Calculus

Expressions

• In algebra, we use letters like ${x,}$ ${y,}$ ${n,}$ and ${m}$ to represent some object (e.g., numbers). These letters are called variables. When we combine these variables with mathematical operations (e.g., addition, subtraction, multiplication, etc.), we obtain an expression.

  • example. The following are all expressions:

    $$a + 2 \\[1em] x^2 \\[1em] \dfrac{x^{1/3} + y}{xy}$$

  • example. Of course, expressions need not have variables. These are expressions, too:

    $$1 + 1 \\[1em] 2/3 \\[1em] \pi$$
• When we place an equal sign between two expressions, we have an equation. This is just a symbolic way of saying, "These two expressions are the same."
  • example. Here is a famous equation: $$a^2 + b^2 = c^2.$$

The Domain of an Expression

• All mathematical expressions have a domain – the set of all values for which the expression is defined.

  • example. In the expression ${a^2 + b^2 = c^2,}$ we assume the expression is defined for real numbers. That is, the expression only makes sense if we plug real numbers into the variables. It would not make sense if the domain included, say, dogs. Of course, it very well could, but we'd have to define what it means to square and add dogs:

    $$\text{rottweiler}^2 + \text{greyhound}^2 = \text{mastiff}^2.$$

    The point is, outside the expression's domain, the expression is undefined.

• definition. Given an expression ${u,}$ the notation

  $$\text{dom}(u)$$

  denotes the domain of ${u.}$

  • example. In the expression ${1/((t-1)(t-3)),}$ we define the domain to be all real numbers (${\reals}$), not including ${1}$ and not including ${-3.}$ If we let ${t = 1}$ or ${t = -3,}$ then we have a zero denominator. And as we all know, we cannot divide by zero.

    $$\text{dom} \left( \dfrac{1}{(t-1)(t+3)} \right) = \set{t \in \reals | t \neq 1, t \neq -3}.$$

Solution Sets

• Because an expression may contain variables, an equation is true or false only if we substitute values into those variables. Until then, the equation is neither true nor false.
• If a value is substituted and the equation is true, we call that value the solution, or root, of the equation.
• The set of all roots to an equation is called the equation's truth set or solution set.
• If the equation's solution set is the equation's domain, we call the equation an identity.
  • example. ${x^2 - 9 = (x - 3)(x + 3).}$
  • example. ${\dfrac{x^2}{x} = x.}$

Numbers

• We will start by giving informal definitions of numbers.
• informal definition. A real number is any number that can be expressed in decimal form. We denote the set of real numbers with the symbol ${\reals.}$
  • The real numbers can be represented as points on a number line.
  • A fundamental fact: There is a one-to-one correspondence between the set of real numbers and the set of points on a line. For each given point, the real number associated with that point is called the point's coordinate.
• informal definition. A natural number is any number of the set ${\{ 0, 1, 2, 3, \ldots \}.}$ We denote this set with the symbol ${\N.}$
  • The naturals are the most basic numbers. These are the numbers we use for counting.
• informal definition. An integer is any number of the set ${\{ \ldots, -2, -1, 0, 1, 2, \ldots \}.}$ We denote this set with the symbol ${\Z.}$
  • The naturals with the negative numbers comprise the set of integers.
  • As its name suggests, a rational number is any number that can be written as the ratio of two integers.
• informal definition. Let ${a \in \Z}$ and let ${b \in \Z.}$ Then a rational number is any number of the form ${a/b.}$ We denote the set of all such numbers with the symbol ${\mathbb{Q}.}$
• informal definition. Let ${x}$ be some number. If ${x \notin \mathbb{Q},}$ then ${x}$ is an irrational number.
  • Any number that does not satisfy the definition of rational numbers is called an irrational number.
  • Examples of irrational numbers include ${\sqrt{2},}$ ${\pi,}$ and Euler's number ${e.}$ None of these numbers can be written as a ratio of integers.

Relations

• In mathematics, a relation is the definition of some relationship between mathematical objects.
• We can relate one real to another in terms of their position on the real line.
  • example. Viewing the real line, we can say that ${0}$ is "to the left" of ${1,}$ or, more succinctly, ${0 \lt 1.}$ We can talk about the reals in terms of their relations to other reals because of the fact that the real numbers are ordered: ${2}$ comes after ${1,}$ ${3}$ comes after ${2,}$ ${0}$ comes before ${1,}$ ${\pi}$ comes before ${4,}$ etc.
• Below is a table of common notations for relations.

