone is thrown vertically upward with a velocity , then its height after seconds is
(a) What is the maximum height reached by the stone?
(b) What is the velocity of the stone when it is above the ground on its way up? On its way down?
(a)
(b)
The time is when the height of the stone is 5.
42. If is the total value of the production when there are workers in a plant, then the average productivity is .
Find . Explain why the company wants to hire more worker if ?
If , since , then .
is the rate of productivity.
This means , rate of productivity is larger than , average productivity.
Thus, the company wants to hire more.
43. Let be the population of bacteria colony at time hours. Find the growth rate of the bacteria after 10 hours.
44. The angular displacement of simple pendulum is given by with the angular amplitude , the angular frequency and a phase constant . Find .
45. Show that .
Let . Then , and as , .
The Chain Rule and Inverse Functions
13. Find the differential .
1.
2.
3.
45. Find of these functions.
4.
5.
6. Find where .
, ,
,
Therefore, .
7. Find for an integer if .
8. Show that the curves and are orthogonal.
Thus, the intersections of two curves are and .
At , it is not defined.
At ,
orthogonal
9. Use differentiation to show that
1013. Find of the following expressions.
10.
11.
12.
1/2*(2*sqrt(x)*cos(x)/sin(x)+log(sin(x))/sqrt(x))*sin(x)^sqrt
13.
(log(arccos(x))/(x^2+1)arctan(x)/(sqrt(x^2+1)*arccos(x)))*arccos(x)^arctan(x)
14. Find if .
(log(x)/(x^2+1)+arctan(x)/x)*x^arctan(x)
(log(x)/(x^2+1)+arctan(x)/x)^2*x^arctan(x)(2*x*log(x)/(x^2+1)^2+arctan(x)/x^22/((x^2+1)*x))*x^arctan(x)
15. Use differentiation to show that for all .
for all
16. Find an equation of the tangent line to the curve at for an arbitrary value .
, so we have to find the tangent line at .
17. Show the following identities.
(a)
(b)
(c)
(a)
(b)
(c)
1830. Prove the following identities.
18.
19.
20.
21.
22.
23.
24.
By mathematical induction, if , trivial.
Let
25.
26.
Let . Then , so .
.
Note that , but .
Thus, .
27.
Let . Then , so .
.
Note that , but .
Thus, for .
28.
Let .
Then
.
29.
Let . Then , .
Since and , we have
,
so .
30.
Let . Then
31. Let be a point in the first quadrant on the hyperbola . Then and can be parameterized by , , . Let be the area of the region bounded by axis, axis, the straight line and . Let be the area of the region bounded by axis, axis, the straight line and . Show that . (Hint : Integration will be helpful)
,
,
,
Approximation and Related Rates
13. Use differential to approximate the followings.
1.
http://math2.skku.ac.kr/home/pub/42
Let’s define to find an approximation of . Viz, .
2.00062460974081
And Find .
2.00062500000000
Also, We can use Sage functions.
2.00062460974081
2.
1/2*dx/sqrt(x + 27)
5.19615242270663
3.
1/3*dx/(x + 60)^(2/3)
3.93649718310217
4. The height of a circular cone is the same as the radius of its circular bottom. The height and radius were measured and found to be 5cm with a possible error in measurement of at most 0.02cm. What is the relative error in using these value to compute the volume?
5. Find the approximation of the difference between surface areas of two spheres whose radii are 4cm and 4.05cm, respectively.
If
6. The period of the pendulum is given by the formula , where is the length of the pendulum measured in meters and is the gravitational constant. If the length of the pendulum is measured to be 3m with a possible error in measurement 1cm. What is the approximate percentage error in calculating the period ?
, ,
The approximate percentage error
= (relatively error)100%=.
7. A ladder 10 meter long is leaning against a wall. If the foot of the ladder is being pulled away from the wall at 3m/s, how fast is the top of the ladder sliding down the wall when the foot of the ladder is 6 meter away from the wall?
Since so
8. A ladder 10 meter long is leaning against a wall. If the top of the ladder is sliding down the wall at 3m/s, how fast is the foot of the ladder being pulled away from the wall when the foot of the ladder is 6 meter away from the wall?
Since , .
9. A ladder 10 meter long is leaning against a wall. If the top of the ladder is sliding down the wall at 3m/s, how fast is the angle between the top of the ladder and the wall changing when the foot of the ladder is 6 meter away from the wall?
Since ,
10. Two cars start moving from the same point. One travels south at km/hour and the other travels west at km/hour. How fast is the distance changing between the two cars?
Distance of travel south distance of travel west, and distance of two travelers at time .
11. Water is being pumped at a rate of 20 liters per minute into a tank shaped like a frustrum of a right circular cone. The tank has an altitude of 8 meters and lower and upper radii of 2 and 4 meters, respectively. How fast is the water level rising when the depth of the water is 3 meters?
.
Let .
12. Water is being pumped at a rate of 20 liters per minute into a tank shaped like a hemisphere. The tank has a radius of 8 meters. How fast is the
water level rising when the depth of the water is 3 meters?
13. A snowball melts at a rate proportional to its surface area. Does the radius shrink at a constant rate? If it melts to 1/2 its original volume in one hour, how long does it take to melt completely?
. Let .
Since ,
: constant
Let the volume of first time , after an hour, .
Extreme values of a function
14. Determine the following statement is True or False. Explain your answer.
1. If is a continuous function, then has a maximum at only one point in .
False. Consider .
2. One can apply the Mean Value Theorem for
on .
False.
3. Of a continuous function has a extreme value on , then has a absolute maximum or minimum value on .
False.
4. For a continuous function , has only one zero provided is strictly decreasing.
True.
