[Mobile Math] Calculus with Sage
by SGLee e.t.a.l.
Preface
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 Sage-Math. 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 open-source 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 Sage-Math 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 : Sang-Gu Lee, Jihoon Lee, Gi-Sang Cheon, Soonhak Kwon, Changbum Cheon, Mee-Kyoung Kim, Seki Kim, Jeong Hyeong Park, In Sung Hwang, Young Do Chai.
1) Introduction and Use of Sage-Math
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. Sage-Math 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.
Sage-Math 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 Web-Mathematica, 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 pre-existing command to modify for your own problems.
Figure 1. We can use others existing codes as well in Sage
Korean Sage-Math model was developed and relevant experiments were done by BK21 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, PW: math)
http://math2.skku.ac.kr (Make your own accounts)
http://math3.skku.ac.kr (Mobile Server, No need for login)
Figure 2. Sage-Math Community:
Figure 3. Sage-Math Korean Version:
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
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 4. Sage-Math Worksheet
2) The development of Korean Version of Sage-Math
You now have Quick Reference (simple manual) of Sage-Math in English and Korean. Those Quick References can be downloaded from the Web site address in the below.
1) http://wiki.sagemath.org/quickref
2) http://matrix.skku.ac.kr/2010-Album/Sage-QReference-SKKU.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 Sang-Gu Lee and Jae Hwa Lee (Sungkyunkwan University).
Korean Version at http://matrix.skku.ac.kr/2010-Album/Sage-QReference-SKKU.pdf
Notebook (and Command line)
- evaluate cell: <shift-enter>
- 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: ℝ
RR
e.g. .5 0.001 3.14 -42.
- Complex: ℂ
CC
e.g. 1+i 2.5-3*i
Builtin constants and functions
- Constants: 
=pi 
=e 
=I=I 
=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: t, u, v, y, z = var('t u v y z')
- Define functions(
):
f(x)=x^2 or f=lambda x: x^2
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^3-y^3).factor()
- List of (factor, exponent) pairs: (x^3-y^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)
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(
), (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.
[CAS Examples] Sage Problem: Exercise 22 in Section 9.3 (Polar Coordinates)
★ Sketch the curve with the given polar equation.
‣‣ Sol)
Published at
http://math1.skku.ac.kr/home/pub/77 
http://math1.skku.ac.kr/home/pub/78
Simulate at
http://matrix.skku.ac.kr/cal-lab/cal-0-3-1.html
|
theta=var('theta'); polar_plot(sin(4*theta), (0, 2*pi)).show(aspect_ratio=1, xmin=-1, xmax=1, ymin=-1, ymax=1) |
[CAS Examples] Sage Problem: Exercise 16 in Section 11.6 (Cylinders and Quadric Surfaces)
★ Sketch the region bounded by the surfaces 
and 
for 
.
‣‣ Sol)
Published at
.http://math1.skku.ac.kr/home/pub/80
Simulate at
http://matrix.skku.ac.kr/cal-lab/cal-0-3-2.html
|
var('x, y, z') z=(x^2+y^2)^(1/2) plot3d(z, (x, -2, 2), (y, -2, 2), (z, 2, 4))+implicit_plot3d(x^2+y^2==4,(x, -2, 2), (y, -2, 2), (z, 2, 4), opacity=0.5) |
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 Sage-Math in learning and teaching Calculus.
3) Internet resources
- Visualization of LA with Sage
- William Stein demos for Sage-Math
- Linear Algebra with Mobile Sage
- Sage Interact / ODE and Mandelbrot
- Sage Multivariable Calculus (1 of 2) by Travis
- Sage Multivariable Calculus (2 of 2) by Travis
- How to do LU-Decomposition with Sage
- http://bkmath.skku.ac.kr/bk21/index.html
- http://matrix.skku.ac.kr/sglee
Chapter 1. Functions
[1.1] 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?
‣‣ Sol)
(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 
?
‣‣ Sol)
(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.
‣‣ Sol)
(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?
‣‣ Sol)
(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 
.
‣‣ Sol)
(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 
.
