[WIP] Calculus — Notes

12.10.2023 — summary, book — 11 min read

Intro

Chapter 1: Functions and Graphs

A function f consists of a set of inputs, a set of outputs, and a rule for assigning each input to exactly one output.
The set of inputs is called the domain of the function.
The set of outputs is called the range of the function.
For any function, when we know the input, the output is determined, so we say that the output is a function of the input.
Given two sets A and B, a set with elements that are ordered pairs (x, y), where x is an element of A and y is an element of B, is a relation from A to B.
A function is a special type of relation.
For a general function f with domain D, we often use x to denote the input and y to denote the output associated with x.
When doing so, we refer to x as the independent variable and y as the dependent variable, because it depends on x.
Using function notation, we write y = f(x), and we read this equation as “y equals f of x.”
A function can be visualized by plotting points (x, y) in the coordinate plane where y = f(x). The graph of a function is the set of all these points.
Every function has a domain. However, sometimes a function is described with no specific domain given. In this case, the domain is taken to be the set of all real numbers x for which f(x) is a real number.
When describing a set with an infinite number of elements, it is often helpful to use:
- Set-builder notation: { x | 1 > x < 5 } or
- Interval notation: (1, 5)
Piecewise functions are defined using different equations for different parts of their domain. Eg: $f(x) = \begin{cases} 3x + 1 &\text{if } x \geq 2 \\ x^2 &\text{if } x < 2 \end{cases}$
Functions can be represented through:
- Tables
- Formulas
- Graphs
Given an algebraic formula for a function f, the graph of f is the set of points (x, f(x)), where x is in the domain of f and f(x) is in the range.
Those values of x where f(x) = 0 are called the zeros of a function.
The zeros determine where the graph of f intersects the x-axis.
The y-intercept is given by (0, f(0)).
Since a function has exactly one output for each input, the graph of a function can have, at most, one y-intercept.
Vertical line test: Given a function f, every vertical line that may be drawn intersects the graph of f no more than once. If any vertical line intersects a set of points more than once, the set of points does not represent a function.
We say that a function f is increasing on the interval I if for all x₁, x₂ ∈ I, f(x₁) ≤ f(x₂) when x₁ < x₂.
We say that a function f is strictly increasing on the interval I if for all x₁, x₂ ∈ I, f(x₁) < f(x₂) when x₁ < x₂.
We say that a function f is decreasing on the interval I if for all x₁, x₂ ∈ I, f(x₁) ≥ f(x₂) when x₁ < x₂.
We say that a function f is strictly decreasing on the interval I if for all x₁, x₂ ∈ I, f(x₁) > f(x₂) when x₁ < x₂.
To combine functions using mathematical operators, write the functions with the operator and simplify. E.g:
- Sum: $(f + g)(x) = f(x) + g(x)$
- Difference: $(f − g)(x) = f(x) − g(x)$
- Product: $(f · g)(x) = f(x)g(x)$
- Quotient: $(f / g)(x) = f(x) / g(x)$ for $g(x) ≠ 0$
Linear functions have the form f(x) = ax + b, where a and b are constants.
The graph of any linear function is a line.
The slope is the change in y for each unit change in x.
The slope measures both the steepness and the direction of a line:
- If the slope is zero, the line is horizontal.
- The magnitude of the slope determines the slope's steepness.
- The sign of the slope determines the slope's direction:
  - If the slope is positive, the line points upward when moving from left to right.
  - If the slope is negative, the line points downward when moving from left to right.
To calculate the slope of a line, we need to determine the ratio of the change in y versus the change in x. To do so, we choose any two points (x₁, y₁) and (x₂, y₂) on the line and calculate $Slope (m) = \frac{Δy}{Δx} = \frac{y_2 - y_1}{x_2 - x_1}$ .
The slope of a line is independent of the choice of points (x₁, y₁) and (x₂, y₂) on the line.
The relationship between slope and the formula for a linear function f(x) = ax + b:
- Evaluating the function f at x = 0, we see that (0, b) is a point on this line.
- Evaluating this function at x = 1, we see that (1, a + b) is also a point on this line.
- Using the slope formula, this gives us: $Slope = \frac{(a + b) − b}{1 − 0} = a$
- We conclude that the formula f(x) = mx + b tells us the slope, m, and the y-intercept, (0, b). This form of expressing a linear function is called the slope-intercept form.
Consider a line passing through the point (x₁, y₁) with slope m. The equation y − y₁ = m (x - x₁) is the point-slope form for that linear function.
The standard form of a line is given by the equation ax + by = c, where a and b are both not zero. This form is more general because it allows for a vertical line, x = k.
A linear function is a special type of a more general class of functions: polynomials.
A polynomial function is any function consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables.
A univariate polynomial function can always be written (or rewritten) in the form: $f(x) = a_{n}x^{n} + a_{n-1}x^{n-1} + … + a_{1}x^{1} + a_{0}x^{0}$ $f (x) = a_{n} x^{n} + a_{n - 1} x^{n - 1} + \dots + a_{1} x^{1} + a_{0} x^{0}$ Where:
- $a_0$ , ..., $a_n$ are constants that are called the coefficients of the polynomial and
- x is the variable.