58. Find all critical numbers of given functions.
5.
[x == 2, x == (3/2), x == (5/3)]
6.
7. .
[x == 1, x == 1]
8.
[x == 1, e^x == 0]
913. Find all local maximum and minimum if it exists.
9.
No local minimum and maximum
10.
No local maximum, local minimum
11.
sin(x)*cos(x)/abs(sin(x))
[x == 0, x == 1/2*pi]
12.
if , , then has a local minimum at .
if , , then has a local minimum at .
if , , then has a local maximum at .
13. ,
then has a local maximum at
1416. Find the intervals of increase or decrease.
14.
intervals of increase:
intervals of decrease:
15.
intervals of increase:
intervals of decrease:
[x == 1]
16.
intervals of increase:
intervals of decrease:
1719. Prove the inequality.
17. .
for
18. ,
19.
20. Let . Find the number of zeros and give a proper interval containing all zeros.
7 zeros in
21. Suppose and for all . Using the Mean Value Theorem show for all .
,
22. Show that for .
By Excercise (24),
Then, .
2324. Prove the inequality using the Mean Value Theorem.
23.
24.
25. Prove that the inequality for all and using the Mean Value Theorem.
,
26. Let be a continuous function. Prove there is only the function satisfying for all using the Mean Value Theorem.
27. Suppose is continuous on and differentiable on . Prove that there exists such that .
,
By Roll’s theorem,
there exist such that
.
28. Prove Fermat’s Theorem.
Suppose, for the sake of definiteness, that has a local maximum at , Then, if is sufficiently close to . This implies that if is sufficiently close to 0, with being positive or negative, then and therefore
. (＊)
We can divide both sides of an inequality by a positive number. Thus, if and is sufficiently small, we have
.
Taking the righthand limit of both sides of this inequality, we get
But since exists, we have
and so we have shown that .
If , then the direction of the inequality (＊) is reversed when we divide by :
.
So, taking the lefthand limit, we have
We have shown that and also that . Since both of these inequalities must be true, the only possibility is that
29. Prove Rolle’s Theorem.
Suppose then that the maximum is obtained at an interior point of . We shall examine the above right and lefthand limits separately.
For a real such that is in , the value is smaller or equal to because attains its maximum at . Therefore, for every ,
,
hence
,
where the limit exists by assumption, it may be minus infinity.
Similarly, for every , the inequality turns around because the denominator is now negative and we get
,
hence
,
where the limit might be plus infinity.
Finally, when the above right and lefthand limits agree, then the derivative of at must be zero.
30. Prove Increasing and Decreasing Test.
(a) Let and be any two numbers in the interval with . According to the definition of an increasing function we have to show that .
Because we are given that , we know that is differentiable on . So, by the Mean Value Theorem, there is a number between and such that
. (＊)
Now by assumption and because . Thus the right side of (＊) is positive, and so
or .
This shows that is increasing.
Part (b) us proved similarly.
31. Prove the cases (b), (c) of the First Derivative Test.
(b) Let us choose at sufficiently near such that . By the Mean Value Theorem, there exist with and such that
and
.
Since for and for , is decreasing on and increasing on . Hence has a local minimum at .
(c) Let us choose at sufficiently near such that . By the Mean Value Theorem, there exist with and such that
and
.
Since for and for . Hence has neither local maximum nor local minimum at .
32. Prove the case (b) of the Second Derivative Test.
(b) Suppose we have . Then
Thus, for sufficiently small we get which means that if so that is decreasing to the left of , and that if so that is increasing to the right of . Now, by the first derivative test we know that has a local maximum at .
The Shape of a Graph
15. Find the local maximum and minimum values of . In addition, find the intervals on which is increasing and decreasing, and the intervals of concavity and the inflection points, sketch a graph of .
1.
(a) local maximum:
local minimum:
(b) increasing on
decreasing on
(c) inflection point at
CD on
CU on
2.
(a) local minimum:
(b) increasing on
decreasing on
(c) inflection point at
CD on
CU on
4*x^368*x
[x == sqrt(17), x == sqrt(17), x == 0]
12*x^268
[x == 1/3*sqrt(3)*sqrt(17), x == 1/3*sqrt(3)*sqrt(17)]
3.
(a) local maximum:
local minimum:
(b) increasing on
decreasing on
(c) inflection point at
CD on
CU on
4.
(a) local maximum: No
local minimum:
(b) increasing on
decreasing on
(c) inflection point: No
CU on
5.
(a) Maximum: 1
Minimum: 0
(b) interval of increase:
interval of decrease:
(c) inflection point at
CD on
2*(tanh(x^2)^2  1)*x
611. Find the inflection points of In addition, find where the graph of is concave upward or concave downward.
6.
(a) inflection point at
(b) CU on
CD on
7.
(a) inflection point at
(b) CU on
CD on
8. ,
(a) inflection point at
(b) CD on
CU on
9.
[x == (1/3)]
(a) inflection point at
(b) CD on
CU on
10.
(a) inflection point at
(b) CU on
CD on
11.
(a) inflection point :
(b) CU on
CD on
1215. Find the vertical and horizontal asymptotes of .
12.
vertical asymptote
horizontal asymptote
13.
vertical asymptote
horizontal asymptote
x^2/(x^4  6*x^2 + 9)
14.
horizontal asymptote
15.
vertical asymptote :
horizontal asymptote : No
log(sec(x))
1618. Sketch the graph of using the following information.
(a) Find the local maximum and minimum values of .
(b) Find the intervals of increase or decrease.
(c) Find the inflection points of and intervals of
concavity.
(d) Find the vertical and horizontal asymptotes.
16.