  Notation	Definition	Example
  ${a \lt b}$	${a}$ is less than ${b.}$	${0 \lt 1.}$
  ${a \leq b}$	${a}$ is less than or equal to ${b.}$	${1 \leq 1,}$ ${0 \leq 1.}$
  ${a \gt b}$	${a}$ is greater than ${b.}$	${1 \gt 0.}$
  ${a \geq b}$	${a}$ is greater than or equal to ${b.}$	${1 \gt 1, 1 \gt 0.}$

Intervals

• When we work with ${\reals,}$ we often want to talk about sections of the real line (or, in terms of set theory, subsets of ${\reals}$). These sections, or subsets, can be defined in terms of intervals.
• definition. The open interval ${(a,b)}$ consists of all reals ${x}$ such that ${a \lt x \lt b.}$ The closed interval ${[a,b]}$ consists of all reals ${x}$ such that ${a \leq x \leq b.}$ The left-inclusive interval ${[a,b)}$ consists of all reals ${x}$ such that ${a \leq x \lt b.}$ The right-inclusive interval ${(a,b]}$ consits of all reals ${x}$ such that ${a \lt x \leq b.}$
• In some cases, we may want talk about a subset of the reals that extends indefinitely in some direction. Such subsets are called unbounded intervals.
• Below are common notations for intervals.

  Notation	Inequality
  ${(a, \infty)}$	${x \gt a}$
  ${[a, \infty)}$	${x \geq a}$
  ${(-\infty, a)}$	${x \lt a}$
  ${(-\infty, a]}$	${x \leq a}$
  ${(-\infty, \infty)}$

Absolute Value

• For some problems, we want to think about how far a given real number ${x}$ is from ${0.}$ The operator for computing this distance is the absolute value.
• definition. Given ${x \in \reals,}$ the absolute value of ${x,}$ denoted ${| x |,}$ is defined as: $$| x | = \begin{cases} -x & \text{if} ~ x \lt 0 \\ x & \text{else} \end{cases}.$$
  • example. ${|6 - 7| = |-1| = 1.}$
• Given that real numbers can be interpreted as points on a number line, the distance between two real numbers is the absolute value of their difference.
• property. Given ${a, b \in \R,}$ the distance between ${a}$ and ${b}$ is ${\vert a - b \vert = \vert b - a \vert.}$
• Below are some additional properties of the absolute value.

  Property	Comment
  ${\vert x \vert \geq 0}$	Absolute values are always nonnegative.
  ${x \leq \vert x \vert}$
  ${-x \leq \vert x \vert}$
  ${\vert x \vert^2 = x^2}$
  ${\sqrt{x^2} = \vert x \vert}$
  ${\vert ab \vert = \vert a \vert \vert b \vert}$
  ${\left\vert \dfrac{a}{b} \right\vert = \dfrac{\vert a \vert}{\vert b \vert}}$	${b \neq 0}$
  ${\vert a + b \vert \leq \vert a \vert + \vert b \vert}$	This is a rather famous property, called the triangle inequality.

Equations

• An equation is a mathematical statement that claims: "The expression on the left of the "${=}$" is the same, in every way, as the expression on the right of the "${=.}$"
• example. ${a = b}$ means "${a}$ is the same, in every way, as ${b.}$"
• When equations contain variables, they belong to a type of mathematical statement called an open sentence — a statement that contains one or more variables, and becomes a proposition when each variable is substituted with an element from the expression's domain.
• example. The equation ${a = b}$ is true when ${a = 1}$ and ${b = 1,}$ but false when ${a = 0}$ and ${b = 1.}$
• When a value is substituted for a variable in an equation such that the equation is true, we say that the value satisfies the equation, and that the value is a solution to the equation.
• When two equations have the exact same set of solutions, we say that the two equations are equivalent.
• A common procedure in algebra is solving for a given variable. This is done by writing a sequence of equivalent equations until we get to the form ${\text{variable} = \text{constant}}$ (e.g., ${x = 1}$ or ${z = \pi}$).
• The following principles apply to generating equivalent equations:
  1. Adding or subtracting the same quantity on both sides of an equation produces an equivalent equation.
  2. Multiplying or dividing both sides of an equation by the same nonzero quantity produces an equivalent equation.
  3. Simplifying an expression on either side of an equation produces an equivalent equation.
• Equations — and more generally, expressions — are simply succinct ways of expressing sentences that would otherwise be too cumbersome to write by prose.

