Dr. I. Mechai & Dr. M. Al Ghamdi Calculus I Jazan ...colleges.jazanu.edu.sa/sci/math/Documents/Calculus/CalculusI/Lecture... · Dr. I. Mechai & Dr. M. Al Ghamdi Calculus I Jazan University

Dr. I. Mechai & Dr. M. Al Ghamdi Calculus I Jazan University Department Of Mathematics

Contents

Appendix A: Numbers, Inequalities, and Absolute Values . . . . . . . . . . . . . . . . . . . . . 11. Intervals: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22. Inequalities: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33. Absolute Value: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Appendix B: Coordinate Geometry And Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . 61. Distance between two points: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72. Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73. Parallel And Perpendicular Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Appendix D: Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111. The trigonometric functions: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122. Trigonometric identities: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Appendix A: Numbers, Inequalities, and Absolute Values

Calculus is based on the real number system:Integers:

Rational numbers (ratios of integers): r = mn; where m and n are integers and n 6= 0:

Examples:

Note that division by 0 is always ruled out, so expressions like 30and 0

0are unde�ned.

Irrational numbers: A real numbers that cannot be expressed as a ratio of integers are called irrationalnumbers.Examples:

The set of all real numbers is denoted by R. Every number has a decimal representation.If the number is rational, then the corresponding decimal is repeating.Examples:

= 0:5000::: = 0:50 = 0:66666::: = 0:6

= 0:317171717::: = 0:317 = 1:285714285714::: = 1:285714

On the other hand, if the number is irrational, the decimal is non repeating:

= 1:414213562373095::: = 3:141592653589793:::

The real numbers can be represented by points on a line as:

- 1 - Lecture Week 1


The real numbers are ordered: We say a is less than b and write a < b if b � a is apositive number.

Equivalently, we say b is greater than a and write b > a. The symbol a � b (or b � a )means that either or and is read � is less than or equal to.�For instance, the following

are true inequalities:

7 < 7:4 < 7:5 3 < �p2 � 2 2 � 2

1. Intervals:

Intervals correspond geometrically to line segments.

Examples: if a < b, the open interval from a to b consists of all numbers between a and band is denoted by the symbol (a; b), and also written as

(a; b) = fxj a < x < bg

Open interval

the closed interval from a to b is the set

[a; b] = fxj a � x � bg

Closed interval

For other possible types of intervals are shown in the following Table:



Note: In�nite intervals can be write also as

2. Inequalities:

We have the following rules:

Example 1: Solve the inequality 1 + x < 7x+ 5:Solution:

Example 2: Solve the inequalities 4 � 3x� 2 < 13:Solution:



Example 3: Solve the inequality x2 � 5x+ 6 � 0:Solution:

Another method is to use test values:

Example 4: Solve x3 + 3x2 > 4x:Solution:

3. Absolute Value:

The absolute value of a number a, denoted by jaj, is the distance from a to 0 on the realnumber line, and we have

jaj � 0 for every number a:Examples:



In general, we have

Example 5: Express j3x� 2j without using the absolute-value symbol.Solution:

Note that:pa2 = :

Properties of Absolute Values: Suppose a and b are any real numbers and n is an integer. Then

1. jabj = jaj jbj 2.��ab

�� = jajjbj (b 6= 0) 3. janj = jajn

For solving equations or inequalities involving absolute values, it�s often very helpful touse the following statements

Example 6: Solve j2x� 5j = 3.Solution:



Example 7: Solve jx� 5j < 2.Solution:

Example 8: Solve j3x+ 2j � 4.Solution:

The Triangle Inequality: If a and b are any real numbers, then

ja+ bj � jaj+ jbj :

Example 9: If jx� 4j < 0:1 and jy � 7j < 0:2, use the Triangle Inequality to estimate j(x+ y)� 11j.Solution:

Appendix B: Coordinate Geometry And Lines

In Appendix A, we identi�ed the points on a line with real numbers by assigning them coordinates.Similarly, the points in a plane can be identi�ed with ordered pairs of real numbers as follows:- Draw two perpendicular coordinate lines that intersect at the origin O: The horizontal line is called thex-axis and the vertical line is called the y-axis.- We say that P is the point with coordinates (a; b), and we denote the point by the symbol P (a; b).This coordinate system is called the rectangular coordinate system or the Cartesian coordinatesystem divide the Cartesian plane into four quadrants, which are labeled I, II, III, and IV in Figure 1.



Example 1: Describe and sketch the regions given by the following sets:a) f(x; y)jx � 0g ; b) f(x; y)j y = 1g, c) f(x; y)j jyj < 1g :Solution:

1. Distance between two points:

The distance between two points P1 (x1; y1) and P2 (x2; y2) is

jP1P2j =

Example 2: Find the distance between the points (1;�2) and (5; 3) :Solution: See Textbook.Exercise 3 (Page 1181): Find the distance between the points (6;�2) and (�1; 3) :



Solution:

2. Lines

De�nition 2: Let the nonvertical line L that passes through the points P1 (x1; y1) and P2 (x2; y2) : Wede�ne the slope of the line L by

m =�y

�x=

Note that the slope of a vertical line is not de�ned.