The exponent on a variable in a term is called the degree of that variable in that term.
The degree of a polynomial is the largest degree of any term with nonzero coefficient.
The leading coefficient is the coefficient of the term with the largest degree.
A linear function of the form f(x) = mx + b is a polynomial of degree 1 if m ≠ 0 and degree 0 if m = 0.
A polynomial of degree 0 is also called a constant function.
The function f(x) = 0 is called the zero function and its degree is undefined.
A polynomial function of degree 2 is called a quadratic function and has the form f(x) = ax2 + bx + c, where a ≠ 0.
A polynomial function of degree 3 is called a cubic function.
A power function is any function of the form f(x) = axᵇ, where a and b are any real numbers.
A power function is also a polynomial function if b is a positive integer.
If b is even, then f(x) = axᵇ is an even function because f(−x) = a(−x)ᵇ = axᵇ if b is even.
If b is odd, then f(x) = axᵇ is an odd function because f(−x) = a(−x)ᵇ = −axᵇ if b is odd.
The end behavior of a function f(x) is what happens to the values of f(x) as x → ∞ and as x → −∞.
The value of f(x) either:
- Approaches a finite number c and we say “f(x) approaches c as x goes to infinity,” and we write f(x) → c as x → ∞. Also, the line y = c is a horizontal asymptote for the function. E.g, f(x) = 2 + 1/x.
- Approaches ±infinity and we say “f(x) approaches infinity as x approaches infinity,” and we write f(x) → ∞ as x → ∞. E.g, f(x) = 3x².
End behavior of polynomials:
- For a quadratic function f(x) = ax² + bx + c.
  - If a > 0, the values f(x) → ∞ as x → ±∞.
  - If a < 0, the values f(x) → −∞ as x → ±∞.
  - Since the graph of a quadratic function is a parabola, the parabola opens upward if a > 0; the parabola opens downward if a < 0.
- For a cubic function f(x) = ax³ + bx2 + cx + d.
  - If a > 0, then f(x) → ∞ as x → ∞ and f(x) → −∞ as x → −∞.
  - If a < 0, then f(x) → −∞ as x → ∞ and f(x) → ∞ as x → −∞.
- The behavior for higher-degree polynomials can be analyzed similarly.
The zeros of a polynomial function are where the function intersects the x-axis. To determine where a function f intersects the x-axis, we need to solve the equation f(x) = 0 for x:
- For a linear function f(x) = mx + b, the x-intercept is given by (−b/m, 0).
- For a quadratic function, we can find the zeroes of the quadratic equation ax² + bx + c = 0:
  - Sometimes it's possible to factor the equation
  - Make use of the quadratic formula x = (−b ± √(b2 − 4ac)) ÷ 2a. The discriminant part of the formula is b² − 4ac.
    - If the discriminant b² − 4ac > 0, there are two real numbers that satisfy the quadratic equation.
    - If b² − 4ac = 0, there is only real number one solution.
    - If b² − 4ac < 0, no real numbers satisfy the quadratic equation.
- In the case of higher-degree polynomials, it may be more complicated to determine where the graph intersects the x-axis.
A mathematical model is a method of simulating real-life situations with mathematical equations.
Physicists, engineers, economists, and other researchers develop models by combining observation with quantitative data to develop equations, functions, graphs, and other mathematical tools to describe the behavior of various systems accurately.
Models are useful because they help predict future outcomes.
An algebraic function is one that involves addition, subtraction, multiplication, division, rational powers, and roots.
The two types of algebraic functions are
- Rational functions are any function of the form f(x) = p(x) / q(x), where p(x) and q(x) are polynomials.
- Root functions are a power function of the form f(x) = x^1/n, where n is a positive integer greater than one.
For root functions, If n is even, the domain of f(x) = x^1/n is [0, ∞).
For root functions, If n is odd, the domain of f(x) = x^1/n is [-∞, ∞) and the function is an odd function.
Transcendental functions cannot be described by basic algebraic operations. They are said to “transcend,” or go beyond, algebra.
The most common transcendental functions are trigonometric, exponential, and logarithmic functions.
A piecewise-defined function is defined by different formulas on different parts of its domain. An example is the absolute-value function: $f(x) = \begin{cases} a &\text{if } b \\ c &\text{if } d \end{cases}$
To graph a piecewise-defined function, we graph each part of the function in its respective domain, on the same coordinate system. If the formula for a function is different for x < a and x > a, we need to pay special attention to what happens at x = a when we graph the function. Sometimes the graph needs to include an open or closed circle to indicate the value of the function at x = a. An open circle is used to denote that a graph doesn’t define the function output at x = a. A closed circle is used for the opposite effect.
Given the base function y = f(x), we can apply multiple function transformations to it: y = cf(a(x + b)) + d. When function transformations are combined, you can follow the order below to get the transformed function’s graph from the base function’s graph:
- Horizontal shift of the graph of y = f(x).
- Horizontal scaling of the graph of y = f(x + b) by a factor of |a|. If a < 0, reflect the graph about the y-axis.
- Vertical scaling of the graph of y = f(a(x + b)) by a factor of |c|. If c < 0, reflect the graph about the x-axis.
- Vertical shift of the graph of y = cf(a(x + b)).

