MATH 221: First Semester Calculus - Comprehensive Study Material

Chapter 1: Numbers and Functions

The foundational subject of this calculus course is "functions of one real variable." To understand functions, it is essential to first grasp the concept of a real number.

What is a Number?

Numbers are categorized into several types:

Positive Integers: The counting numbers {1, 2, 3, 4, ...}.
Zero: The number 0.
Negative Integers: The numbers {... , -4, -3, -2, -1}.
Integers (Whole Numbers): The combination of positive integers, zero, and negative integers.
Rational Numbers: Numbers obtained by dividing one integer by another non-zero integer (e.g., 1/2, 2/3, -1/4). Rational numbers can be added, subtracted, multiplied, and divided, with the result always being a rational number (excluding division by zero). They can be represented as finite or repeating infinite decimal expansions (e.g., 279/25 = 11.16, 1/3 = 0.333..., 1/7 = 0.142857...).
Irrational Numbers: Numbers that cannot be expressed as a simple fraction of two integers. The classic example is the square root of two (\sqrt{2}), which cannot be written as m/n such that (m/n)^2 = 2. These numbers have unending and non-repeating decimal expansions (e.g., \sqrt{2} \approx 1.41421356...). The existence of \sqrt{2} can be geometrically demonstrated using the Pythagorean theorem on a right triangle with sides of length 1.
Real Numbers: The set of all rational and irrational numbers, represented by possibly unending decimal expansions. All numbers used in this first semester of calculus are real numbers.
Imaginary Numbers: Numbers whose square is -1 (e.g., \sqrt{-1}, denoted as i). These are introduced in second-semester calculus.

The Real Number Line and Intervals

Real numbers are visualized as points on a straight line, with a designated origin (0). Directions are typically labeled "left" for negative values and "right" for positive values. The distance between two numbers x and y on the number line is given by |x - y|, which is always non-negative.

A collection of all real numbers between two given real numbers forms an interval. Notation for intervals includes:

Open Interval (a, b): The set of all real numbers x such that a < x < b.
Half-Open/Half-Closed Intervals [a, b) or (a, b]: Include one endpoint but not the other (e.g., a \le x < b or a < x \le b).
Closed Interval [a, b]: Includes both endpoints, a \le x \le b.

Endpoints can also be \infty or -\infty (e.g., (-\infty, 2] includes all real numbers less than or equal to 2). Intervals are drawn with a filled dot for included endpoints and an open dot for excluded endpoints.

Set Notation

A set is a collection of mathematical objects, often real numbers. Common notation:

Set-Builder Notation: A = \{x | x \text{ satisfies this or that condition}\}. For example, (a, b) = \{x | a < x < b\}\}.
Union (\cup): The union of sets A and B, denoted A \cup B, contains all numbers belonging to A or B (or both).
Intersection (\cap): The intersection of sets A and B, denoted A \cap B, contains all numbers belonging to both A and B.

Examples of sets include D = \{1, 2, 3\}\} or the set of solutions to an equation like x^3 - 4x^2 + 1 = 0.

Chapter 2: Derivatives (1)

This chapter introduces the concept of derivatives by examining two classical problems: finding the tangent to a curve and determining instantaneous velocity. Both problems lead to the idea of a limit.

The Tangent to a Curve

Given a function y = f(x) and its graph, finding the tangent line at a point P on the graph involves a limiting process. A secant line is drawn through point P and another point Q on the graph. As Q is moved closer and closer to P, the secant lines approach a specific line, which is defined as the tangent line. The challenge is that Q cannot be equal to P, as two distinct points are needed to define a line. This problem is resolved by the concept of a limit.

Example: Tangent to a Parabola

Consider the function f(x) = x^2 and the point P(1, 1). A secant line through P and another point Q(x, x^2) has a slope m_{PQ} given by the "rise over run" formula:

m_{PQ} = \frac{\Delta y}{\Delta x} = \frac{x^2 - 1}{x - 1}

By factoring the numerator, x^2 - 1 = (x - 1)(x + 1), the slope simplifies to m_{PQ} = x + 1 (for x \ne 1). As x approaches 1, the slope m_{PQ} approaches 1 + 1 = 2. This is written as \lim_{x \to 1} m_{PQ} = 2. The tangent line at (1, 1) therefore has the equation y - 1 = 2(x - 1), or y = 2x - 1. It is crucial to note that direct substitution of x = 1 into the original slope formula would result in 0/0, which is undefined; algebraic simplification is necessary before taking the limit.