(a) local maximum: No
local minimum: No
(b) decreasing on
(c) inflection point at
CU on
CD on
(d) vertical asymptote:
horizontal asymptote:
17.
,
(a) local minimum at
(b) increasing on
decreasing on
(c) inflection point: No
CU on
(d) vertical asymptote:
horizontal asymptote: No
18. ,
,
(a) local minimum at
local maximum:
(b) increasing on
decreasing on
(c) inflection point at
CU on
CD on
(d) vertical asymptote: No
horizontal asymptote:
19. Find the range of on which has different roots.
20. Find the minimum constant on which
for all real .
,
local minimum at
21. Find that has two
inflection points and .
22. Let .
(a) Find and .
(b) Find the vertical and horizontal asymptotes of .
(c) Sketch the graph of using (a) and (b).
(a)
(b) ,
vertical asymptote
horizontal asymptote
(c)







23. Let and be increasing functions. Prove that is increasing function.
, ,
Therefore, is increasing.
24. Prove the concavity test.
(a) By Increasing Test
If on an interval then is increasing on .
So is concave upward on .
Part (b) us proved similarly.
2527. Use CAS.
25. Find and when .
[df == 4*x^(1/3) + 2/x^(2/3), dff == 4/3/x^(2/3)  4/3/x^(5/3)]
26. Let . Find the local maximum, minimum values and inflection points of . Sketch the graph of .
local maximum:
local minimum: No
inflection point:
27. Let . Find the local maximum, minimum values and inflection points of .
local maximum:
local minimum: No
inflection point:
The Limit of Indeterminate
Forms and L’Hospital’s Rule
111. Evaluate the limits of given indeterminate forms.
1.
by L’Hospital Rule.
2.
3.
4.
5.
By L’Hospital Rule
6.
2
7.
8.
By L’Hospital Rule
9.
10.
11.
12. Let . Use L’Hospital’s rule to show that .
13. For what values of and , is the following equation correct?
By L’Hospital Rule
By L’Hospital Rule
14.
by L’Hospital Rule
15.
1625. Find the limit by using L’Hospital’s rule. If you can not apply L’Hospital’s rule, explain why and then find the limits some other way.
16.
, Not exists.
17.
18.
19.
20.
21.
22.
, Not exists.
23.
24.
0
25.
26. The function are defined by
and .
Answer the following questions.
(a) Find .
(b) Does exist ?
(a)
(b)
does not exist.
27. Let be a continuous function with and .
Find .
28. Let be a angle of a sector of a circle. Find where and are the area of the segment between the chord and the area of the triangle respectively.
,
Optimization Problems
1. Determine where the point between and should be located to maximize the angle .
Let , ,
.
Then
,
when . Since
2. A shop sells 200 mp3 players per week while each costs . According to the market research, sales will increase by 20 mp3 players per week for discount. How much should they discount to maximize profits?
If discount,
profit
profits is maximum when .
They should discount $120 or $130.
3. The height of a safe in and its bottom is in the shape of a square whose side is . It costs won per to make the top and the bottom, and won per to make the side. Find the maximum volume of the safe which can be made by won.
4. Two particles and are in motion in the plane. Find the minimum distance between and .
5. A closed cylindrical can is to hold of liquid. Find the height and radius that minimize the amount of material needed to manufacture the can.
6. Find a point on the curve that is closest to the point .
The distance from to an arbitrary point on the curve is
and the square of the distance is
,
.
Graphing on gives us a zero at , and so . The point on that is closest to is .
7. Find the largest area of a triangle which is inscribed in the circle of radius .
The area of the triangle is
Then,
or
Now
the maximum occurs where ,
so the triangle has height
and base .
8. Let be the volume of the right circular cone and be the volume of the right circular cylinder that can be inscribed in the cone. Find the ratio when the cylinder has the greatest volume.
,
By similar triangles, , so .
The volume of the cylinder is
.
Now .
So or . The maximum clearly occurs when and then the volume is
910. Consider an ellipse .
9. Find the area of the rectangle of greatest area that can be inscribed in the ellipse.
Without loss of generality, choose
on ellipse in first quadrant.
10. Find the minimum length of the tangent line which is cut by the axis and the axis.
,
11. A conshaped paper cup is to be made to hold of water. Find the height and radius of the cup that minimize the amount of paper used to make.
The volume and surface area of a cone with radius and height are given by and .
We’ll minimize subject to
so
, so and hence has an absolute minimum at these values of and .
12. A pipe is being carried horizontally around a corner from a hallway wide into a hallway wide. What is the longest length that the pipe can have?
Let be the length of the line going from wall to wall touching the inner corner . As or , we have and there will be an angle that makes a minimum. A pipe of this length will just fit around the corner. From the diagram,
when
. Then and
, so the longest pipe has length
.
13. Find the length of the shortest ladder that reaches over an 8ft high fence to a wall which is 3ft behind the fence.
,
when
when ,
when ,
so has an absolute minimum when , and the shortest ladder has length
,
Newton’s Method
13. Compute , the third approximation to the root of the given equation using Newton’s method with the specified initial approximation .
1.
2.
3.
4. Calculate the second approximation to the root of the equation by using Newton’s method with initial approximation to .
510. Approximate the given number using Newton’s method.
5.
6.
7.
8.
9.
(ⅰ)
(ⅱ)
10.
(ⅰ)
(ⅱ)
11. Apply Newton’s method to the equation to derive the following reciprocal algorithm:
. Hence compute .
(ⅰ)
12. Use Newton’s method to find the absolute minimum value of the function .
Areas and Distances
1. Find the area under the curve from 0 to 2.
2. Find the exact area of the region under the graph of from 0 to 2.
3. Find the exact area under the sine curve from to , where .
(Since , the value what we evaluate is equal to the area.)