‣‣ Sol)
(a) 
(b) 
(c)
[1.2] Symmetry
1-5. Piecewise-Defined Functions : Graph the following functions.
1. 
‣‣ Sol)
2. 
‣‣ Sol)
3. 
‣‣ Sol)
4. 
‣‣ Sol)
5. Find a formula for given graphed.
‣‣ Sol)
6-10. Graph the functions in Exercises 6-10. 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. 
‣‣ Sol)
symmetric with respect to (w.r.t.) the origin.
decreasing on 
7. 
‣‣ Sol)
symmetric w.r.t. the origin.
decreasing on 
increasing on 
8. 
‣‣ Sol)
symmetric with respect to the 
-axis.
decreasing on 
increasing on 
9. 
‣‣ Sol)
symmetric w.r.t the origin.
increasing on 
10. 
‣‣ Sol)
it has no symmetry.
decreasing on 
11-16.Say whether the functions is even, odd, or neither. Give reasons for your answer.
11. 
‣‣ Sol)
even function
12. 
‣‣ Sol)
even function
13. 
‣‣ Sol)
odd function
14. 
‣‣ Sol)
even function
15. 
‣‣ Sol)
neither odd or even function
16. 
‣‣ Sol)
neither odd or even function
[1.3] Common Functions
1. If 
, find 
and 
.
‣‣ Sol)
2. Find the domain of the function.
(a) 
(b) 
(c) 
(d) 
(e) 
‣‣ Sol)
(a) 
(b) 
(c) 
(d) 
(e) 
3. Find the domain and sketch the graph of the function.
(a) 
(b) 
(c) 
‣‣ Sol)
(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) 
‣‣ Sol)
(a) even
(b) odd
(c) neither
(d) even
(e) odd
(f) neither
[CAS] 5. Draw the original and given functions graphs together.
‣‣ Sol)
http://matrix.skku.ac.kr/cal-lab/cal-1-3-5.html 
|
var('x,y'); p1=plot(exp(-(x^2)/2),x,-3,3,color='blue'); show(p1, aspect_ratio=1, ymax=2) |
[1.4] Translation, stretching and rotation of functions
1-6. Exercises 1-6 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 
.
‣‣ Sol)
2. 
. Left 
, down 
.
‣‣ Sol)
3. 
. Left 
.
‣‣ Sol)
4. 
. Up 
.
‣‣ Sol)
5. 
. Up 
, right 
.
‣‣ Sol)
6. 
. Down 
, right 
.
‣‣ Sol)
7-16. Graph the functions in Exercises 7-16.
[CAS] 7. 
‣‣ Sol)
http://matrix.skku.ac.kr/cal-lab/cal-1-4-7.html 
|
var('x'); plot(sqrt(x+4), x, -4, 2, color='blue') |
[CAS] 8. 
‣‣ Sol)
http://matrix.skku.ac.kr/cal-lab/cal-1-4-8.html 
|
var('x'); plot(abs(x-2), x, 0, 4, color='blue') |
[CAS] 9. 
‣‣ Sol)
http://matrix.skku.ac.kr/cal-lab/cal-1-4-9.html 
|
var('x'); plot(1+sqrt(x-1), x, 1, 4, color='blue') |
[CAS] 10. 
‣‣ Sol)
http://matrix.skku.ac.kr/cal-lab/cal-1-4-10.html 
|
var('x'); plot((x+1)^(2/3), x, -1, 4, color='blue') |
[CAS] 11. 
‣‣ Sol)
http://matrix.skku.ac.kr/cal-lab/cal-1-4-11.html 
|
var('x'); plot(1-x^(2/3), x, 0, 4, color='blue') |
[CAS] 12. 
‣‣ Sol)
http://matrix.skku.ac.kr/cal-lab/cal-1-4-12.html 
|
var('x'); plot((x-1)^(1/3)-1, x, 1, 4, color='blue') |
[CAS] 13. 
‣‣ Sol)
http://matrix.skku.ac.kr/cal-lab/cal-1-4-13.html 
|
var('x'); plot(1/(x-2)-1, x, 1, 4, color='blue') |
[CAS] 14. 