Families of Expressions

• When we examine the variety of expressions found in mathematics, we find that certain groups of expressions share similiar patterns and properties. To try and organize this vast variety, we will organize these expressions into families of expressions, or kinds of expressions.

Linear Expressions

• The simplest family of expressions is the linear expression in one variable. As simple as such expressions are, many real world problems can be reduced to linear expressions, or more specifically, linear equations.

• definition. A linear expression in one variable is an expression that can be written in the form

  $$ax + b,$$

  where ${a}$ and ${b}$ are real numbers and ${a \neq 0.}$

• In the definition above, we call the expressions ${ax}$ and ${b}$ the terms — or, more generally, the subexpressions — of the linear expression.

Monomial Expressions

• The terms of a linear expression are themselves a special type of expression called monomial expressions.
• The simplest monomial expression is the univariate monomial expression, or the monomial expression in one variable.
• definition. Let ${u}$ be an expression. We say that ${u}$ is a univariate monomial expression in a single variable ${x}$ if ${u}$ satisfies one of the following rules:
  1. ${u}$ is an integer or fraction.
  2. ${u = x.}$
  3. ${u = x^n,}$ where ${n \in \Z}$ and ${n \gt 1.}$
  4. ${u}$ is a product with two operands that satisfies either (1), (2), or (3).

Quadratic Expressions

• definition. A quadratic expression in one variable is an expression that can be written in the form $$ax^2 + bx + c,$$ with ${a,b,c \in \reals}$ and ${a \neq 0.}$
• property. ${pq = 0}$ if and only if ${p = 0}$ or ${q = 0.}$

The Quadratic Formula

• We can also solve quadratic equations directly with the quadratic formula:
• formula. The solutions of the quadratic equation ${ax^2 + bx + c = 0,}$ where ${a \neq 0,}$ are given by $$x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}.$$

Polynomial Expressions

• Quadratic expressions can be generalized as polynomial expressions in one variable. I.e., they are a specific type of polynomial expression in one variable.

• definition. A polynomial ${u}$ in one variable is an expression of the form:

  $$u_nx^n + u_{n-1}x^{n-1} + \ldots + u_1x + u_0,$$

  where the coefficients ${u_j}$ are rational numbers, and ${n}$ is a non-negative integer.

• If ${u_n \neq 0,}$ we call ${u_n}$ the leading coefficient of ${u}$ and ${n}$ the degree of ${u.}$ The expression ${u = 0}$ is called the zero polynomial.

• By convention, the zero polynomial has leading coefficient ${0}$ and, according to mathematical convention, has degree ${-\infty.}$

• definition. Let ${u}$ be a polynomial expression in a single variable ${x.}$ Then:

  $$\text{deg}(u,x)$$

  corresponds to the degree of ${u}$ and

  $$\text{lc}(u,x)$$

  corresponds to the leading coefficient of ${u.}$ If the variable ${x}$ is clear from context, we use the shorthand notation ${\text{deg}(u)}$ and ${\text{lc}(u).}$

Multivariate Polynomials

• A multivariate polynomial is a polynomial that contains one or more variables.

• definition. A multivariate polynomial ${u}$ in the set of symbols ${\lbrace x_1, x_2, \ldots, x_m \rbrace}$ is a finite sum with one or more monomial terms of the form

  $$c x_1^{n_1} x_2^{n_2} \ldots x_m^{n_m},$$

  where the coefficient ${c}$ is a rational number and the exponents ${n_j}$ are non-negative integers.

• example. ${x^2 - y^2}$

• example. ${mc^2}$

• example. ${a + \dfrac{1}{2}ax^2 + aby.}$

• example. ${ax^2 + bx + 3c.}$

• example. ${3x^2 + 4.}$

  • In this example, we see that all univariate polynomials are multivariate polynomials, since a multivariate polynomial is a polynomial that contains one or more variables.