4. (a) Let be the area of a polygon with equal sides inscribed in a circle with radius . By dividing the polygon into congruent triangles with central angle , show that
.
(b) Show that .
(Hint: Use Equation 3.4.2)
(a) The area of one piece of
consists of such pieces so the area of
(b) Using ,
The Definite Integral
14. Find the Riemann sum by using Midpoint rule with given value of to approximate the integral.
1. ,
Let . With the interval width is and midpoints are for . So the Riemann sum is
2. ,
Let . With the interval width is and midpoints are for . So the Riemann sum is
3. ,
4. ,
1.04754618408
58. Express the limit as a definite integral on the given interval.
5. ,
6. ,
7. ,
8.
918. Determine whether the statement is true or false. If it is true, explain why. If it is false, give a counter example.
9. If and are continuous on , then
.
True.
10. If and are continuous on , then
.
False.
11. If and are continuous on and for all , then
.
False.
12. If is continuous on , then
.
True.
13. If is continuous on , then
.
False.
14. If is continuous on and , then
.
False.
Let . Then and .
Hence .
15. If then for all .
False.
16. If and are continuous and and then .
True.
17. If and are differentiable and for , then for .
False.
18. All continuous functions have derivatives.
False.
1922. Evaluate the integral.
19.
20.
21.
22.
2326. Evaluate the integral by interpreting in terms of the areas.
23.
24.
25.
,
26.
Let . Then .
Since and ,
we have .
27. Prove that .
By using the end point rule,
.
Hence .
28. Prove that .
By using the end point rule,
Hence .
29.If and , find .
30. If and , find .
31. Find if
Since , is continuous.
3235. Verify the inequality without evaluating the integrals.
32.
Since for , we have .
Hence .
33.
Since for , we obtain .
Hence .
34.
Since , we obtain
.
So .
35.
Since and for , we obtain
.
The Fundamental Theorem of
Calculus
1. Let , where is the function whose graph is shown.
(a)Evaluate and .
(b)Estimate , and .
(c) On what interval is increasing?
(d) Where does have a maximum value?
(e) Sketch a rough graph of .
(f) Use the graph in part
(e) to sketch the graph of . Compare with the graph of .
(a)
(b)
(c)
(d)
(e)
(f)
23. Draw the area represented by . Then find in two ways:
(a)by using Part 1 of the FTC and
(b)by evaluating the integral using Part 2 and then differentiating.
2.
(a)
(b)
3.
(a)
(b)
47. Find the derivative of the function using part 1 of the FTC.
4.
5.
[Hint: ]
6.
7.
810. Evaluate the integral using Part 2 of FTC.
8.
9.
10.
11. Let . Use Part 1 of FTC to find .
so .
12. Give a nonpolynomial function () such that and .
For any function set .
Then clearly and so
For example .
13. Let and . Find .
and .
Hence so .
14. Let defined on
. Find .
.
So .
Note that and for . Hence should be .
.
15. Let .
Find.
Differentiate both sides to get .
.
Hence
1617. Evaluate the integral and interpret it as a difference of areas.
16.
17.
18. If , where
, find .
19. Find the value of if , is continuous, and .
20. If is continuous and and are differentiable functions, find a formula for .
Indefinite Integrals and the Net Change Theorem
14. Verify by differentiation that the formula is correct.
1.
Let . Then .
That is
2.
Let and .
Then and .
That is
.
3.
Let then .
So
.
Also change of variable fo , then . .
That is
4.
Let then .
So
.
Also change of variable for , then .
That is
513. Find the general indefinite integral.
5.
6.
7.
8.
9.
10.
11.
12.
13.
So,
1422. Evaluate the integral.
14.
15.
16.
17.
18.
19.
20.
1/2*sqrt(2)*log(2*sqrt(6) + 5) + sqrt(2)*sqrt(6)
21.
22.
Let . Then .
Thus
.
23. Estimate the area of the region that lies under the curve and above the axis.
0.876956449003
24. Water is being added to a tank at a rate of per minute. How much water is added to the tank from to ?
The amount of water added for is
where so that .
25. Find the area of the region that lies to the right of the axis and to the left of the parabola .
Since the parabola meets the axis at and , the area is given by
.
26. A particle is moving along a line with the velocity function . Find the displacement and the distance traveled by the particle during the time .
The displacement=,
the distance= .
27. A honeybee population starts with 30 bees and increase at a rate of bees per week. How many honeybees are there after 10 weeks?
Since the net change in population during 10 weeks is , the total number of honeybees after 10 weeks is .
28. The acceleration function (in ) of a particle is given by and the initial velocity is . Find the velocity of the particle at time and determine the total distance traveled for .
Since , the velocity at time is . Therefore the distance traveled for is .
The Substitution Rule
13.Evaluate the integral by making the given substitution.
1.
2.
3.
410. Evaluate the indefinite integral.
4.
5.
6.
7.
8.
9.
10.
1123. Evaluate the definite integral, if it exists.
11.
( is odd function)
12.
13.
14.
15.
4
16.
17.
18.
Since
,
first we have
Then
( is even function.)
Hence
19.
because is odd function.
20.
Let
21.
22.
23.
because is odd function.
2425. Find the exact area.
24. ,
25. ,
4
26. Let be a function symmetrical with respect to .
If , find .
Since ,
.
Hence .
The Logarithm Defined as an Integrals
1. (a) By comparing areas, show that .
(b) Use the Midpoint Rule with to estimate .
(a)
From above figure, we have
.
Since ,
and , we have .