‣‣ Sol)
http://matrix.skku.ac.kr/cal-lab/cal-1-4-14.html 
|
var('x'); plot(1/x +2, x, -2, 2, color='blue') |
[CAS] 15. 
‣‣ Sol)
http://matrix.skku.ac.kr/cal-lab/cal-1-4-15.html 
|
var('x'); plot(1/(x-1)^2, x, -2, 2, color='blue') |
[CAS] 16. 
‣‣ Sol)
http://matrix.skku.ac.kr/cal-lab/cal-1-4-16.html 
|
var('x'); plot(1/x^2 +1, x, -2, 2, color='blue') |
Chapter 2. Limits and Continuity
[2.1] Limits of functions
1-7. Find the following limits or explain why the limit does not exist.
1. 
‣‣ Sol)
2. 
‣‣ Sol)
does not exist.
[CAS] 3.
‣‣ Sol)
http://matrix.skku.ac.kr/cal-lab/cal-2-1-3.html
You can draw 
to use Sage. Then 
diverge to 
at 
.
|
var('x') p1=plot(sin(x)/abs(cos(x)), x, -pi,pi/2, color='blue'); p2=plot(sin(x)/abs(cos(x)), x, pi/2,pi, color='red'); show(p1+p2, ymax=50, ymin=-10) |
|
limit(sin(x)/abs(cos(x)), x=pi/2, dir='plus') |
+Infinity
4. 
‣‣ Sol)
[CAS] 5. 
‣‣ Sol)
http://matrix.skku.ac.kr/cal-lab/cal-2-1-5.html
|
P1=plot(1/abs(x)-1/x, x, -5,-0.1, color='blue') P2=plot(1/abs(x)-1/x, x, 0.1,5, color='red') show(P1+P2) |
6. 
‣‣ Sol)
[CAS] 7. 
‣‣ Sol)
http://matrix.skku.ac.kr/cal-lab/cal-2-1-7.html
|
p1=plot(x*sin(3/x), x, -0.5,0, color='blue'); p2=plot(x*sin(3/x), x, 0,0.5, color='red'); show(p1+p2, ymax=0.3, ymin=-0.3, aspect_ratio=1) |
|
limit(x*sin(3/x), x=0) |
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.
‣‣ Sol)
(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.
‣‣ Sol)
(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 
.
‣‣ Sol)
(a) 
(b) Since 
, we have
.
so, 
(c) We know that 
and
, so by the Squeeze Theorem,
.
[CAS] 12. Use squeeze Theorem to find
.
‣‣ Sol)
http://matrix.skku.ac.kr/cal-lab/cal-2-1-12.html
|
P1=plot(sqrt(5*x)*(4-cos(3/sqrt(x))), 0.01,2) P2=plot(sqrt(5*x)*(5), 0.01,2, color='red') P3=plot(sqrt(5*x)*(3), 0.01,2, color='red') show(P1+P2+P3) |
[CAS] 13. Use squeeze Theorem to find 
.
‣‣ Sol)
http://matrix.skku.ac.kr/cal-lab/cal-2-1-13.html
|
P1=plot(sin(x)/x,-4,4, color='blue') P2=plot(cos(x), -4,4, color='red') P3=plot(1, -4,4, color='red') show(P1+P2+P3) |
(※We can use Squeeze Theorem around 0.)
14. Let 
. Find all positive integer 
such that 
‣‣ Sol)
ⅰ) 
; 
ⅱ) 
; 
ⅲ) 
; 
15. Find all the asymptotes (vertical, horizontal, and oblique) of the function 
.
‣‣ Sol)
,
so 
,
,
and 
.
Thus, 
and 
are vertical asymptotes.
,
Thus, 
is oblique asymptotes.
16. Find the limit 
‣‣ Sol)
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.
‣‣ Sol)
(a) 
, so 
. 
.
Thus, 
.
18.Find 
such that 
whenever 
.
‣‣ Sol)
19. Use the 
argument to prove that 
.
‣‣ Sol)
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 
.
‣‣ Sol)
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.
21-26. Prove the statements using the 
argument.
21. 
‣‣ Sol)
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. 
‣‣ Sol)
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. 
‣‣ Sol)
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. 
