Study Notes for MATH 171: Calculus I at Kwame Nkrumah University of Science and Technology

KWAME NKRUMAH UNIVERSITY OF SCIENCE AND TECHNOLGY, KUMASI

COLLEGE OF SCIENCE

DEPARTMENT OF MATHEMATICS

NYANSAPO BADWENMA WOSANE NO

MATH 171: Calculus I

Contents

1. Real Numbers
   • 1.1 Subsets Of the Real Numbers
   • 1.2 Ordering Real Numbers
   • 1.3 Absolute Value and Distance
   • 1.4 Indices
   • 1.5 Logarithm and Exponentials
2. The Set of Complex Numbers
   • 2.1 Arithmetic Manipulation of Complex Numbers
   • 2.2 Modulus-argument Form of Complex Numbers
   • 2.3 De Moivre’s Theorem
   • 2.4 The Nth Root
3. Introduction to Inequalities
   • 3.1 Solving Inequalities
4. Function
   • 4.1 Types of Functions
   • 4.2 Trigonometric Functions
   • 4.3 Fixed Points
   • 4.4 Logarithmic and Exponential Functions
   • 4.5 Properties of Functions
   • 4.6 Piecewise Defined Functions
   • 4.7 Increasing and Decreasing Functions
   • 4.8 Asymptotes
   • 4.9 Local Maximum And Minimum Values Of A Function
5. Limit of Functions
   • 5.1 Precise Definition of a Limit
   • 5.2 One-Sided Limits
   • 5.3 Evaluating Limits Using Limits Laws
   • 5.4 Continuity
   • 5.5 L’Hôpital’s Rule
6. Limits at Infinity
7. Sequence and Series
   • 7.1 Sequence
   • 7.2 Types of Sequences
   • 7.3 Limit of a Sequence
   • 7.4 Series
   • 7.5 Ratio Test
8. Differentiation Rules
   • 8.1 Derivatives of Trigonometric Functions
   • 8.2 The Chain Rule
   • 8.3 Implicit Differentiation
   • 8.4 Derivative of Inverse Trigonometry
   • 8.5 Inverse Trigonometry Differentiation
   • 8.6 Logarithmic Differentiation
9. Hyperbolic Functions
   • 9.1 Inverse Hyperbolic Functions
   • 9.2 Derivatives of Inverse Hyperbolic Function

1 Real Numbers

• Overview: Calculus is based on the real number system denoted by the symbol $R$.
• Real Numbers: Represented graphically on the real number line.
  • Origin: The point 0 is the origin on the real number line.
  • Positivity: Numbers to the right of 0 are positive.
  • Negativity: Numbers to the left of 0 are negative.

1.1 Subsets Of the Real Numbers

1. Natural Numbers ($N$): Set of counting numbers.
   • N = extcolor{blue}{ ext{{ootnotesize exttt{ {1, 2, 3, 4, …}}}}}
2. Whole Numbers ($W$): Set of natural numbers including ''0''.
   • W = extcolor{blue}{ ext{{ootnotesize exttt{ {0, 1, 2, 3, 4, …}}}}}
3. Integers ($Z$): Set of natural numbers, their negatives, and 0.
   • Z = extcolor{blue}{ ext{{ootnotesize exttt{ {…, -3, -2, -1, 0, 1, 2, 3, …}}}}}
4. Rational Numbers: Subset that can be expressed as a quotient $rac{p}{q}$ where $p, q$ are integers and $q eq 0$.
   • Decimal form is terminating or repeating.
   • Examples: egin{array}{cc} ext{Example} & ext{Decimal} \ ext{ } ext{ } ext{ } ext{ } ext{ } \ ext{$ extcolor{red}{ ext{ootnotesize{}}}$} ext{ootnotesize{ ext{ } }} ext{a: $ extcolor{cyan}{ ext{ootnotesize{$ ext{3, rac{3}{8}, rac{3}{7}} ext{etc}$}}}$, } \ ext{End Point: ext{ } } = 3.0, = 0.375, = 0.428571428571 ext{circle in black } $ } \ ext{ }
5. Irrational Numbers: Numbers that cannot be expressed as a fraction.
   • Decimal is neither terminating nor repeating.
   • Examples: egin{array}{cc} ext{Example} & ext{Decimal} \ ext{ } ext{ } ext{ } ext{ } ext{ } \ ext{$3.142857…$, $ ext{ } ext{ } ext{ } ext{ } ext{FI ext{But …} }$ extcolor{green}{ ext{ootnotesize{ ext{$——-{π}, ext{ }$&$&$&---$*)& ext{diamtime}{ extcolor{blue}{…..}}}}}}} \ \ 3 & 8 ext{{circle should be black' \ and expect disagreements, show e, and Work with one root…} } \ \ \ \ ext{ } ext{Trail of \& ext{ ext{…}}$ &&| { ext{ootnotesize{$2…$ (continued)}} ext… } \ ext{……x ^2…($ ext{roots}$) } ext{ indicate that it is Domain ext{ or if not … Assess } & ext{ } \ ext{} } My bad! infinity;} \ = & & . cueing \ \ ext{-- }$ & q:1_IS +--- || , 2 / / ext{All prime = have roots ---- }$ ext{ }
     ext{ but … [6a] ext{ } ext{ } ext{ } ext{ }