Instantaneous Velocity

Similar to the tangent problem, defining instantaneous velocity also involves a limit. Average velocity is calculated as \text{distance traveled} / \text{time it took}. However, instantaneous velocity refers to the velocity at a specific moment, where both distance traveled and time elapsed are zero, leading to an undefined 0/0 expression.

To formalize this, let s(t) be the distance covered at time t. The average velocity over a short time interval \Delta t, from t to t + \Delta t, is:

\frac{s(t + \Delta t) - s(t)}{\Delta t}

The instantaneous velocity v(t) at time t is defined as the limit of this average velocity as \Delta t approaches zero:

v(t) = \lim_{\Delta t \to 0} \frac{s(t + \Delta t) - s(t)}{\Delta t}

Rates of Change

Both the tangent and instantaneous velocity problems are examples of a more general concept: the rate of change of a function. For a function y = f(x), if x changes by \Delta x, then y changes by \Delta y = f(x + \Delta x) - f(x). The average rate of change is \frac{\Delta y}{\Delta x} = \frac{f(x + \Delta x) - f(x)}{\Delta x}.

The rate of change of f at x, or the derivative of f at x, denoted f'(x), is the limit of the average rate of change as \Delta x approaches zero:

f'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x}

Understanding this limit is fundamental to understanding derivatives, which will be the focus of the first half of the semester.

Examples of Rates of Change

Acceleration: The rate at which velocity changes. If v(t) is velocity, then instantaneous acceleration a(t) is \lim_{\Delta t \to 0} \frac{v(t + \Delta t) - v(t)}{\Delta t}. Units are typically miles/hour^2 or meters/second^2.
Reaction Rates in Chemistry: The rate of change of the amount of a substance [A] in a chemical reaction with respect to time t is given by the derivative [A]'(t) or d[A]/dt. For example, the law of mass-action states d[A]/dt = k[A][B]^2, where k is the reaction constant.

Chapter 3: Limits and Continuous Functions

This chapter delves into the precise definition and properties of limits, which are crucial for understanding derivatives and continuity.

Informal Definition of Limits

The concept of a limit, \lim_{x \to a} f(x) = L, means that if values of x are chosen close to (but not equal to) a, then f(x) will be close to L. Furthermore, f(x) gets closer to L as x gets closer to a. Alternative notation is f(x) \to L \text{ as } x \to a.

Examples of Informal Limit Evaluation

For f(x) = x + 3, \lim_{x \to 4} f(x) = 7, as substituting values near 4 yields results near 7.
For f(x) = \frac{x^2 - 2x}{x^2 - 4} as x \to 2, direct substitution gives 0/0. However, substituting values close to 2 (e.g., 2.001) suggests the limit is 0.5.

Caution: Substituting numbers can sometimes suggest the wrong answer, especially for functions like g(x) = \frac{101000x}{100000x + 1} as x \to 0. Small values might suggest a limit of 1, but even smaller values reveal the limit is 0. This highlights the need for a formal definition.

Formal, Authoritative Definition of Limit (ε-δ Definition)

The formal definition of \lim_{x \to a} f(x) = L is as follows:

f(x) need not be defined at x = a, but it must be defined for all other x in some interval containing a.
For every \varepsilon > 0 (how close you want f(x) to be to L), one can find a \delta > 0 (how close x must be to a) such that for all x in the domain of f, |x - a| < \delta implies |f(x) - L| < \varepsilon.

The absolute values |x - y| represent the distance between x and y on the number line.