‣‣ Sol)
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. 
‣‣ Sol)
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. 
‣‣ Sol)
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 
‣‣ Sol)
(a) 
(b) 
[2.2] Continuity
[CAS] 1. If 
and 
are continuous functions
with 
and 
, find 
.
‣‣ Sol)
http://matrix.skku.ac.kr/cal-lab/cal-2-2-1.html
|
var('g') solve(3^2-4*3*g==5, g ) |
g(4)=1/3
2. If 
and 
are continuous functions with 
and 
, find 
.
‣‣ Sol)
Since 
is continuous, 
.
[CAS] 3. Show that the function 
is discontinuous at 
.
‣‣ Sol)
http://matrix.skku.ac.kr/cal-lab/cal-2-2-3.html
|
p1=plot(abs(ln((x-2)^2)), x, -3,2, color='blue'); p2=plot(abs(ln((x-2)^2)), x, 2,3, color='red'); show(p1+p2, ymax=5, ymin=-1) |
|
limit(abs(ln((x-2)^2)), x=2) |
+Infinity
4-7. 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 
.
[CAS] 4. 
‣‣ Sol)
http://matrix.skku.ac.kr/cal-lab/cal-2-2-4.html
|
plot(Piecewise([[(-2*pi,0),sin(x)],[(0,pi),cos(x)],[(pi,2*pi),-1]])) |
5. 
‣‣ Sol)
We see that 
exists for all a except 
. Notice that the right and left limits are different 
.
6. 
‣‣ Sol)
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. 
‣‣ Sol)
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 
.
8-10. For what values of the constant 
is the function 
continuous on 
?
8. 
‣‣ Sol)
Thus, for 
to be continuous on 
.
[CAS] 9. 
‣‣ Sol)
http://matrix.skku.ac.kr/cal-lab/cal-2-2-9.html
|
var('c') solve(16-c^2==4*c+20, c) |
c=-2
10. 
‣‣ Sol)
Since 
is not continuous at 
, solution is 
.
http://matrix.skku.ac.kr/cal-lab/cal-2-2-10.html
|
var('c') solve(10/(c-2)==2*c+4, c) |
c=-3, c=3
11-13. Show that the following functions 
has the removable discontinuity at 
. Also find a function 
that agrees with 
for 
and is continuous on ℝ.
11. 
‣‣ Sol)
for 
. The discontinuity is removable and 
agrees with 
for 
and is continuous on ℝ.
12. 
‣‣ Sol)
for 
. The discontinuity is removable and 
agrees with 
for 
and is continuous on ℝ.
13. 
‣‣ Sol)
for 
. The discontinuity is
removable and 
agrees with
for 
and is continuous on ℝ.
14. Let 
. Is 
removable discontinuity?
‣‣ Sol)
Since 
is not removable discontinuous.
15. If 
, show that there is a number 
such that 
.
‣‣ Sol)
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 
.
‣‣ Sol)
Let 
. 
is continuous on the interval 
, 
and 
. Since 
, there is a number 
in 
such that 
by the Intermediate Value Theorem.
17-22. Prove that there is a root of the given equation in the specified interval by using the Intermediate Value Theorem.
[CAS] 17. 
‣‣ Sol)
http://matrix.skku.ac.kr/cal-lab/cal-2-2-17.html
|
var('x'); f(x)=x^4+x-3 plot(f,1,2) |
|
f(x=1) |
|
f(x=2) |
[CAS] 18. 
, 
‣‣ Sol)
http://matrix.skku.ac.kr/cal-lab/cal-2-2-18.html
|
var('x'); f(x)=sqrt(6*x-x^2)-1 P=plot(f,0,6); show(P, aspect_ratio=1) |
|
bool(f(0)<0) |
True
|
bool(f(2)<0) |
True
[CAS] 19. 
‣‣ Sol)
http://matrix.skku.ac.kr/cal-lab/cal-2-2-19.html
|
var('x'); f(x)=x^(1/3)-1+x plot(f,0,1) |
|
f(x=0) |
-1
|
f(x=1) |
1
[CAS] 20. 