General Monomial Expressions

• definition. Let ${c_1, c_2, \ldots, c_r}$ be algebraic expressions and let ${x_1, x_2, \ldots, x_m}$ be algebraic expressions that are not integers or fractions. A general monomial expression (GME) in ${\lbrace x_1, x_2, \ldots, x_m \rbrace}$ is an expression of the form

  $$c_1 c_2 \ldots c_r x_1^{n_1} x_2^{n_2} \ldots x_m^{n_m},$$

  where the exponents ${n_j}$ are non-negative integers and each ${c_i}$ does not contain an expression ${x_j}$ for ${j = 1,2,\ldots,m.}$

• In the definition above, the expressions ${x_j}$ are called generalized variables, and the expressions ${c_i}$ are called generalized coefficients. The expression ${x_1^{n_1} \ldots x_m^{n_m}}$ is called the variable part of the monomial. If there aren't any generalized coefficients in the monomial, the coefficient is ${1.}$

General Polynomial Expression

• definition. A general polynomial expression (GPE) is a GME or a sum of GMEs.

Cartesian Plane

• The Cartesian plane is a plane constructed from two perpendicular real lines, called axes.
  • The horizontal line is called the ${x}$-axis.
  • The vertical line is called the ${y}$-axis.
  • If we place a point ${p}$ on this plane, then that point can be represented as a pair ${(x_p,y_p),}$ where ${x_p}$ is the point's coordinate on the ${x}$-axis and ${y_p}$ is the point's coordinate on the ${y}$-axis.

• The point of intersection of the two axes is called the origin, usually denoted by the letter ${O.}$
• The plane is divided into four regions called quadrants, often labeled ${\text{I}}$ through ${\text{IV}.}$
• The pair ${(x_p, y_p)}$ is an example of an ordered pair.
  • We call the coordinate ${x_p}$ the ordered pair's abscissa and ${y_p}$ the ordinate.
  • For brevity, the notation ${p(x, y)}$ means that the point ${p}$ has the coordinates ${(x,y).}$

The Distance Between Two Points

• Suppose we're given two points ${P(x_P, y_P)}$ and ${Q(x_Q, y_Q).}$ We want to know the distance between these two points.

• We can solve this problem by employing one of the oldest theorems in mathematics:

• pythagorean theorem. In a right triangle, the lengths of the sides are related by the equation

  $$a^2 + b^2 = c^2,$$

  where ${a}$ and ${b}$ are the lengths of the sides forming the right triangle and ${c}$ is the length of the hyptonuse (the side opposite the right angle).

• So, if we suppose that the red line in our diagram is ${c,}$ then the length of that line is found by solving for ${c:}$

  $$c = \sqrt{a^2 + b^2}.$$

• But what are ${a}$ and ${b}$? They must be the lengths of the other sides of the right triangle:

• Thus, we have the following formula.

  • distance formula. The distance ${d}$ between points ${(x_1, y_1)}$ and ${(x_2, y_2)}$ is given by
  $$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.$$

Midpoint Formula

• Given two points ${P(x_1, y_1)}$ and ${Q(x_2, y_2),}$ what is the midpoint of the line segment joining ${P}$ and ${Q}$?
• For example, consider the figure below. The midpoint is marked by the red point. What is the coordinate of this point?

• Here, we apply the midpoint formula:

  • midpoint formula. Given points ${P(x_1, y_1)}$ and ${Q(x_2, y_2),}$ the midpoint of the line segment joining ${P}$ and ${Q}$ has the coordinates:
  $$\left( \dfrac{x_1 + x_2}{2}, \dfrac{y_1 + y_2}{2} \right).$$

Real Functions

• A real function is a function mapping real numbers to real numbers.

Affine Functions

• In the sea of real functions, affine functions are among the most desirable. They're easy to understand because they're easy to gut — internally, they have at most two operations: multiplication and addition.
• definition. Suppose ${A}$ and ${B}$ are real number constants. A function of the form $$f(x) = ax + b$$ is called an affine function. In the case where ${a = 0,}$ then ${f(x) = b,}$ and we call ${f}$ a constant function. In the case where ${b = 0}$ and ${a \neq 0,}$ then ${f(x) = ax,}$ and we call ${f}$ a linear function.
  • The graph of an affine function ${f}$ is a continuous straight line.
  • If ${a \gt 0,}$ then ${f}$ is strictly increasing.
  • If ${a \lt 0,}$ then ${f}$ is strictly decreasing.
  • If ${a \neq 0,}$ then then ${f}$ has a unique zero ${x = -\dfrac{b}{a}.}$
  • To demonstrate these facts, consider the interactive figure below. Adjust the ${a}$ and ${b}$ values to see what happens to our line.

