[Calculus] Summary for Exam

1 Set

• A set is a collection of distinct objects.

• Notation:

\(S_1 = \{1,2,3,4,5\}\)

\(\Rightarrow\) "x can only take values of 1, 2, 3, 4 or 5.

• Set builder notation: \(\{\text{variable | conditions}\}\) or \(\{\text{variable : conditions}\}\)

\(S_2 = \{\text{x | x is a letter in the English alphabet}\}\)

\(\Rightarrow\) “Collection of all letter x such that x is in the English alphabet”.

\(S_3 = \{\text{x : x > 2 and x is odd}\}\)

\(\Rightarrow\) “Collection of all numbers x such that x is greater than 2 and x is odd”.

\(S_3 = \{\text{x | x} \in S_1\}\)

\(\Rightarrow\) "Collection of all numbers x such that x can only take values from the set \(S_1\).

1.1 Element

• An object inside a set. Notation: \(1 \in \mathbb{N}\).
  
• If an object is not inside a set, the notation is \(\sqrt{3} \notin \mathbb{Q}\).

1.2 Subset

• If every element of a set A is also an element of a set B, then we say that A is a subset of B. Notation: \(A \subseteq B\).

• If A and B are sets such that \(A \subseteq B\) but \(A \neq B\), then we say that A is a proper subset of B. In other words, a set A is a proper subset of a set B. Notation: \(A \subset B\), if (1) \(A \subseteq B\) and (2) there exists at least one element in B that is not in A. The second condition states that the set A is properly “smaller” than the set B.

• B is not a subset of A if there exists at least one element in B that is not in A. Notation: \(B \not\subset A\).

1.3 Empty set

• The set that contains no elements is called the empty set and is denoted by \(\varnothing\).

• The empty set, \(\varnothing\), is a subset of every set. \(\varnothing\) has no elements and therefore contains no element that is not also in any set A.

1.4 Universal set

• A universal set is a set which includes all elements being considered in the problem. Notation: U.

1.5 Complement

• A complement of set A is the set of all elements in the universal set U that are not in A. Notation: \(A^c\) or \(A'\).
  
• \(A^c = \{x|x \in U \text{ and } x \notin A\}\).

1.6 Common number sets

• Natural numbers: \(\mathbb{N} = \{1,2,3,4,...\}\)

• Integer numbers: \(\mathbb{Z} = \{...,-4,-3,-2,-1,0,1,2,3,4,...\}\)

• Rational numbers: \(\mathbb{Q} = \{x|x=\frac{p}{q}\text{ and }p\in\mathbb{Z}\text{ and }q\in\mathbb{Z}\}\)

• Real numbers: \(\mathbb{R} = \{x|x\in\mathbb{Q}\text{ and }x\notin\mathbb{Q}\}\)
  Example: \(\sqrt{2}\) or \(\pi\).

1.7 Real number line

1.8 Intervals

• An interval is a set of all real numbers between two boundary numbers.

1.8.1 Inequalities on number lines

1.9 Venn diagram

• A Venn diagram is a visual representation of sets

1.10 Operations

1.11 Element counting

• The number of elements in a set is called the cardinality of the set.

• If A and B are disjoin sets, then \(n(A\cup B) = n(A) + n(B)\).

• If A and B are finite sets, then \(n(A\cup B) = n(A) + n(B) - n(A \cap B)\).

2 Function

• A function is a relation between a set of inputs and a set of outputs.
  

• \(y = f(x)\)

  (\(y\) is output, \(x\) is input, \(f\) is function)

• \(f : X \rightarrow Y\)

  (\(Y\) is the set of output, \(X\) is the set of input, \(f\) is function)

• Each input value can only give one output value.

2.1 Operations

• A function can be replaced with its full expression.

Ex1: \(f(x) = \frac{1}{2}x\), \(g(x) = f(x) - 5\) or \(g(x) = \frac{1}{2}x - 5\)

Ex2: \(f(x) = \frac{1}{2}x\), \(g(x) = f(x) + 10\) or \(g(x) = \frac{1}{2}x + 10\)

Ex3: \(f(x) = 3x\), \(h(x) = 2x\), \(g(x) = f(x)h(x)\) or \(g(x) = 6x^2\)

Ex4: \(f(x) = \frac{1}{2}x\), \(g(x) = \frac{1}{4}f(x)\) or \(g(x) = \frac{1}{8}x\)

2.2 Composite function

• The result of a function can become the input to another function.