, 
‣‣ Sol)
http://matrix.skku.ac.kr/cal-lab/cal-2-2-20.html
|
var('x'); f(x)=exp(x^3)-x^6 plot(f,-1,1) |
|
bool(f(-1)<0) |
True
|
bool(f(1)>0) |
True
21. 
‣‣ Sol)
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 
.
22. 
‣‣ Sol) 
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 
.
23-26. Show that each of the following equation has at least one real root.
[CAS] 23. 
‣‣ Sol)
http://matrix.skku.ac.kr/cal-lab/cal-2-2-23.html
We can draw 
and 
at the same time. There are real roots at 
or 
.
|
p1=plot(exp(x), x, -2,4, color='blue'); p2=plot(4*sin(x), x, -2,4, color='red'); show(p1+p2, ymax=5, ymin=0) |
|
find_root(exp(x)==4*sin(x),0,1) |
0.37055809596982464
|
find_root(exp(x)==4*sin(x),1,2) |
1.3649584337330951
24. 
‣‣ Sol)
Let 
. Then 
and 
. So by the Intermediate Value Theorem. There is a number
in 
such that 
. This implies that 
.
25. 
‣‣ Sol)
Let 
. Then 
and 
. So by the Intermediate Value Theorem. There is a number 
in 
such that 
. This implies that 
.
26. 
‣‣ Sol)
Let 
. Then 
and 
, and 
is continuous 
So by the Intermediate Value Theorem. there is a number 
in 
such that 
. This implies that 
.
27-28. Find the values of 
for which 
is continuous.
27. 
‣‣ Sol)
The value of 
don’t exist.
28. 
‣‣ Sol)
is continuous at 
.
Chapter 3. Theory of Differentiation
[3.1] Derivatives of Polynomials, Exponential Functions, Trigonometric Functions, The Product Rule
1-5. Find the derivative 
where 
is
1. 
‣‣ Sol)
2. 
‣‣ Sol)
3. 
‣‣ Sol)
4. 
‣‣ Sol)
5. 
‣‣ Sol)
[CAS] 6. Find 
where 
.
‣‣ Sol)
http://matrix.skku.ac.kr/cal-lab/cal-3-1-6.html
|
var('x'); diff(x^(15/14)+5*e^x,x) |
15/14*x^(1/14) + 5*e^x
[CAS] 7. Find the equation of the tangent line to the curve 
at 
.
‣‣ Sol)
http://matrix.skku.ac.kr/cal-lab/cal-3-1-7.html
. So the slope of the tangent line is 20.
(
passes through 
).
|
f(x)=x^2*sqrt(x); df(x)=diff(f(x),x); y(x)=df(4)*(x-4)+32; y(x) |
20*x-48
|
p1=plot(f(x),x,0,10, color='blue'); p2=plot(y(x),x,0,10, color='red'); show(p1+p2,ymax=50,ymin=-10) |
[CAS] 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 
.
‣‣ Sol)
http://matrix.skku.ac.kr/cal-lab/cal-3-1-8.html
So the slope of the tangent line is 
. (
)
Then the slope of the normal line is 
.
Thus, 
.
The normal line is 
.
|
f(x)=1+e^x; df(x)=diff(f(x),x); y(x)=-1/df(0)*x+2; y(x) |
-x+2
|
p1=plot(f(x),x,-5,5, color='blue'); p2=plot(y(x),x,-5,5, color='red'); show(p1+p2,ymax=5,ymin=-5) |
9. where is the function 
differ-entiable? Give a formula for 
.
‣‣ Sol)
,
then 
.
So 
is continuous on 
.
,
then 
is not differentiable at 
.
is differentiable on 
.
10. Let 
. Find the values of 
and 
that make 
differentiable everywhere.
‣‣ Sol)
To 
be differentiable at 
, 
, so 
.
And also 
have to be continuous at 
.
, since 
, 
.
[CAS] 11. Evaluate 
.
‣‣ Sol)
http://matrix.skku.ac.kr/cal-lab/cal-3-1-11.html
Method (1) Let 
, and 
. Then by the definition of a derivative,
.
Method (2) Note that
.
So 
|
lim((x^2020-1)/(x-1),x=1) |
2020
12-13. Differentiate the following functions.
12. R
‣‣ Sol)
13. 
‣‣ Sol)
14-17. Find 
14. 
‣‣ Sol)
15. 
‣‣ Sol)
16. 
‣‣ Sol)
17. 
‣‣ Sol)
It is appear that the denominator terms is 
, and numerator is 
.
18. Find the 
th derivative of 
.
‣‣ Sol)
19. Prove if 
, then 
satisfies the identity 
.
‣‣ Sol)
20. If 
satisfies the identity 
for all 
and 
and 
, then show that 
satisfies 
for all 
.
‣‣ Sol)
Thus, 
is 
or 
.
Since 
and 
, 
is 
for all 
.
Now,
Since 
, then
.
Therefore, 
.
21-23. Find the following derivatives.
21. 
‣‣ Sol)
22. 
‣‣ Sol)
23. 
‣‣ Sol)
24-25. Find the following limit.
24. 
‣‣ Sol)
(by L’Hospital)
or 
25. 
‣‣ Sol)
(by L’Hospital)
or 
26. Show that the curve 
has no tangent line with slope 
.
‣‣ Sol)
Since 
is always positive, there is no 
such that 
.
So 
has no tangent line with slope 0.
27-28. Find 
and 
of the followings.
27. 
‣‣ Sol)
28. 
‣‣ Sol)
29. Given 
, show that it satisfies the following identity.
Using this identity, find 
.
‣‣ Sol)
Therefore,
.
Therefore, 
for 
is even,
for 
is odd.
30. Given 
, show that 
.
‣‣ Sol)
Since 
so 
is continuous on 
.
And 
31-33. Find 
of the following expressions.
31. 
‣‣ Sol)
32. 
‣‣ Sol)
33. 
‣‣ Sol)
34. If 
, where 
and 
are three times differentiable, find expressions for 
and 
.
‣‣ Sol)
35. Given 
, find 
at the point 
.
‣‣ Sol)
,
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?
‣‣ Sol)
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?
‣‣ Sol)
(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 
.
‣‣ Sol)
(a) 
(b) 
(c) When 
, 
(d) 
(e) 
39. The population of the bacteria colony after 
hours is 
. Find the growth rate when 
.
‣‣ Sol)
40. A cost function is given by 
.
(a) Find the marginal cost function.
(b) Find 
.
‣‣ Sol)
(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?
‣‣ Sol)
(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 
?
‣‣ Sol)
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.
‣‣ Sol)
44. The angular displacement 
of simple pendulum is given by 
with the angular amplitude 
, the angular frequency 
and a phase constant 
. Find 
.
‣‣ Sol)
45. Show that 
.
‣‣ Sol)
Let 
. Then 
, and as 
, 
.
[3.2] The Chain Rule and Inverse Functions
1-3. Find the differential 
.
1. 
‣‣ Sol)
2. 
‣‣ Sol)
3. 
‣‣ Sol)
4-5. Find 
of these functions.
4. 
‣‣ Sol)
5. 
‣‣ Sol)
6. Find 
where 
.
‣‣ Sol)
, 
,
, 
Therefore, 
.
7. Find 
for an integer 
if 
.
‣‣ Sol)
8. Show that the curves 
and 
are orthogonal.
‣‣ Sol)
Thus, the intersections of two curves are 
and 
.
At 
, it is not defined.
At 
, 
orthogonal
9. Use differentiation to show that 
‣‣ Sol)
10-13. Find 
of the following expressions.
10. 
‣‣ Sol)
11. 
‣‣ Sol)
[CAS] 12. 
‣‣ Sol)
http://matrix.skku.ac.kr/cal-lab/cal-3-2-12.html
|
var('x'); diff((sin(x))^(sqrt(x))) |
1/2*(2*sqrt(x)*cos(x)/sin(x)+log(sin(x))/sqrt(x))*sin(x)^sqrt
[CAS] 13. 
‣‣ Sol)
|
var('x'); diff((arccos(x))^arctan(x)) |
http://matrix.skku.ac.kr/cal-lab/cal-3-2-13.html
(log(arccos(x))/(x^2+1)-arctan(x)/(sqrt(-x^2+1)*arccos(x)))*arccos(x)^arctan(x)
[CAS] 14. Find 
if 
.