• If ${f}$ is an affine function, then ${f}$ has a constant rate of change equal to the slope of its graph.
• Conversely, if a function ${f}$ has a constant rate of change, then ${f}$ must be an affine function.

Quadratic Functions

• definition. Let ${a,}$ ${b,}$ and ${c}$ be real number constants, with ${a \neq 0.}$ A function of the form

  $$f(x) = ax^2 + bx + c$$

  is called a quadratic function with leading coefficient ${a.}$

• The graph of a quadratic function is called a parabola.
• Given the quadratic function ${f(x) = ax^2 + bx + c:}$
  • If ${a \gt 0,}$ then the parabola has a local minimum at ${x = -\dfrac{b}{2a}}$ and the parabola is convex.
  • If ${a \lt 0,}$ then the parabola has a local maximum at ${x = - \dfrac{b}{2a}}$ and the parabola is concave.
• All parabolas have a special point called the vertex. This point is given by: $$\left( -\dfrac{b}{2a}, \dfrac{4ac - b^2}{4a} \right).$$
• Below is an interactive plot of a quadratic.

• A few observations playing with this plot:
  1. The graph of ${y = ax^2}$ is a parabola with its vertex at the origin. It is similar in shape to ${y = x^2.}$
  2. The parabola ${y = ax^2}$ opens upward if ${a \gt 0,}$ and downward if ${a \lt 0.}$
  3. The parabola ${y = ax^2}$ is narrower than ${y = x^2}$ if ${\lvert a \rvert \gt 1,}$ and wider than ${y = x^2}$ if ${\lvert a \rvert \lt 1.}$
• Importantly, through completing the square, the equation of the parabola ${y = ax^2 + bx + c}$ can always be written in the form $$y = (x - h)^2 + k,$$ where ${(h,k)}$ is the parabola's vertex and the axis of symmetry is the line ${x = h.}$

The Discriminant

• The quantity ${b^2 - 4ac}$ is called the discriminant of ${ax^2 + bx + c.}$
• The equation $$y = a\left(x + \dfrac{b}{2a}\right)^2 + \dfrac{4ac - b^2}{4a}$$ is called the canonical equation of the parabola ${y = ax^2 + bx + c.}$
• Notice that the parabola is symmetric about the vertical line ${x = -\dfrac{b}{2a},}$ passing through the vertex. This line, called the axis of symmetry, is parallel to the ${y}$-axis.

Quadratic Formula

• The quadratic formula provides a quick and easy way to find the roots of a polynomial equation.
• formula. Let ${a,}$ ${b,}$ and ${c}$ be constants in ${\reals.}$ The roots of the equation ${ax^2 + bx + c = 0}$ are given by the formula $$x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}.$$
• One can gain many insights about a particular quadratic by simply examining its determinant.
• corollary. Suppose ${u}$ is the quadratic ${ax^2 + bx + c = 0}$ with ${a \neq 0,}$ ${b,}$ and ${c}$ as real constants. Then the following are true:
  1. If ${b^2 - 4ac = 0,}$ then ${u}$ is tangent to the ${x}$-axis and has one (repeated) real root.
  2. If ${b^2 - 4ac \gt 0,}$ then the parabola has two distinct real roots.
  3. If ${b^2 - 4ac \lt 0,}$ then the parabola has two complex roots.
  4. If ${b^2 - 4ac \gt 0,}$ then ${u}$ has the same sign as ${A.}$

The Square Function

• definition. The square function is a function of the form $$f(x) = x^2.$$
• Suppose ${f}$ is the square function ${f(x) = x^2.}$ Then:
  • ${f}$ is an even function; its graph is symmetric about the ${y}$-axis.
  • ${f}$ is an increasing function for ${x \gt 0.}$
  • ${f}$ is a decreasing function for ${x \lt 0.}$
  • ${f}$ has a minimum value at ${x = 0.}$
  • ${f(x)}$ becomes unbounded positive as ${x}$ becomes unbounded positive or negative.