An enumeration of the different function transformations and their related effects on the graph of a function: | Transformation of f; Where c > 0 | Effect on the graph of f | |--------------------------------------|--------------------------------------------------------------| | f(x) + c | Vertical shift up c units | | f(x) − c | Vertical shift down c units | | f(x + c) | Shift left by c units | | f(x − c) | Shift right by c units | | cf(x) | Vertical stretch if c > 1;
Vertical compression if 0 < c < 1 | | f(cx) | Horizontal stretch if 0 < c < 1;
Horizontal compression if c > 1 | | −f(x) | Reflection about the x-axis | | f(−x) | Reflection about the y-axis |

Trigonometric Functions

Almost any repetitive or cyclical motion can be modeled by some combination of trigonometric functions.
Radians are a more natural measurement of angles compared to degrees because they are related directly to the unit circle, a circle with radius 1.
The radian measure of an angle θ is the arc length s of the associated arc on the unit circle. I.e. the angle corresponding to the arc of length 1 has radian measure 1.
Since an angle of 360° corresponds to the circumference of a circle, or an arc of length 2π (on the unit circle), we conclude that an angle with a degree measure of 360° has a radian measure of 2π.
Use the fact that 180° is equivalent to π radians as a conversion factor: 1 = π rad / 180° = 180° / π rad
Let P = (x, y) be a point on the unit circle centered at the origin O. Let θ be an angle with an initial side along the positive x-axis and a terminal side given by the line segment OP. The trigonometric functions are then defined as:
- sinθ = y
- cscθ = 1/y
- cosθ = x
- secθ = 1/x
- tanθ = y/x
- cotθ = x/y
  If x = 0, secθ and tanθ are undefined. If y = 0, then cotθ and cscθ are undefined.
For a point P = (x, y) on a circle of radius r with a corresponding angle θ, the coordinates x and y satisfy:
- cosθ = x / r
- x = r / cosθ
- sinθ = y / r
- y =r / sinθL