(b) Let . Then we get
2. By comparing areas, show that
Note that . Let . If we use the left endpoint rule with subinterval to estimate , then th height is . If we use the right endpoint rule with subinterval to estimate , th height is . Since of both rules is same as 1 and , we obtain the desired one.
3. (a) By comparing areas, show that .
(b) Deduce that .
(a) By similar way in Ex1,
and
.
Hence .
(b) Since is an increasing function and , we have .
4. Deduce the following laws of logarithms.
(a)
(b)
(c)
Let and .
Then and .
(a)
(b)
(c)
5. Show that by using the integral.
6. Evaluate
7. Evaluate
Let .
Then .
Hence we have .
8. Evaluate .
The above limit of sum can be written as a definite integral namely . To evaluate the integral, we first use the substitution . Then
The last integral above is computed as follows:
Hence
Additional problem
9. Find .
We know that as and as while the second factor approaches , which is 0.
The integral has as its upper limit of integration(Part 1 FTC) suggesting that differentiation might be involved.
Then the denominator reminds us the definition of the derivative with . So this becomes .
Thus define then .
Now
(FTC 1)
Note Alternatively use l’Hospital’s Rule.
Areas between Curves
121. Find the area of the region.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
2223. Find the integral and interpret is as the area of a region.
22.
23.
2425. Approximate the area of the region bounded by the given curves using the Midpoint Rule with n=4.
24.
25.
2629. Determine the area of the region bounded by the curves.
26.
27.
28.
29.
30. Find the region defined by the inequalities .
31. Find the area enclosed by the loop of the curve with equation .
(Tschirnhausen’s cubic.)
32. Find the area of the region bounded by the parabola the tangent line to this parabola at and the axis.
33. Find the number such that the line divides the region bounded by the curves and into two regions with equal area.
34. (a) Find the number such that the line bisects the area under the curve .
(b) Find the number such that the line bisects the area in part (a).
(a) ,
(b) ,
35. Find the values of such that the area of the region enclosed by the parabolas and is 1944.
For and , is another solution.
Volumes
117. Calculate the volume of the solid obtained by rotating the region bounded by the given curves about the specified line.
1. about the axis
2. about the axis
3. about the axis
4. about the axis
5. about the axis
6. about the axis
,
7. about the axis
8. about the axis
9. about the axis
10. about
11. about
12. about
13. about
14. about
15. about
,
16. about
17. about
1823. Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line.
18. about
19. about
20. about
21. about
22. about
23. about
2427. Describe the solid whose volume is rspresented by the integral.
24.
25.
26.
27.
2829. Find the volume of the described solid .
28. A right circular cone with height and base radius .
29. A frustum of a right circular cone with height , lower base radius , and top radius .
30. A tetrahedron with three mutually perpendicular faces and three mutually perpendicular edges with lengths 6cm, 8cm and 10cm.
,
31. The base of is an elliptical region with boundary curve . Crosssections perpendicular to the axis are isosceles right triangles with hypotenuse in the base.
32. A bowl is shaped like a hemisphere with diameter 40cm. A ball with diameter 20cm is placed in the bowl and water is poured into the bowl to a depth of h centimeters. Find the volume of water in the bowl.
Volumes by Cylindrical Shells
118. Find the volume generated by rotating the region bounded by the given curves about the axis using the method of cylindrical shells.
1.
(You may do it with Sage in http://math1.skku.ac.kr/. Open resources in http://math1.skku.ac.kr/pub/)
2.
3.
4.
5.
6. about
7. about
8. about the axis
9. about
10.
11.
12.
13.
14.
15. about the axis
16. about
17. about the axis
18. about the axis
Work
1. Find the work done in pushing a car a distance of 10 while exerting a constant force of .
Since ,
2. How much work is done by a weightlifter in raising a barbell from the floor to a height of ?
Since ,
.
So .
3. A particle is moved along the axis by a force that measures pounds at a point feet from the origin. Find the work done in moving the particle from the origin to a distance of .
. Let change of variable to , then . So
4. Shown is the graph of a force function (in newtons) that increases to its maximum value and then remains constant. How much work is done by the force in moving an object a distance of ?
5. A force of is required to hold a spring stretched beyond its natural length. How much work is done in stretching it from its natural length to beyond its natural length?
Since and ,
.
By Hooke’s Law, .
So .
6. A spring has a natural length of . If a force is required to keep it stretched to a length of , how much work is required to stretch it from to ?
By Hooke’s Law, ,
so .
.
7. Suppose that of work is needed to stretch a spring from its natural length of to a length , how much work is needed to stretch it from to .
.
That is .
Hence .
816. Find the work done.
8. A heavy rope, long, weighs and hangs over the edge of a building high.
(a) How much work is done in pulling the rope to the top of the building?
(b) How much work is done in pulling half the rope to the top of the building?
(a)
(b)
9. A chain lying on the ground is long and its mass is . How much work is required to raise one end of the chain to a height of ?
, .
So, .
10. A cable that weight is used to lift of coal up a mine shaft deep. Find the wore done.
Since the coal’s weight is , the length of coal is .
So
11. A bucket that weighs and a rope of negligible weight are used to draw water from a well that is deep. The bucket is filled with of water and is pulled up at a rate of , but water leaks out of a hole in the bucket at a rate of . Find the work done in pulling the bucket to the top of the well.
Since ,
12. A chain weighs and hangs from a ceiling. Fing the work done in lifting the lower end of the chain to the ceiling so that it’s level with the upper end.
, .
So .
13. An aquarium long, side and deep is full of water. Find the work needed to pimp half of the water out of the aquarium.
14. A circular swimming pool has a diameter of , the sides are high and the depth of the water is . How much work is required to pump all of the water out over the side?