Exercise 9 (Page 1181): Find an slope of the line through P (�3;�3) and Q (�1;�6) :Solution:

De�nition 3 (Point-Slope Form Of The Equation Of A Line): An equation of the line passing throughthe point P1 (x1; y1) and having slope m is

y � y1 = m (x� x1)

Example 3: Find an equation of the line through (1;�7) with slope 12:

Solution: See Textbook.Exercise 22 (Page 1181): Find an equation of the line through (�1; 4) with slope �3:Solution:



Example 3: Find an equation of the line through the points (�1; 2) and (3;�4).Solution: See Textbook.Exercise 26 (Page 1181): Find an equation of the line through the points (1;�2) and (4; 3).Solution:

De�nition 4 (Slope-Intercept Form Of The Equation Of A Line): An equation of the line with the slopem and y-intercept b is

y = mx+ b:

Note: A line has y-intercept b, means it intersects the y-axis at the point (0; b) :

Exercise 28 (Page 1181): Find an equation of the line with the slope2

5and y-intercept 4.

Solution:

Note: The equation of every line can be written in the form

Ax+By + C = 0;

and called a linear equation or the general equation of a line. And we have the following formulas:



Example 5: Sketch the graph of the equation 3x� 5y = 15Solution:

Example 6: Graph the inequality x+ 2y > 5:Solution:

3. Parallel And Perpendicular Lines

De�nition 6: � Two nonvertical lines are parallel if and only if they have the same slope.� Two lines with slopes m1 and m2 are perpendicular if and only if m1m2 = �1; that

is, their slopes are negative reciprocals:

m1 = �1

m2

:

Example 7: Find an equation of the line through the point (5; 2) that is parallel to the line 4x+6y+5 = 0.Solution: See Textbook.Exercise 34 (Page 1181): Find an equation of the line through the point y-intercept 6 and parallel tothe line 2x+ 5y + 8 = 0.Solution:



Example 8: Show that the lines 2x+ 3y = 1 and 6x� 4y � 1 = 0 are perpendicular.Solution:

Appendix D: Trigonometry

Angles: Angles can be measured in degrees or in radians (abbreviated as rad), and related by thefollowing relation

�rad = 180� (1)

and1rad = ( )� � 57:3� 1� =

�

180rad � 0:017rad (2)

Example 1:

(a) Find the radian measure of 60�. (b) Express5�

4rad in degrees.

Solution: See textbook.Exercise 3 & 9 (Page 1235):

(3) Find the radian measure of 9�. (9) Express5�

12rad in degrees.

Solution:

Note: In calculus we use radians to measure angles except when otherwise indicated.



The correspondence between degree and radian measures of some common angles:

Relation between an arc length a, angle �, and radius r of a circle:

(3)

and � is measured in radians.

Example 2:(a) If the radius of a circle is 5 cm, what angle is subtended by an arc of 6 cm?(b) If a circle has radius 3 cm, what is the length of an arc subtended by a central angle

of3�

8rad?

Solution: See textbook for more details.Use Equation 3 for (a) with r = 5 and a = 6 cm to �nd �, and for (b) with r = 3 and � =

3�

8to �nd a.

Standard position of an angle � and direction: The vertex of � placed at the origin of a coordinatesystem and its initial side on the positive x-axis. A positive angle is obtained by rotating the initial sidecounterclockwise, and negative angle by rotating the initial side clockwise.

� � 0 � < 0



Notice that di¤erent angles can have the same terminal side and 2� rad represents a complete revolution.

1. The trigonometric functions:

For an acute angle � the six trigonometric functions are de�ned as follows:

Let P (x; y) any point in the terminal side of the standard angle � and r is the distance jOP j, then we

de�ne

Notes: - tan � and sec � are unde�ned when x = 0- cot � and csc � are unde�ned when y = 0- � is measured in radian- The exact trigonometric ratios for certain angles can be read from the triangles:



- The signs of the trigonometric functions for angles in each quadrants can be remembered bymeans of the rule �All Students Take Calculus�as shown in the following �gure:

Example 3: Find the exact trigonometric ratios for2�

3:

Solution: See textbook.Exercise 23 (Page 1235): Find the exact trigonometric ratios for

3�

4:

Solution:

Therefore: taking x = �1, y = 1 and r =p2 in the de�nitions of the trigonometric ratios, we have



Values of sin � and cos � for some angles

Example 4: If cos � =2

5and 0 < � <

�

2, �nd the other �ve trigonometric functions of �.

Solution: See textbook.Exercise 29 (Page 1235): If sin � =

3

5and 0 < � <

�

2, �nd the other �ve trigonometric functions of �.

Solution:

Example 5: Use a calculator to approximate the value of x in the following �gure:

Solution:

2. Trigonometric identities:

A trigonometric identity is a relationship among the trigonometric functions.



sin2 � + cos2 � = 1 tan2 � + 1 = sec2 � 1 + cot2 � = csc2 �

sin (��) = � sin � cos (��) = cos � sin (� + 2�) = sin � cos (� + 2�) = cos �

sin (x+ y) = sinx cos y + cosx sin y cos (x+ y) = cosx cos y � sin x sin ysin (x� y) = sin x cos y � cosx sin y cos (x� y) = cosx cos y + sinx sin ytan (x+ y) =

tan x+ tan y

1� tan x tan y tan (x� y) = tan x� tan y1 + tanx tan y

sin 2x = 2 sinx cosx cos 2x = cos2 x� sin2 xcos 2x = 2 cos2 x� 1 cos 2x = 1� 2 sin2 xcos2 x =

1 + cos 2x

2sin2 x =

1� cos 2x2

2 sin x cos y = sin (x+ y) + sin (x� y) 2 cos x cos y = cos (x+ y) + cos (x� y)2 sin x sin y = cos (x� y)� cos (x+ y)

Note: These identities can be deduced from the identity of sin (x+ y) and cos (x+ y) :Example 6: Find all values of x in the interval [0; 2�] such that sin x = sin 2x.Solution:
