# [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.

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://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:

http://www.sagemath.org

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

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 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.

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: ℝRR

e.g.  .5  0.001  3.14  -42.

- Complex: ℂCC

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

- 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)

- 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

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

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.

# 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)

 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.

 var('x'); plot(sqrt(x+4), x, -4, 2, color='blue')

[CAS] 8.

 var('x'); plot(abs(x-2), x, 0, 4, color='blue')

[CAS] 9.

Sol)

 var('x'); plot(1+sqrt(x-1), x, 1, 4, color='blue')

[CAS] 10.

Sol)

 var('x'); plot((x+1)^(2/3), x, -1, 4, color='blue')

[CAS] 11.

Sol)

 var('x'); plot(1-x^(2/3), x, 0, 4, color='blue')

[CAS] 12.

Sol)

 var('x'); plot((x-1)^(1/3)-1, x, 1, 4, color='blue')

[CAS] 13.

Sol)

 var('x'); plot(1/(x-2)-1, x, 1, 4, color='blue')

[CAS] 14.

Sol)

 var('x'); plot(1/x +2, x, -2, 2, color='blue')

[CAS] 15.

Sol)

 var('x'); plot(1/(x-1)^2, x, -2, 2, color='blue')

[CAS] 16.

Sol)

 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)

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)

 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)

 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)

 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)

 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)

 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)

 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)

 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 .

 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)

 var('x'); f(x)=x^4+x-3 plot(f,1,2)

 f(x=1)

 f(x=2)

[CAS] 18. ,

Sol)

 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)

 var('x'); f(x)=x^(1/3)-1+x plot(f,0,1)

 f(x=0)

-1

 f(x=1)

1

[CAS] 20. ,

Sol)

 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)

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)

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.