.
15. Newton’s Law of Gravitation states that two bodies with masses and attract each other with a force where is the distance between the bodies and is the gravitational constant. If one of the bodies is fixed, find the work needed to move the other from to .
16. Use Newton’s Law of Gravitation to compute the work required to launch a satellite vertically to center. Take the radius of Earth to be and .
Use the result of problem 15, then
.
Average Value of a Function
17. Find the average value of the function on the given interval.
1. ,
2. ,
3. ,
4. ,
To calculus this integral, change of variable to , then . So,
.
5. ,
6. ,
8. ,
Use change of variable to then . So
.
810. (a) Find the average value of on the given interval.
(b) Find such that .
(c) Sketch the graph of and a rectangle whose area is the same as the area under the graph of .
8. ,
(a)
(b)
9. ,
(a)
(b)
(c)
10. ,
.
Use change of variable to , then . So
.
Also change of variable to ,
then .
11. If is continuous and , show that takes on the value at least once on the interval .
Since is continuous and , there exists a constant in such that
.
12. Find the numbers such that the average value of on the interval is equal to .
That is
13. In a certain city the temperature (in ) hours after A.M. was modeled by the function
.
Find the average temperature during the period from A.M. to P.M.
14. If a cup of coffee has temperature in a room where the temperature is then, according to Newton's Law of Cooling, the temperature of the coffee after minutes is . What is the average temperature of the coffee during the first half hour?
15. The linear density in a rod long is , where is measured in meters from one end of the rod. Find the average density of the rod.
Let . Then
. Change of variable to , then . So
16. If denotes the average value of on the interval and , show that
1721. Find the area of the region bounded by the given curves.
17. ,
Since ,
18. ,
Since
19. ,
Assume , then
Thus,
20. ,
21. , ,
2225. Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.
22. , ; about the axis
23. , ; about axis
24. , ; about
25. , ; about
26. Find the volume of the solids obtained by rotating the region bounded by the curves and about the following lines:
(a) the axis
(b) the axis
(c)
(a)
(b)
(c)
27. Let be the region in the first quadrant bounded by the curves and . Calculate the following quantities.
(a) The area of
(b) The volume obtained by rotating about the axis
(c) The volume obtained by rotating about the axis
(a)
(b)
(c)
Integration by Parts
1.
(log(x)^2  2*log(x) + 2)*x
2.
2/49*(7*x1)*e^(7*x) + 1/343*(49*x^2  14*x + 2)*e^(7*x) + 1/7*e^(7*x)2/49*(7*x  1)*e^(7*x) + 1/343*(49*x^2  14*x + 2)*e^(7*x) + 1/7*e^(7*x)
3.
4.
5.
6.
by Example 1 section 7.3.
7.
8.
9.
10.
11.
12.
13.
Since ,
14.
15.
16.
17.
18.
(use 13)
19.
1/2*x^2*arccos(x)  1/4*sqrt(x^2 + 1)*x + 1/4*arcsin(x)
20.
1/10*(2*sin(2*x)  cos(2*x) + 5)*e^(x)
21.
x*log(sqrt(x) + 1)  1/2*x + sqrt(x)  log(sqrt(x) + 1)
22.
1
23.
1/4*pi  1/2*log(2)
24.
2*pi  4
25.
4*log(2)^2  2*log(2) + 15/32
26.
1/2*a^2*arcsin(x/sqrt(a^2)) + 1/2*sqrt(a^2  x^2)*x1/2*a^2*arcsin(x/sqrt(a^2)) + 1/2*sqrt(a^2  x^2)*x
27. (a) Establish the reduction formula.
(b) Use part (a) to evaluate
(c) Use parts (a) and (b) to evaluate .
(a)
(b)
(c)
28. (a) Establish the reduction formula.
(b) Find the integral explicitly for .
29. (a) Establish the reduction formula
(b) Find the integral for .
Trigonometric Integrals
1.
2.
1/9*cos(x)^9 + 2/7*cos(x)^7  1/5*cos(x)^5
3.
1/2*x  1/16*sin(8*x)
4.
1/3*(3*cos(x)^2  1)/cos(x)^3
5.
1/48*sin(2*x)^3 + 1/16*x  1/64*sin(4*x)
6.
1/3*(3*tan(x)^2  1)/tan(x)^3 + arctan(tan(x))
7.
2*sqrt(cos(x)) + 2/3/cos(x)^(3/2)
8.
1/10*cos(5*x) + 1/2*cos(x)
9.
1/5*cos(t)^5 + 2/3*cos(t)^3  cos(t)
10.
1/27*cos(3*x)^9  1/7*cos(3*x)^7 + 1/5*cos(3*x)^5  1/9*cos(3*x)^3
11.
3/128*t  1/256*sin(8*t) + 1/2048*sin(16*t)
12.
1/45*(5*cos(3*y)^2  3)/cos(3*y)^5
13.
1/3/sin(x)^3
14.
1/2/tan(t)^2
15.
1/18*cos(9*y) + 1/2*cos(y)
16.
1/6*(3*tan(2*x)^2  1)/tan(2*x)^3 + 1/2*arctan(tan(2*x))
17.
1/24
18.
1/9*sin(x)^9 + 1/7*sin(x)^71/9*sin(x)^9 + 1/7*sin(x)^7
19.
5/12
20.
1/2*(4*x*sin(2*x)*sin(x) + 4*x*cos(2*x)*cos(x) + (sin(2*x)^2 +cos(2*x)^2 + 2*cos(2*x) + 1)*log(sin(x)^2 + cos(x)^2  2*sin(x) + 1) (sin(2*x)^2+ cos(2*x)^2 + 2*cos(2*x) + 1)*log(sin(x)^2 + cos(x)^2 + 2*sin(x) + 1) + 4*x*cos(x))/(sin(2*x)^2 + cos(2*x)^2 + 2*cos(2*x) + 1)
21.