Examples of ε-δ Proofs

\lim_{x \to 5} (2x + 1) = 11: We want |(2x + 1) - 11| < \varepsilon. This simplifies to |2x - 10| = 2|x - 5|. So, if 2|x - 5| < \varepsilon, then |x - 5| < \varepsilon/2. We can choose \delta = \varepsilon/2.
\lim_{x \to 1} x^2 = 1: We want |x^2 - 1| < \varepsilon. Factoring gives |x^2 - 1| = |(x - 1)(x + 1)| = |x + 1||x - 1|. To bound |x + 1|, we agree to choose \delta \le 1. If |x - 1| < \delta \le 1, then 0 < x < 2, so |x + 1| < 3. Thus, |x^2 - 1| < 3|x - 1|. To ensure |x^2 - 1| < \varepsilon, we need 3|x - 1| < \varepsilon, or |x - 1| < \varepsilon/3. So, we choose \delta = \min(1, \varepsilon/3).
\lim_{x \to 4} (1/x) = 1/4: We want |1/x - 1/4| < \varepsilon. This is |\frac{4 - x}{4x}| = \frac{|x - 4|}{|4x|}. Again, choose \delta \le 1, so |x - 4| < 1 implies 3 < x < 5. Then 1/|4x| is at most 1/(4 \cdot 3) = 1/12. So |1/x - 1/4| \le \frac{1}{12}|x - 4|. We need \frac{1}{12}|x - 4| < \varepsilon, or |x - 4| < 12\varepsilon. So, \delta = \min(1, 12\varepsilon).

Propagation of Errors: The ε-δ definition can be interpreted in terms of measurement error. If you want to know an area A with an error of at most \varepsilon, and the area is a function of a measured radius x (e.g., f(x) = \pi x^2), then \delta represents the maximum allowable error in the radius measurement |x - R| to achieve the desired precision in the area |f(x) - A| < \varepsilon.

Variations on the Limit Theme

Left and Right Limits: When x approaches a, it can do so from values larger than a (right limit, \lim_{x \downarrow a} f(x) or \lim_{x \to a^+} f(x)) or from values smaller than a (left limit, \lim_{x \uparrow a} f(x) or \lim_{x \to a^-} f(x)). A limit \lim_{x \to a} f(x) exists if and only if both the left and right limits exist and are equal (L^+ = L^-).
Limits at Infinity: We can ask what happens to f(x) as x becomes arbitrarily large (x \to \infty) or arbitrarily small (large negative, x \to -\infty). \lim_{x \to \infty} f(x) = L means that for every \varepsilon > 0, there exists an A such that x > A implies |f(x) - L| < \varepsilon. For example, \lim_{x \to \infty} (1/x) = 0.

Properties of the Limit

These properties allow for computation of limits without resorting to the ε-δ definition:

Limits of Constants and x: \lim_{x \to a} c = c and \lim_{x \to a} x = a.
Limits of Sums, Products, and Quotients: If \lim_{x \to a} F_1(x) = L_1 and \lim_{x \to a} F_2(x) = L_2, then:
- \lim_{x \to a} (F_1(x) \pm F_2(x)) = L_1 \pm L_2
- \lim_{x \to a} (F_1(x) \cdot F_2(x)) = L_1 \cdot L_2
- \lim_{x \to a} \frac{F_1(x)}{F_2(x)} = \frac{L_1}{L_2} (provided L_2 \ne 0).

Examples of Limit Computations

Polynomials: \lim_{x \to 2} x^2 = (\lim_{x \to 2} x) \cdot (\lim_{x \to 2} x) = 2 \cdot 2 = 4.
Rational Functions: For \lim_{x \to 2} \frac{x^2 - 2x}{x^2 - 4}, direct substitution yields 0/0. Algebraic simplification \frac{x(x - 2)}{(x - 2)(x + 2)} = \frac{x}{x + 2} allows for substitution, giving 2/(2 + 2) = 1/2.
Square Roots: To find \lim_{x \to 2} \sqrt{x}, if we assume the limit L exists, then L^2 = \lim_{x \to 2} x = 2, so L = \sqrt{2} (rejecting -\sqrt{2} as square roots are positive).
Limits of Rational Functions at Infinity: For \lim_{x \to \infty} \frac{3x^2 + 3}{5x^2 + 7x - 39}, divide numerator and denominator by the highest power of x in the denominator (x^2): \lim_{x \to \infty} \frac{3 + 3/x^2}{5 + 7/x - 39/x^2} = \frac{3 + 0}{5 + 0 - 0} = 3/5.

When Limits Fail to Exist

A limit \lim_{x \to a} f(x) does not exist if there is no number L such that the definition of limit is satisfied.