The ratios of the side lengths of a right triangle can be expressed in terms of the trigonometric functions evaluated at either of the acute angles of the triangle. Let θ be one of the acute angles. Let A be the length of the adjacent leg, O be the length of the opposite leg, and H be the length of the hypotenuse. By inscribing the triangle into a circle of radius H, we see that A, H, and O satisfy the following relationships with θ:
- sinθ = O / H
- cosθ = A / H
- tanθ = O / A
- cscθ = H / O
- secθ = H / A
- cotθ = A / O
A trigonometric identity is an equation involving trigonometric functions that is true for all angles θ for which the functions are defined. We can use the identities to help us solve or simplify equations.
The main trigonometric identities are:
- Reciprocal identities:
  - tanθ = sinθ / cosθ
  - cotθ = cosθ / sinθ
  - cscθ = 1 / sinθ
  - secθ = 1 / cosθ
- Pythagorean identities:
  - sin²θ + cos²θ = 1
  - 1 + tan²θ = sec²θ
  - 1 + cot²θ = csc²θ
- Addition and subtraction formulas:
  - sin(α ± β) = sinα cosβ ∓ cosα sinβ
  - cos(α ± β) = cosα cosβ ∓ sinα sinβ
- Double-angle formulas:
  - sin(2θ) = 2sinθ cosθ
  - cos(2θ) = 2cos²θ − 1 = 1 − 2sin²θ = cos²θ −sin²θ
The trigonometric functions are periodic functions.
The period of a function f is defined to be the smallest positive value p such that f(x + p) = f(x) for all values x in the domain of f.
The sine, cosine, secant, and cosecant functions have a period of 2π. Since, the angle θ and θ +2π correspond to the same point on the unit circle.
Since the tangent and cotangent functions repeat on an interval of length π, their period is π. [Figure1.34]
Transformations can be applied to trigonometric functions. E.g. f(x) = A sin(B(x − α)) + Cwhere:
- The constant α causes a horizontal or phase shift.
- The factor B changes the period. This transformed sine function will have a period 2π / |B|.
- The factor A results in a vertical stretch by a factor of |A|. We say |A| is the amplitude of f.
- The constant C causes a vertical shift. Figure1.35
The graph of y = cosx is the graph of y = sinx shifted to the left π / 2 units: cosx = sin(x + π/2).
Similarly, we can view the graph of y = sinx as the graph of y = cosx shifted right π/2 units: sinx = cos(x − π/2).

Inverse functions

A function maps elements in the domain of f to elements in the range of f. The inverse function maps each element from the range of f back to its corresponding element from the domain of f.
Given a function f with domain D and range R, its inverse function (if it exists) is the function f⁻¹ with domain R and range D such that f⁻¹(y) = x if f(x) = y. In other words, for a function f and its inverse f⁻¹:
- f⁻¹(f(x)) = x for all x in D, and
- f(f⁻¹(y)) = y for all y in R.
f⁻¹ is read as f inverse. The -1 is not used as not used as an exponent.
The range of f becomes the domain of f⁻¹ and the domain of f becomes the range of f⁻¹.
A function f is a one-to-one function if f(x1) ≠ f(x2) when x1 ≠ x2.
The horizontal line test: A function f is one-to-one if and only if every horizontal line intersects the graph of f no more than once.
A strategy to find the inverse function of function f:
- Solve the equation y = f(x) for x.
- Interchange the variables x and y and write y = f⁻¹(x). This is because by convention, x represent the independent variable and y represent the dependent variable. Representing the inverse function in this way is also helpful when graphing a function f and its inverse f⁻¹ on the same axes.
The relationship between the graph of a function f and the graph of its inverse:
- Consider the graph of f and a point (a, b) on the graph.
- Since b = f(a), then f⁻¹(b) = a.
- Therefore, when we graph f⁻¹, the point (b, a) is on the graph.
- As a result, the graph of f⁻¹ is a reflection of the graph of f about the line y = x.
For a non one-to-one function f we can choose a subset of the domain of f such that the function is one-to-one. This subset is called a restricted domain. By restricting the domain of f, we can define a new function g such that the domain of g is the restricted domain of f and g(x) = f(x) for all x in the domain of g. Then we can define an inverse function for g on that domain.
The inverse sine function, denoted sin⁻¹ or arcsin, and the inverse cosine function, denoted cos⁻¹ or arccos, are defined on the domain D = {x| − 1 ≤ x ≤ 1} as follows:
- sin⁻¹(x) = y if and only if sin(y) = x and −π/2 ≤ y ≤ π/2;
- cos⁻¹(x) = y if and only if cos(y) = x and 0 ≤ y ≤ π.
The inverse tangent function, denoted tan⁻¹ or arctan, and inverse cotangent function, denoted cot⁻¹ or arccot, are defined on the domain D = {x| − ∞ < x < ∞} as follows:
- tan⁻¹(x) = y if and only if tan(y) = x and − π 2 < y< π 2;
- cot⁻¹(x) = y if and only if cot(y) = x and 0 < y < π.
The inverse cosecant function, denoted csc⁻¹ or arccsc, and inverse secant function, denoted sec⁻¹ or arcsec, are defined on the domain D = {x |x| ≥ 1} as follows:
- csc⁻¹(x) = y if and only if csc(y) = x and − π 2 ≤ y≤ π 2, y ≠ 0;
- sec⁻¹(x) = y if and only if sec(y) = x and 0 ≤ y ≤ π, y ≠ π/2.
The sine function is one-to-one on an infinite number of intervals, but the standard convention is to restrict the domain to the interval [−π/2, π/2]. By doing so, we define the inverse sine function on the domain [−1, 1] such that for any x in the interval [−1, 1], the inverse sine function tells us which angle θ in the interval [−π/2, π/2] satisfies sinθ = x.
When evaluating an inverse trigonometric function, the output is an angle.