4/35
Trigonometric Substitution
1.
sqrt(x^2 + 1)/x  arcsin(x)
2.
1/16*sqrt(2)*arcsinh(2*x/abs(2*x + 4)  4/abs(2*x + 4)) +1/16*sqrt(2)*arcsinh(2*x/abs(2*x  4) + 4/abs(2*x  4))
3.
1/9*sqrt(x^2 + 9)/x
4.
x/sqrt(x^2 + 1)
5.
1/3*sqrt(3)*log(2*sqrt(3*x^2  2*x  1)*sqrt(3)+6*x 2)
6.
7.
1/9*x/sqrt(x^2+4*x + 13) + 2/9/sqrt(x^2 + 4*x + 13)
8.
x/sqrt(x^2+4)arcsin(1/2*x)
9.
1/3*arcsinh(3/abs(x))
10.
sqrt(x^2 + 9)  3*log(6*sqrt(x^2 + 9)/abs(x) + 18/abs(x))
11.
1/2*a^2*log(2*x + 2*sqrt(a^2 + x^2)) + 1/2*sqrt(a^2 + x^2)*x
12.
1/6*(2*x^2  2)^(3/2)/x^3
13.
1/2/(x^2  1)
14.
arcsin(x)
15.
log(2*x + 2*sqrt(x^2 + 3*x + 1) + 3)
16.
3*sqrt(x^2x+5)+1/2*arcsin(1/21*(2*x+ 1)*sqrt(21))
17.
sqrt(x^2  1) + arcsin(1/abs(x))
18.
1/2*sqrt(e^(2*x) + 9)*e^x + 9/2*arcsin(1/3*e^x)
19. (a) Prove that
.
(b) Use to prove that
.
(a) ,
1/2*a^2*arcsinh(x/sqrt(a^2)) + 1/2*sqrt(a^2 + x^2)*x
(b) Use to prove that
Integration of Rational Functions by the method of Partial Fractions
1.
x/(x^2  2*x + 1) + 2*log(x  1)  log(x)
2.
1/2*log(x1)+1/2*log(x+1)log(x)
3.
2/3*log(x  5) + 1/3*log(x + 1)
4.
4/(x^2  4) + 3*log(x  2) + 2*log(x + 2)
5.
2/3*arctan(1/2*x)  1/3*arctan(x)
6.
1/2*x^2 + 2*x + 3/2*log(x  1)  1/2*log(x + 1)
7.
1/2*log(x^2 + 1) + log(x)
8.
1/3*sqrt(3)*arctan(1/3*(2*x1)*sqrt(3))+1/3*log(x+1)1/6*log(x^2x+1)
9.
5/(x  3) + 3*log(x  3)
10.
1/6*sqrt(3)*arctan(1/3*(2*x1)*sqrt(3))1/6*sqrt(3)*arctan(1/3*(2*x+1)*sqrt(3))+1/6*log(x1)1/6*log(x+1)+1/12*log(x^2x+1)1/12*log(x^2+x+1)
11.
1/8*sqrt(2)*log(x^2sqrt(2)*x+1)+1/8*sqrt(2)*log(x^2+sqrt(2)*x+1)+1/4*sqrt(2)*arctan(1/2*(2*xsqrt(2))*sqrt(2))+1/4*sqrt(2)*arctan(1/2*(2*x+sqrt(2))*sqrt(2))
12.
3/5*arctan(1/3*x)  2/5*arctan(1/2*x)
13.
1/2*x/(x^2 + 1) + 1/2*arctan(x)
14.
1/4*(x1)/(x^2+1)+1/4*log(x1)1/8*log(x^2+1)1/2*arctan(x)
15.
16.
4*log(x + 1) + 13/2*log(2*x + 3)
17.
sqrt(x^2 + 9)  3*arcsinh(3/abs(x))
18.
1/2*(12*x^3  18*x^2 + 4*x + 1)/(x^4  2*x^3 + x^2)  6*log(x  1)+ 6*log(x)
19.
A^3*log(x) + 1/6*(2*x^(3*n) + 18*x^n*A^2 + 9*A*x^(2*n))/n
20.
1/9*pi*sqrt(3) + 1/3*log(2) + 1/2
21.
1/2*log(x^2 + 4) + log(x)  1/2*arctan(1/2*x)
22.
1/2*x/(x^2 + 1) + 1/2*arctan(x)
23.
1/3*sqrt(3)*log((sqrt(x + 2)  sqrt(3))/(sqrt(x + 2) + sqrt(3)))1/3*sqrt(3)*log((sqrt(x + 2)  sqrt(3))/(sqrt(x + 2) + sqrt(3)))
24.
4*sqrt((x + 2)/(x  2))/((x + 2)/(x  2)  1) + 4*arctan(sqrt((x +2)/(x  2)))
25.
6*x^(1/6)  3*log(x^(1/6)  1) + 3*log(x^(1/6) + 1)
26.
27.
[Hint: Substitute ]
1/2*sqrt(2)*arcsinh(x/abs(x) + 2/abs(x))
Guidelines for Integration
1.
2/3*(a + x  x/a)^(3/2)/(1/a  1)
2.
2*(sqrt(x + 5)  1)*e^(sqrt(x + 5))
3.
log(sqrt((x + 1)/(x  1))  1) + log(sqrt((x + 1)/(x  1)) + 1) 2*arctan(sqrt((x + 1)/(x  1)))
4.