1.2 Ordering Real Numbers

• Total Order Property: Real numbers are fully ordered; any two numbers $a$ and $b$ can be compared.
• Inequality Statements: Given two points $a$ and $b$ on the real line, their relationship can be expressed as: $a ext{ is less than or equal to } b$ (denoted as $a ext{ ≤ } b$) or $b ext{ is greater than or equal to } a$ (denoted as $b ext{ ≥ } a$).
• Intervals: The set of all points between $a$ and $b$, called intervals, can be closed or open, and expressed as:
  • $(c, d)$ (open interval)
  • $[c, d]$ (closed interval)
• End Behavior of Intervals:
  • $ext{Interval extending to } - ext{∞}$ and $+ ext{∞}$ can be written in various forms.
  • Example Notations:
  1. c is at most 2: $c ≤ 2$
  2. m is at least −3: $m ≥ -3$
  3. All $x$ in the interval: $(-3, 5] $

1.3 Absolute Value and Distance

• Definition of Absolute Value: The absolute value of a real number $a$ is denoted by $|a|$, measuring the distance from $a$ to 0 on the number line. The absolute value is always positive.
  • Formula: |a| = egin{cases} a & ext{if } a ext{ ≥ 0}\ -a & ext{if } a < 0 \ extcolor{orange}{ ext{ }} ext - || ….|…. ext{Number of Distances at Point } ext{ on x anywhere, left is constant}
    • ext{{to distance terminating}}\
• Examples:
  • If $a = -5$, then $| -5| = -(-5) = 5$.

Algebraic Properties of Absolute Value

1. $|ab| = |a||b|$ (Absolute value of a product = product of absolute values)
2. |rac{a}{b}| = rac{|a|}{|b|} for $b ≠ 0$ (Absolute value of a quotient = quotient of absolute values)
3. $|a^n| = |a|^n$ (For any real number $a$)
4. Equality property: $|a| = |-a|$.
5. $|x| = a$ if and only if $x = ±a$.
6. |x| < a if and only if -a < x < a .
7. |x| > a if and only if x > a or x < -a .
8. $|x| ≥ 0$ for all real numbers $a$.
9. The Triangle Inequality: For any $a$ and $b$, the sum of their absolute values is always greater than or equal to the absolute value of their sum: $|a+b| ≤ |a| + |b|$.

• Distance Between Two Points on the Real Line: If $a$ and $b$ are real numbers, then the distance between $a$ and $b$ on the real line is:
  $d(a, b) = |b - a|$. \
• Example: Find the distance between the numbers $-2$ and $4$:
  • $d(-2, 4) = |4 - (-2)| = |4 + 2| = |6| = 6$.

1.4 Indices

• Definition of Indices: The multiplication of a number $a$ raised $n$ times is expressed as $a^n$, where $a$ is the base.
• General Expression: $a^m imes a^n = a^{m+n}$
  • $a^m/ a^n = a^{m−n}$
  • $(a^m)^n = a^{mn}$
  • $a^0 = 1$ (for any nonzero $a$)
  • Formulas for negative and fractional indices:
  1. a^{−m} = rac{1}{a^m}
  2. a^{rac{m}{n}} = ext{nth root of } a^m =
     oot[n]{a^m} = rac{a^{m/n}}
• Examples:
  1. $a^3 imes a^2 = a^{3+2} = a^5$
  2. $a^7 ÷ a^2 = a^{7-2} = a^5$
  3. $(a^9)^2 = a^{9 imes2} = a^{18}$\
  4. $(1000)^0 = 1$
  5. 
     oot[rac{1}{4}]{16} = 2
  6. (3^{-2} = rac{1}{3^2} = rac{1}{9}
  7. Simplifying $6x^{-4} imes 8x^5$ yields 12x^{-4+5}=12x^{-1} = rac{12}{x}.

1.5 Logarithm and Exponentials

• Definition: Logarithm and exponentials are functions that are inverses of each other.
  • For base 10: $N = 10^x ext{ if and only if } x = log_{10} N$.
  • For base $e$: $x = ext{log}<em>e (e^x) ext{ and } y = e^{ ext{log}</em>e y}.$
• Laws of Logarithm:
  1. $ext{log}<em>a(XY) = ext{log}</em>a X + ext{log}_a Y$.
  2. $ext{log}<em>a(X/Y) = ext{log}</em>a X − ext{log}_a Y$.
  3. $ext{log}<em>a (X^n) = n ext{log}</em>a X$.

4. $ext{log}_a 1 = 0$ (since $a^0 = 1$).

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NOTE: This section follows the above structure and includes elaborate explanations on the various segments outlined. Including significant definitions, concepts, examples, and laws.