Exponential and logarithmic functions

Any function of the form f(x) = bˣ, where b > 0, b ≠ 1, is an exponential function with base b and exponent x.
The domain of the exponential function is (−∞, ∞) and the range is (0, ∞).
For b > 1, f(x) = bˣ is increasing on (−∞, ∞) and bˣ → ∞ as x → ∞, whereas bˣ → 0 as x → −∞.
For 0 < b < 1, f(x) = bˣ is decreasing on (−∞, ∞) and bˣ → 0 as x → ∞, whereas bˣ → 0 as x → −∞.
Exponential functions have constant bases and variable exponents.
Note that a function of the form f(x) = xᵇ for some constant b is not an exponential function but a power function.
An exponential function grows faster than a power function.
Properties of exponents:
- If x is a positive integer, then bˣ = b · b ⋯ b (with x factors of b).
- If x is a negative integer, then bˣ = 1/bˣ.
- If x is zero, then bˣ = 1.
- If x is a rational number (x = c/d; where c and d are integers), then bˣ = ᵈ√bᶜ.
Laws of Exponents (For any constants a > 0, b > 0, and for all x and y,)
- bˣ · bʸ = bˣ⁺ʸ
- bˣ / bʸ = bˣ⁻ʸ
- (bˣ)ʸ = bˣʸ
- (ab)ˣ = aˣbˣ
- aˣ / bˣ = (a/b)ˣ
The function f(x) = eˣ is called the natural exponential function.
e ≈ 2.718282.
The function f(x) = eˣ is the only exponential function with tangent line at x=0 that has aslope of 1.
As we see later in the text, having this property makes the natural exponential function the most simple exponential function to use in many instances.
The inverse of exponential functions are logarithm functions.
These come in handy when we need to consider any phenomenon that varies over a wide range of values, such as pH in chemistry or decibels in sound levels.
For any b > 0, b ≠ 1, the logarithmic function with base b, denoted logb, has domain (0, ∞) and range (−∞, ∞), and satisfies logb(x) = y if and only if bʸ = x.
since y = logb(x) and y = bx are inverse functions, logb(bx) = xandb logb(x) = x.
The most commonly used logarithmic function is the function loge. Since this function uses natural e as its base, it is called the natural logarithm. Here we use the notation ln(x) or lnx to mean loge(x).
Properties of Logarithms If a,b,c>0,b≠1, and r is any real number, then 1. logb(ac)=logb(a)+logb(c) (Product property) 2. logb⎛ ⎝ a c⎞ ⎠=logb(a)−logb(c) (Quotient property) 3. logb(ar)=rlogb(a) (Power property)
The common logarithm is log10 or log.
Change-of-Base Formulas Let a > 0, b > 0, and a ≠ 1, b ≠1. 1. ax =bxlogba for any real number x. If b = e, this equation reduces to ax = e xlogea = exlna. 2. logax = logbx logba for any real number x > 0. If b = e, this equation reduces to logax = lnx Proof lna .
The hyperbolic functions are defined in terms of certain combinations of ex and e−x. These functions arise naturally in various engineering and physics applications, including the study of water waves and vibrations of elastic membranes. Another common use for a hyperbolic function is the representation of a hanging chain or cable, also known as a catenary.