6*x^(1/6)  3*log(x^(1/6)  1) + 3*log(x^(1/6) + 1)
5.
1/315*(63*cos(x)^4  90*cos(x)^2 + 35)/cos(x)^9
6.
1/15*(5*tan(x)^2 + 3)/tan(x)^5
7.
3/2*(x+1)^(2/3)3*(x+1)^(1/3)+3*log((x+1)^(1/3)+1)
8.
6*x^(1/6)  6*arctan(x^(1/6))
9.
1/2*sqrt(2)*log(sqrt(2)*x^(1/4) + sqrt(x) + 1) 1/2*sqrt(2)*log(sqrt(2)*x^(1/4) + sqrt(x) + 1) 
sqrt(2)*arctan(1/2*(sqrt(2)  2*x^(1/4))*sqrt(2)) sqrt(2)*arctan(1/2*(sqrt(2) + 2*x^(1/4))*sqrt(2)) + 4*x^(1/4)
10.
11.
20/(x + 1) + x^3  7/2*x^2 + 20*x  34*log(x + 1)
12.
4*(x^(3/4)  3*sqrt(x) + 6*x^(1/4)  6)*e^(x^(1/4))
13.
1/6*sqrt(3)*log((sqrt(log(x) + 3)  sqrt(3))/(sqrt(log(x) + 3) +sqrt(3)))  sqrt(log(x) + 3)/log(x)
14.
1/2*sqrt(x^2 + 3*x  2)*x  3/4*sqrt(x^2 + 3*x  2) + 1/8*arcsin(2*x 3)
15.
1/3*sqrt(x^2 + 1)*x^2  2/3*sqrt(x^2 + 1)
16.
1/12*x*cos(3*x)  1/4*x*cos(x) + 1/36*sin(3*x) + 1/4*sin(x)
17.
5/11*(x + 1)^(11/5)  5/6*(x + 1)^(6/5)
18.
1/2*sqrt(x^2 + 1)*x  1/2*arcsinh(x)
19.
1/2*arctan(sqrt(x^4 + 3)/x^2)
Integration Using Tables
1.
1/6*sqrt(3*x^2 + 7)*x + 7/18*sqrt(3)*arcsin(1/7*sqrt(21)*x)
2.
5*(x^4  12*x^2 + 24)*sin(x)  (x^5  20*x^3 + 120*x)*cos(x)
3.
4.
1/3*(x^2  4*x + 7)^(3/2) + sqrt(x^2  4*x + 7)*x  2*sqrt(x^2  4*x+7) + 3*arcsinh(1/3*(x  2)*sqrt(3))1/3*(x^2  4*x + 7)^(3/2) + sqrt(x^2  4*x + 7)*x  2*sqrt(x^2  4*x + 7) + 3*arcsinh(1/3*(x  2)*sqrt(3))
5.
1/48*(15*sin(x)^5  40*sin(x)^3 + 33*sin(x))/(sin(x)^6  3*sin(x)^4 +3*sin(x)^2  1)  5/32*log(sin(x)  1) + 5/32*log(sin(x) + 1)
6.
1/4*x^4*arctan(3*x)  1/36*x^3 + 1/108*x  1/324*arctan(3*x)
7.
sqrt(e^(2*x) + 1)*e^(x)
8.
1/4*(3*e^(2*x) + 2)^(3/2)*e^x + 3/4*sqrt(3*e^(2*x) + 2)*e^x +1/2*sqrt(3)*arcsin(1/2*sqrt(6)*e^x)
9.
log(log(x) + 1) + log(log(x))
10.
1/2*(2*sin(x)^2  1)/(sin(x)^4  sin(x)^2)  log(sin(x)^2  1) +2*log(sin(x))
11.
1/13*(3*sin(3*x) + 2*cos(3*x))*e^(2*x)
12.
log(sqrt(x^2 + 3*x  2)/abs(2*x  3) + 1/2/abs(2*x  3))
Approximate Integration
16. Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson's Rule to approximate the given integral with the specified value of .
1. ,
(a) 0.235977 (b)0.232771 (c)0.233760
2. ,
(a) 2.612462 (b) 2.576704 (c) 2.588559
3. ,
(a) 0.919952 (b) 0.927027 (c) 0.925237
4. ,
5. ,
(a) 0.457277 (b) 0.458528 (c) 0.458114
6. ,
(a) 1.182973 (b) 1.160116 (c) 1.169130
7. (a) Determine the approximations and for .
(b) Find the errors involved in the approximations of part (a).
(c) Determine how large do we have to choose so that the approximations and to the integral in part (a) are accurate to within 0.00001?
(a) ,
(b)
(c)
For , we choose so that solving this,
so that .
For , we choose so that solving this gives,
so that .
8. (a) Determine the approximations and for and for and the corresponding errors and .
(b) Compare the actual errors in part (a) with the error estimates given by 㰊³ and 㰊´.
(c) Determine how large do we have to choose so that the approximations , , and to the integral in part (a) are accurate to within 0.00001?
(a) , , , .
(b)Since (3) and (4) gives
(c)For , find so that , .
For , find so that , .
For , find so that , .
9. Given the function at the following values,

1.8 
1.9 
2.0 
2.1 
2.2 
2.3 
2.4 

0.028561 
0.020813 
0.015384 
0.011525 
0.008742 
0.006709 
0.004079 
approximate using Simpson's Rule.
Plot
And Use Simpson’s Rule
Improper Integrals
1.
ValueError: Integral is divergent
2.
1
3.
pi
4.
ValueError: Integral is divergent.
5.
sin(1)  cos(1)
6.
1/log(2)1/log(2)
7.
1/2
8.
2
9.