• \(y = f(x)\), \(z = g(y)\), then \(z = g(f(x))\)

Ex1: \(f(x) = 2x\), \(g(x) = 5x + 1\), then \(g(f(x)) = 5f(x) + 1 = 5(2x) + 1 = 10x + 1\)

Ex2: \(f(x) = 2x\), \(g(x) = 5x + 1\), then \(f(g(x)) = 2g(x) = 2(5x + 1) = 10x + 2\)

2.3 Inverse function

• If \(y = f(x)\) then its inverse function is \(f^-1(y)\).

• If \(g(x) = f^-1(x)\) and \(f(x) = g^-1(x)\) then \(f(x)\) and \(g(x)\) are inverses of each other.

• The function is invertible if the inverse function exists.

• Not every function has an inverse function.

• To find the inverse function, write the full expression, then rewrite it as a function of x in terms of y (1 single x on the left side and all the y terms on the right side of the equation):

\[ y = 4x - 5 \\ 4x = y + 5 \\ x = \frac{1}{4}y + \frac{5}{4} \\ \Rightarrow \text{The inverse function of } y = 4x - 5 \text{ is } f^-1(y) = \frac{1}{4}y + \frac{5}{4} \]

2.4 Graphing

• The graph of a function shows all valid pairs \((x,y) = (x,f(x))\) on the Cartesian plane.

2.5 Asymptote

• \(y = k\) is a horizontal asymtote for the function as the function \(f(x)\) continually approaches the line \(y = k\) but never reaches it.

• \(x = a\) is a vertical asymtote for the function as the value of the function \(f(x)\) grows indefinitely the more the function approaches \(x = a\).

2.6 Domain & Range

• Domain of a function is the set of all possible input values to the function.

• Range of a function is the set of all possible values that the function can take.

2.7 Monotonicity

• If \(f : X \rightarrow Y\) is a set function from a collection of sets X to an ordered set Y, then \(f\) is said to be monotone if whenever \(A \subseteq B\) as elements of \(X\), \(f(A) \leq f(B)\)

2.8 Roots

• Roots (or zeros) of a function are the intersection(s) of the function with the \(x\) axis.

• Roots are all values of \(x\) such that \(f(x) = 0\).

2.9 Types

2.9.0.1 Attributes of functions

2.9.1 Distance between 2 points

Applying Pythagorean theorem: \(d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}\)

3 Linear equation

3.1 Form

• Slope-intercept form: \(y = mx + b\)

• General form: \(Ax + By + C = 0\)

3.2 Point lies on line

• A point lies on a line if it satisfies its equation.

3.3 Intersection point

3.4 Linear equation from 2 points

3.5 Least-squares line

• The system of normal equations:

\[ \begin{array}{lcl} nb + \left(\sum x_i\right)m = \sum y_i \\ \left(\sum x_i\right)b + \left(\sum x_i^2\right)m = \sum x_iy_i \end{array} \]

• Solving them yields:

\[ m = \dfrac{n \sum x_i y_i - \sum x_i \sum y_i}{n \sum x_i^2 - \left(\sum x_i\right)^2}\\ b = \dfrac{\sum y_i - m\sum x_i}{n} = \bar{y} - m\bar{x} \]

4 System of linear equations

4.1 Types

4.2 Solution

4.2.1 Graphing

4.2.2 Method of elimination

4.2.3 Method of substitution

4.2.4 Methods using matrices

Accepted row operations:
- Row interchange: \(R_i \leftrightarrow R_j\)
- Row multiplication with a non-zero constant: \(cR_i\)
- Row addition then multiplication with a non-zero constant: \(R_i + cR_j\)

4.2.4.1 Row Echelon form & Reduced Row Echolon form

4.2.4.2 Gaussian elimination method to Row Echelon form

4.2.4.3 Gauss-Jordan elimination method

5 Linear inequality

6 Systems of linear inequalities

7 Matrix

• A matrix is an ordered rectangular array of numbers.
  
• An \(n \times m\) matrix has n rows and m columns.
  
• \(a_{ij}\) is located at row \(i\) and column \(j\).

• A vector is a matrix with either single row or single column.

• A scalar is a real number muliplied with a matrix to create a scalar product.

7.1 Operation

7.1.1 Addition and substraction

• Example

7.1.2 Product

7.1.2.1 Scalar product

• Example

7.1.2.2 Matrix product

• \(A_{n \times m} \cdot B_{m \times p} = AB_{n \times p}\)

• Example

7.2 Transpose

• \(A_{n \times m} \rightarrow A^T = B_{m \times n}\)

7.3 Identity matrix

• \(I_nA = A\) for every \(n \times p\) matrix A.
  
• \(BI_n = B\) for every \(m \times n\) matrix B.
  
• \(I_nA = AI_n = A\) for every \(n \times n\) matrix A.