‣‣ Sol)
http://matrix.skku.ac.kr/cal-lab/cal-3-2-14.html
|
var('x'); df(x)=diff(x^(arctan(x)),x); df(x) |
(log(x)/(x^2+1)+arctan(x)/x)*x^arctan(x)
|
ddf(x)=diff(df(x),x); ddf(x) |
(log(x)/(x^2+1)+arctan(x)/x)^2*x^arctan(x)-(2*x*log(x)/(x^2+1)^2+arctan(x)/x^2-2/((x^2+1)*x))*x^arctan(x)
15. Use differentiation to show that 
for all 
.
‣‣ Sol)
for all 
16. Find an equation of the tangent line to the curve 
at 
for an arbitrary value 
.
‣‣ Sol)
, so we have to find the tangent line at 
.
17. Show the following identities.
(a) 
(b) 
(c) 
‣‣ Sol)
(a) 
(b) 
(c) 
18-30. Prove the following identities.
18. 
‣‣ Sol)
19. 
‣‣ Sol)
20. 
‣‣ Sol)
21. 
‣‣ Sol)
22. 
‣‣ Sol)
23. 
‣‣ Sol)
24. 
‣‣ Sol)
By mathematical induction, if 
, trivial.
Let 
25. 
‣‣ Sol)
26. 
‣‣ Sol)
Let 
. Then 
, so 
.
.
Note that 
, but 
.
Thus, 
.
27. 
‣‣ Sol)
Let 
. Then 
, so 
.
.
Note that 
, but 
.
Thus, 
for 
.
28. 
‣‣ Sol)
Let 
.
Then 
.
29. 
‣‣ Sol)
Let 
. Then 
, 
.
Since 
and 
, we have 
,
so 
.
30. 
‣‣ Sol)
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)
‣‣ Sol)
,
,
,
[3.3] Approximation and Related Rates
1-3. Use differential to approximate the followings.
[CAS] 1. 
‣‣ Sol)
http://matrix.skku.ac.kr/cal-lab/cal-3-3-1.html
Let’s define 
to find an approximation of 
. Viz, 
.
|
var('a,b,x,dx'); f(x)=(x+31)^(1/5); f(x); f(1.05) |
2.00062460974081
And Find 
.
|
dy(x)=diff(f(x),x)*dx; dy(x) |
1/5*dx/(x + 31)^(4/5)
|
f(1)+dy(x=1,dx=0.05) |
2.00062500000000
Also, We can use Sage functions.
|
(32.05)^(1/5).n() |
2.00062460974081
[CAS] 2. 
‣‣ Sol)
|
var('a,b,x,dx'); f(x)=(x+27)^(1/2); dy(x)=diff(f(x),x)*dx; dy(x) |
http://matrix.skku.ac.kr/cal-lab/cal-3-3-2.html
1/2*dx/sqrt(x + 27)
|
(f(0)+dy(x=0, dx=0)).n() |
5.19615242270663
3. 
‣‣ Sol)
|
var('a,b,x,dx'); f(x)=(x+31)^(1/5); f(x); f(1.05) |
http://matrix.skku.ac.kr/cal-lab/cal-3-3-3.html
1/3*dx/(x + 60)^(2/3)
|
dy(x)=diff(f(x),x)*dx; (f(1)+dy(x=1, dx=0)).n() |
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?
‣‣ Sol)
5. Find the approximation of the difference between surface areas of two spheres whose radii are 4cm and 4.05cm, respectively.
‣‣ Sol)
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 mea-surement 1cm. What is the approxi-mate percentage error in calculating the period 
?
‣‣ Sol)
, 
,
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?
‣‣ Sol)
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?
‣‣ Sol)
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?
‣‣ Sol)
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?
‣‣ Sol)
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?
‣‣ Sol)
.
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?
‣‣ Sol)
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?
‣‣ Sol)
. Let 
.
Since 
, 
: constant
Let the volume of first time 
, after an hour, 
. 
Authors : Sang-Gu Lee e.t.a.l.