Sign Function (\text{sign}(x)) at x = 0: \text{sign}(x) = -1 for x < 0, 0 for x = 0, and 1 for x > 0. The left limit is -1, and the right limit is +1. Since they are not equal, \lim_{x \to 0} \text{sign}(x) does not exist.
Backward Sine (\sin(\pi/x)) at x = 0: As x \to 0, \pi/x goes to infinity, causing \sin(\pi/x) to oscillate infinitely often between -1 and +1. It never stays close to a single value, so the limit does not exist (even one-sided limits).
1/x at x = 0: As x \to 0, 1/x becomes arbitrarily large (positive or negative depending on the direction). It does not approach a finite number, so the limit does not exist. This can also be shown by contradiction using limit properties.
x as x \to \infty: x does not approach any particular finite number, so \lim_{x \to \infty} x does not exist.

What's in a Name? (Dummy vs. Free Variables)

In a limit expression like \lim_{x \to a} (2x + 1), x is a dummy variable (or placeholder/bound variable), meaning its name can be changed (e.g., to u) without altering the limit's value. a is a free variable, whose value affects the limit's result (e.g., \lim_{x \to a} (2x + 1) = 2a + 1).

Limits and Inequalities

Theorem: If f(x) \le g(x) for all x, and \lim_{x \to a} f(x) and \lim_{x \to a} g(x) exist, then \lim_{x \to a} f(x) \le \lim_{x \to a} g(x).
The Sandwich Theorem (Squeeze Theorem): If f(x) \le g(x) \le h(x) for all x, and \lim_{x \to a} f(x) = \lim_{x \to a} h(x) = L, then \lim_{x \to a} g(x) = L. This theorem is powerful because it guarantees the existence of the limit of g(x). For example, since -|x| \le x \cos(1/x) \le |x| and \lim_{x \to 0} -|x| = \lim_{x \to 0} |x| = 0, then \lim_{x \to 0} x \cos(1/x) = 0.

Continuity

A function g is continuous at a if \lim_{x \to a} g(x) = g(a). A function is continuous on an interval if it is continuous at every point in its domain within that interval.

Polynomials are Continuous: For example, P(x) = x^2 + 3x is continuous at x = 2 because \lim_{x \to 2} (x^2 + 3x) = 2^2 + 3 \cdot 2 = 10 = P(2).
Rational Functions are Continuous: A rational function R(x) = P(x)/Q(x) is continuous at any point a in its domain (where Q(a) \ne 0).
Discontinuous Functions: Functions where \lim_{x \to a} g(x) does not exist (e.g., sign function at x=0) or where \lim_{x \to a} g(x) \ne g(a) (e.g., f(x) = x^2 for x \ne 3 and f(3) = 47) are discontinuous.

Substitution in Limits

For a composite function h(x) = f(g(x)), if \lim_{x \to a} g(x) = L and f is continuous at L, then \lim_{x \to a} f(g(x)) = f(\lim_{x \to a} g(x)) = f(L). This is also written as \lim_{x \to a} f(g(x)) = \lim_{u \to L} f(u) by setting u = g(x).

Two Limits in Trigonometry

Two important trigonometric limits are:

\lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1. This is proven using the Sandwich Theorem by comparing the areas of a circular wedge, an inscribed triangle, and a circumscribed triangle. For 0 < \theta < \pi/2, we have \sin \theta < \theta < \tan \theta, which leads to \cos \theta < \frac{\sin \theta}{\theta} < 1. Since \lim_{\theta \to 0} \cos \theta = 1, the theorem applies.
\lim_{\theta \to 0} \frac{1 - \cos \theta}{\theta^2} = \frac{1}{2}. This is derived using the identity \sin^2 \theta + \cos^2 \theta = 1, rewriting the expression as \frac{1}{1 + \cos \theta} \left(\frac{\sin \theta}{\theta}\right)^2.

Chapter 4: Derivatives (2)

Building on the concept of limits, this chapter formally defines derivatives and introduces rules for their computation.

Derivatives Defined

The derivative of a function f at a point a, denoted f'(a), is the value of the limit:

f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a}

An equivalent definition, often more convenient for computation, uses h = x - a (so x = a + h) as h \to 0:

f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}

If this limit exists, f is said to be differentiable at a. If f is differentiable at every point in an interval, it is differentiable on the interval.

Leibniz Notation: Gottfried Wilhelm von Leibniz introduced the notation \frac{dy}{dx} = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}, where dx and dy are considered