Even Power Functions

• The graphs of ${y = x^{2n + 2},}$ where ${n \in \natnums}$ (e.g., ${y = x^2,}$ ${y = x^4,}$ ${y = x^6,}$ etc.) look similar to one another.
• For ${-1 \leq x \leq 1,}$ the higher the exponent, the flatter the graph is along the ${x}$-axis.
• This follows from the fact that, where ${-1 \leq x \leq 1,}$ ${\lvert x \rvert \implies \ldots \lt x^6 \lt x^4 \lt x^2 \lt 1.}$

Cubic Functions

• definition. Let ${a,}$ ${b,}$ ${c,}$ and ${d}$ be real number constants with ${a \neq 0.}$ A function of the form

  $$f(x) = ax^3 + bx^2 + cx + d$$

  is called a cubic function.

• The graph of a cubic function is a cubic curve. As seen from its general form, the cubic curve depends on four parameters: ${a,}$ ${b,}$ ${c,}$ and ${d.}$ Where ${a = 1,}$ ${b = 0,}$ ${c = 0,}$ ${d = 0,}$ we get the simplest cubic:

  $$f(x) = x^3.$$

• The graph of the cubic function ${f(x) = x^3}$ is concave when ${x \lt 0}$ and convex when ${x \gt 0.}$ Examining the behavior of ${f(x) = x^3,}$ we see that ${f}$ is an increasing odd function.

The Cube Function

• definition. The cube function ${f}$ is defined as $$f(x) = x^3.$$
• Some properties of the cube function ${f:}$
  • ${f}$ is an odd function; its graph is symmetric about the origin.
  • ${f}$ is an increasing function for all real numbers.
  • ${f}$ has no maximum or minimum.
  • ${f(x)}$ becomes unbounded positive as ${x}$ becomes unbounded positive.
  • ${f(x)}$ becomes unbounded negative as ${x}$ becomes unbounded negative.

Odd Power Functions

• The graphs of ${y = x^{2n + 3},}$ ${n \in \natnums}$ (e.g., ${y = x^3,}$ ${y = x^5,}$ ${y = x^7,}$ etc.) look similar to one another. For ${-1 \leq x \leq 1,}$ the higher the exponent, the flatter the graph is to the ${x}$-axis, since $$\lvert x \rvert \lt 1 \implies \ldots \lt \lvert x^7 \rvert \lt \lvert x^5 \rvert \lt \lvert x^3 \rvert \lt 1.$$
• For ${\lvert x \rvert \geq 1,}$ the higher the exponent, the steeper the graph will be. This follows from the fact that $$\lvert x \rvert \gt 1 \implies \ldots \gt \lvert x^7 \rvert \gt \lvert x^5 \rvert \gt \lvert x^3 \rvert \gt 1.$$

Polynomial Functions

• definition. A polynomial function ${f}$ is a function of the form $$f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0,$$ where ${n \in \natnums}$ and each ${a_i}$ is a constant. If ${a_n \neq 0,}$ the degree of the polynomial function is ${n.}$
• Functions of the form ${y = x^n}$ with ${n \in \natnums}$ (e.g., the square and cubic functions) are called power functions.
  • They are the simplest polynomial functions, and can be interpreted as the building blocks of more complex polynomial functions.

Graphs of Polynomial Functions

• Suppose ${f}$ is a polynomial function.
• If ${\deg(u) = 0}$ or ${\deg(u) = 1,}$ then the graph of ${f}$ is a line.
• If ${\deg(u) \geq 2,}$ then the graph of ${f}$ is an unbroken smooth curve.
• The graph of a polynomial function of degree ${n}$ has at most ${n - 1}$ turning points.
• Regardless of the degree, as ${\lvert x \rvert}$ gets very large, ${\lvert y \rvert}$ grows very large (except, of course, for the constant function).
• The following theorem, stated informally, is often helpful for getting a rough idea of a given polynomial's behavior.
• theorem. Suppose ${f}$ is a polynomial function of the form $$f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0$$ with ${a_n \neq 0.}$ Then $$f(x) \approx a_n x^n$$ when ${\lvert x \rvert}$ is very large.

