Errata (2018. Spring) : Calculus

CONTENTS

Part I Single Variable Calculus

Part II Multivariate Calculus

Main Author : Sang-Gu Lee

Co-Authors : Eung-Ki Kim, Yoonmee Ham, Ajit Kumar, Robert Beezer, Quoc-Phong Vu, Lois Simon, Suk-Geun Hwang.

Reviewers : Hyunsoo Kim, Jaedong Sim, Insung Hwang, Mohit Kumbhat,

L. Shapiro, R. Sakthivel, K. Das, Victoria Lang

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An example of a piecewise defined function is shown in the Figure 2.

The algebraic representation is given below.

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For example, the function is odd because

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7. Consider manipulating the code to see changes in the graph

http://matrix.skku.ac.kr/cal-lab/cal-1-2-7.html

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If , find the functions , , (and their domaions).

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http://matrix.skku.ac.kr/cal-lab/cal-2-1-27.html

graph of #12 needs to replace

Find . Let

⇒ If , then

. ⇒

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29.

http://matrix.skku.ac.kr/cal-lab.cal-2-1-28.html → Include

(sol)

Please Follow the proof in 28

Sec 2.2

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In other words, is continuous at if approaches as approaches (see Figure 1.) Thus, a continuous function has the property that a small change in produces a small change in the corresponding value . Intuitively the graph of a continuous function has no break. So the graph can be drawn without raising the pen off the paper.

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Discontinuities fall into two categories: removable and non-removable. A discontinuity at is called removable if can be made continuous by defining (as in Figure 2(a)) or redefining at (as in Figure 2(c.)) The discontinuity in Figure 2(b) and Figure 2(d) is non-removable.

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(b) At has a discontinuity because does not exist, although is defined.

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(ⅰ) The value can be taken on once [as in Figure 4 part (a)] or more than once [as in part (b.)]

(ⅱ) The Intermediate Value Theorem is not true in general for discontinuous functions.

(ⅲ) The Intermediate Value Theorem is useful in locating roots of equations

(see corollary below.)

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제 3장 목차

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That is, the derivative of a quotient is the denominator times the derivative of the numerator minus the munerator times the derivative of the denominator, all divided by the square of the denominator.

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p. 82 Figure 4 --> Figure 7

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Chapter 6

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Figure6 ---> Figure7

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p. 249 Figure

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Chapter 7

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Chapter 8

Applications of integrals to areas, volumes, work done and the average value have been considered in the previous chapters. In this chapter, we consider more applications of integrals to the length of curves, the surface area of a surface of revolution, and center of mass. We also describe applications of integrals to simple differential equations.

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Chapter 9

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Hence, this sequence is convergent.

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In this section we will focus on tests for convergence for series with terms that are

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Thus, by the Alternating Series Test, for , this series is convergent.

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Thus, the given power series converges for

So, and

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interval of convergence is .

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Part 2

Chapter 10

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following three rules will be useful in sketching polar curves.

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Chapter 11

In this chapter, we introduce vectors and coordinate systems for three-dimensional space. We also deal with various conepts in vectors, such as the dot and the cross product of vectors, vector equations of lines and planes in space. We will also discuss standard quadric surfaces.

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is orthogonal to the vector a.

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and , so a

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That is, is orthogonal to the plane

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, then find .

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In the plane , the equation of a line can be uniquely determined when a

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where , and are real numbers and and are not both zero.

Let’s find the equation of a line in . If a line passes through the poin

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of the line. Thus we have vectors

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This is an ellipsoid.

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This is an elliptic paraboloid.

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Chapter 12

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A vector function is said to be continuous at if

(ⅰ) is defined

(ⅱ) exists

(ⅲ) .

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Many surfaces are obtained by revolving a plane curve about various axes. For example if the upper semi-circle with and is rotated about the x-axis, we get the sphere. We can easily write a parametrization of such surfaces. If we take a curve on an interval . Then we think of this as a curve in xz=plane.

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Torus and ICME 11 Logo

Chapter 13

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Graph of a Multivariate Function

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Functions of Three Variables

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derivative and partial derivatives of compositions of functions. Let us consider

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The Gradient and Level Curves

Note that Observation 3 says that in the direction orthogonal to the gradient

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Suppose has continuous first order partial derivatives.

Then the surface

is one of the level surfaces of . Let

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위치 확인

Chapter 14

In this chapter, we extend the definition of definite integral for a single variable function to double and triple integrals of function of two or three variables. We apply the definition of double and triple integrals to compute areas and volumes of plane and solid regions. We introduce the polar coordinate system in plane, which helps us to deal with some special types of regions. In a similar way, we introduce cylinder and spherical coordinate systems in space. This chapter also deals with the change of variable formula, which helps us to transform an integral from on coordinate system to another coordinate system.

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Recall that the area of ellipse is .

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We have already seen an application of double integrals to special types of

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in 1^st octant. (See Figure 3.)

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Find the surface area for the part that was cut by the palne on cylinder in 1^st octant.

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Chapter 15

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partial derivatives and Then is conservative.

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Theorem 2 and 3.

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curve, (see Figure 1.) that is possible to transforms line integrals to double integrals and vice versa.

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endency of the vector field to rotate or to diverge where is the velcoity of fluid (or gas) in physics applications. We can understand the terms in the following discussion, by calculating curl and divergence of vector fields. If at a point , then is called bold at . If div , then is said to be bold