7.4 Inverse

• \(A^{-1}\) or \(A'\) is the inverse of matrix \(A_{n \times n}\) if \(A^{-1}A = AA^{-1} = I_n\).
  
• A matrix is called invertible if it has an inverse.
• A \(n \times m\) matrix may have an \(m \times n\) left inverse if \(A^{-1}A = I_n\).
  
• A \(n \times m\) matrix may have an \(m \times n\) right inverse if \(AA^{-1} = I_m\).

8 Permutation

• A permutation of the set is an arrangement of the set objects in a definite order.

• Permutations of n distinct objects taken r at a time:

\(P(n,r) = _nP_r = P_r^n = P_{n,r} = \dfrac{n!}{(n-r)!}\)

• where \(n! = n(n-1)(n-2) \cdot \ldots \cdot 3 \cdot 2 \cdot 1\).
  
• Remember: \(0! = 1\).

8.1 Permutation with non-distinct objects

• Permutations of n objects, not all distinct taken n at a time:

\(\dfrac{n!}{n_1!n_2! \ldots n_m!}\)
- where \(n_1 + n_2 + \ldots + n_m = n\)

8.2 Circular permutation

\(P(n) = (n-1)!\)

9 Combination

• A combination of a given number of articles is a subset of articles selected from those given where the order of the articles in the subset is not taken into consideration.

\(C(n,r) = _nC_r = C_r^n = C_{n,r} = \dfrac{n!}{r!(n-r)!}\)

• where \(r \leq n\)

10 Probability

10.1 Terminology

• The probability of an event, denoted \(P(E)\), is the likelihood of that event occuring.

• \(P(E) = \dfrac{n(E)}{n(S)}\)

• Experiment is any process that can be repeated in which the results are uncertain.
  

• Outcome is the result of an experiment • Sample event (or point): any single outcome from a probability experiment.
  

• Sample space is a list (or set) of all possible outcomes of a probability experiment.
  

• Event is any collection of outcomes from a probability experiment (a subset of the sample space).

10.2 Properties

• \(0 \leq P(E) \leq 1\)

• \(P(E) = 0\) means impossible event.

• \(P(E) = 1\) means certainty of event.

• if E and F are any 2 events of an experiment, then \(P(E \cup F) = P(E) + P(F) - P(E \cap F)\)

• If E and F are mutually exclusive, then \(P(E \cup F) = P(E) + P(F)\)

• If \(S = \{e_1,e_2,e_3,...,e_n\}\), then \(P(e_1) + P(e_2) + P(e_3) + \ldots + P(e_n) = P(S) = 1\)

• If \(E = \{s_1,s_2,s_3,...,s_n\}\), where \(\{s_1\}\), \(\{s_2\}\), \(\{s_3\}\), …, \(\{s_n\}\) are simple events, then \(P(E) = P(s_1) + P(s_2) + P(s_3) + \ldots + P(s_n)\)

• \(P(E^c) = 1 - P(E)\)

10.3 Diagram

10.3.1 Venn diagram

10.3.2 Tree diagram

• Used to find conditional probabilities in a finite stochastic process.

• Stochastic process: an experiment consisting of a finite number of stages in which the outcomes and associated probabilities of each stage depend on the outcomes and associated probabilities of the preceding stage.

10.4 Conditional probability

• The probability of B given A, which means that Event A has already occurred, is denoted \(P(B|A)\).

• \(P(B|A) = \dfrac{P(B \cap A)}{P(A)}\)

• \(P(B \cap A) = P(A) \cdot P(B|A)\)

• If A and B are independent events, then \(P(B|A) = P(B)\) and \(P(A|B) = P(A)\). In other words, \(P(A \cap B) = P(A) \cdot P(B)\).

10.5 Bayes’ Theorem

10.6 Probability distribution

• A random variable is a rule that assigns a number to each outcome in a chance experiment (random process).
  
• Range is the set of real numbers.
  
• Domain is a sample space from a random experiment.
  
• Probability distribution of a random variable is a list of probabilities associated with each of its possible values.

10.6.1 Histogram

• A Histogram is a graphical display of data distribution using bars.

10.6.2 Mean, Median, Mode

10.6.3 Variance, Standard deviation

• Variance of population

\[ Var(x) = \sigma^2 = \dfrac{\sum(x - \mu)^2}{N} \]

• Standard deviation of population

\[ \sigma = \sqrt{Var(x)} = \sqrt{\dfrac{\sum(x - \mu)^2}{N}} \]

• Standard deviation of sample

\[ \delta = \sqrt{Var(x)} = \sqrt{\dfrac{\sum(x - \bar{x})^2}{n-1}} \]