Rational Functions

• definition. A rational function ${r}$ is a function of the form $$r(x) = \dfrac{f(x)}{g(x)},$$ where ${f}$ and ${g}$ are polynomial functions.
• The graphs of rational functions differ from that of polynomial functions significantly because they employ the operation of division.
• That operation, as we all know, has a strict restriction: We cannot divide by ${0.}$
• To illustrate, consider the graph of ${f(x) = 1/x}$ (drag the slider to change the ${x}$ value of the point on the graph).

• Notice that the graph comprises two separate curves, rather than the single continuous curve we see with polynomials.
• We call these curves the branches or arms of the rational function's graphs. The function ${f(x) = 1/x}$ has the following properties:
  1. ${f}$ is an odd function; its graph is symmetric about the origin.
  2. ${f}$ is decreasing for ${x \lt 0}$ and for ${x \gt 0.}$
  3. ${f}$ has no maximum or minimum.
  4. ${f(x)}$ approaches zero from above as ${x}$ becomes unbounded positive and from below as ${x}$ becomes unbounded negative.
  5. ${f(x)}$ becomes unbounded negative as ${x}$ approaches zero from the left and unbounded positive as ${x}$ approaches zero from the right.
• Now consider the function ${g(x) = \dfrac{1}{x^2}.}$ This function has the following plot:

• Some properties of the graph of ${g(x) = \dfrac{1}{x^2}:}$
  1. ${g}$ is an even function; its graph is symmetric about the ${y}$-axis.
  2. ${g}$ is increasing on ${(-\infty, 0)}$ and decreasing on ${(0, \infty).}$
  3. ${g}$ has no maximum or minimum.
  4. ${g(x)}$ approaches zero from above as ${x}$ becomes unbounded positive or negative.
  5. ${g(x)}$ becomes unbounded positive as ${x}$ approaches zero from the left or right.

Graphs of 1/𝑥ⁿ

• In general, the graph of ${f(x) = 1/x^n}$ where ${n \in \Z^+}$ will resemble either ${1/x^2}$ or ${1/x.}$
• If ${n}$ is an even integer, then ${f}$'s graph will resember ${1/x^2.}$
• If ${n}$ is an odd integer, then ${f}$'s graph will resember ${1/x.}$

Exponential Functions

• definition. Let ${b}$ denote an arbitrary constant, with ${b \gt 0}$ and ${b \neq 1.}$ The exponential function ${f}$ with base ${b}$ is a function of the form $$f(x) = ab^{kx},$$ where ${a,}$ ${b,}$ and ${k}$ are real constants.
• example. ${f(x) = 2^x.}$ Below is ${f}$'s graph.

• Some observations of ${f(x) = 2^x}$ and its graph:
  1. The domain of ${f}$ is ${\reals.}$ The range is ${\reals^+.}$
  2. The ${y}$-intercept is ${1.}$ The graph never crosses the ${x}$-axis, so ${f}$ has no ${x}$-intercept.
  3. For ${x \gt 0,}$ ${f}$ increases rapidly.
  4. For ${x \lt 0,}$ ${f}$ rapidly approaches the ${x}$-axis. The ${x}$-axis is a horizontal asymptote for the graph; ${f}$ approaches zero from above as ${x}$ becomes unbounded negative.
  5. ${f}$ becomes unbounded positive as ${x}$ becomes unbounded positive.
  6. The function is one-to-one.
• More generally, for the function ${f(x) = b^x}$ with ${b \gt 0}$ and ${b \neq 1,}$
  1. For ${b \gt 1,}$ ${f}$ approaches zero from above as ${x}$ becomes unbounded negative, and becomes unbounded positive as ${x}$ becomes unbounded positive.
  2. For ${0 \lt b \lt 1,}$ ${b^x}$ becomes unbounded positive as ${x}$ becomes unbounded negative and approaches zero from above as ${x}$ becomes unbounded positive.

The Exponential Function ${y = e^x}$

• Euler's number ${e}$ is an irrational number: $$e = 2.7182818284\ldots$$
• This number is particularly special because it makes handling exponentials much simpler. The graph of ${f(x) = e^x:}$

Logarithmic Functions

• To study the logarithmic functions, we must first study logarithms as expressions and operators.

Logarithms

• definition. The expression $$\log_b x$$ means, "the exponent to which ${b}$ must be raised to yield ${x.}$" We read the expression, "log base ${b}$ of ${x}$" or "the logarithm of ${x}$ to the base ${b.}$"
• The logarithm is a binary operator. It takes two arguments: First, a base ${b,}$ and second, some real number ${x:}$ $$\log(b,x) \mapsto y$$
• When we evaluate this expression, we get a real number ${y}$ that satisfies the following relation: $$b^y = x.$$
• In other words, the logarithm answers the question, "What is the number I must raise ${b}$ to, such that I get ${x?}$" Following convention, instead of writing ${\log(b,x),}$ we will instead write: $$\log_b x.$$
• Using this notation, we state the following equivalence: $$y = \log_b x \iff x = b^y.$$
• The equation ${y = \log_b x}$ is in logarithmic form, and the equation ${x = b^y}$ is in exponential form. Some logarithms of a particular base are given special symbols as operators:

• In this text, we will use the following conventions: ${\log}$ is assumed to be ${\log_{10},}$ ${\ln}$ is the natural log, ${\lg}$ is the binary log, and all other logarithms will have a specified base.

Manipulating Logarithms

• The following properties are essential to working with logarithms efficiently and cleanly.

  • ${\log_n b = 1.}$
  • ${\log_n 1 = 0.}$
  • ${\log_n ab = \log_n a + \log_n b.}$
  • ${\log_n \frac{a}{b} = \log_n a - \log_n b.}$

  • ${\log_n \frac{1}{x} = - \log_n x.}$
  • ${\log_n x^r = r \log_n x}$ for all ${x \in \reals.}$
  • ${n^{\log_n x} = x.}$
  • ${\log_n n^x = x}$ for all ${x \in \reals.}$

• The following equation is another useful fact to keep in mind. As its name implies, this equation allows us to rewrite logarithms as logarithms of different bases.

  • change of base. ${\log_a x = \dfrac{\log_b x}{\log_b a}.}$

Logarithmic Behavior

• Below are graphs of various logarithmic functions:

• Notice that ${f(x)}$ grows or increases very slowly. It is always increasing as ${x}$ increases, just very, very slowly. Given a function of the form ${f(x) = \log_b x,}$ with ${b \gt 1,}$ we have the following properties:
  1. ${\text{dom}(f) = \reals^+}$ (all positive real numbers).
  2. ${\text{ran}(f) = \reals.}$
  3. There is no ${y}$-intercept.
  4. The ${x}$-intercept is ${1.}$
  5. The assymptote is ${x = 0}$ (i.e., ${y}$-axis).
  6. End behavior: ${f}$ becomes unbounded negative as ${x}$ approaches ${0}$ from the right and becomes unbounded positive as ${x}$ becomes unbounded positive.

Trigonometric Functions

• definition. Let ${\theta}$ be an angle in standard position and let ${P(x,y)}$ be the point where the terminal side of ${\theta}$ intersects the unit circle. The trigonometric functions of ${\theta}$ are defined as follows:

  $$\cos \theta = x \\[2em] \sin \theta = y \\[2em] \tan \theta = \dfrac{y}{x} ~~ (x \neq 0) \\[2em] \sec \theta = \dfrac{1}{x} ~~ (x \neq 0) \\[2em] \csc \theta = \dfrac{1}{y} ~~ (y \neq 0) \\[2em] \cot \theta = \dfrac{x}{y} ~~ (y \neq 0).$$

• To see the relationship between the trigonometric ratios and the ${xy}$-plane, consider the diagram below. Recall that the unit circle is the graph of the equation ${x^2 + y^2 = 1.}$

• Above, we see an imposed triangle with sides ${a,}$ ${b,}$ and ${c.}$ The ${x}$-coordinate is given by the length of ${c,}$ and the ${y}$-coordinate is given by the length of ${b.}$ We know that these lengths are given by the trigonometric ratios, sine and cosine, with respect to ${\theta.}$ Thus, we have:

Trigonometric Notational Conventions

• The expression ${\sin \theta}$ really means ${\sin(\theta),}$ ${\cos \theta}$ means ${\cos(\theta),}$ and so on. For historical reasons, the parentheses are omitted. Parentheses are also omitted in the multiplication of trigonometric functions. E.g., ${(\sin \theta)(\cos \theta)}$ is usually written ${\sin \theta \cos \theta.}$ Similarly, ${a(\sin b)}$ is reduced to ${a \sin b.}$