Difficulty: Introductory | Prerequisites: Pre-calculus (algebra, trigonometry, basic functions)
Calculus is the branch of mathematics that studies how things change and how quantities accumulate. It sits at the heart of a first-year maths sequence, building directly on the algebra and pre-calculus work you have already done with functions, graphs and limits. If you can read a graph and manipulate basic algebraic expressions, you have the prerequisites. Everything in physics, engineering, economics and the life sciences that involves rates, areas or optimisation traces back to the ideas covered here.
Calculus splits into two halves: differential calculus (rates of change, slopes of curves) and integral calculus (accumulation of quantities, areas under curves). The Fundamental Theorem of Calculus ties the two together, showing that differentiation and integration are inverse operations. Master that link and the rest of the course is technique.
Calculus
The mathematical study of change, just as geometry is the study of shape and algebra is the study of operations and equation-solving. The word comes from Latin and originally meant a small stone used for counting. Both branches of calculus deal with infinite sequences and series converging to well-defined limits.
In simple terms, calculus gives you the tools to analyse anything that moves, grows, shrinks or accumulates.
Differential calculus
The branch of calculus concerned with rates of change and slopes of curves. Its primary objects are the derivative of a function, the differential, and their applications.
Think of it as the mathematics of "how fast is something changing right now?"
Integral calculus
The branch of calculus concerned with the accumulation of quantities and the areas under and between curves. It assigns numbers to functions in a way that describes displacement, area, volume and other concepts built from infinitesimal data.
Think of it as the mathematics of "how much has accumulated over this interval?"
Fundamental Theorem of Calculus (FTC)
The theorem linking differentiation and integration. Part 1 states that the definite integral of a function is related to its antiderivative and can be reversed by differentiation; it also guarantees that every continuous function has an antiderivative. Part 2 states that any one of a function's infinitely many antiderivatives can be used to compute its definite integral.
In simple terms, Part 1 says "integration undoes differentiation," and Part 2 says "to evaluate an integral, just find an antiderivative and plug in the endpoints."
Derivative
A measure of how a function's output changes as its input changes. Geometrically, it is the slope of the tangent line to the function's graph at a given point.
Think of it as the instantaneous rate of change.
Antiderivative
A function whose derivative equals the original function. If F'(x) = f(x), then F is an antiderivative of f. Any two antiderivatives of the same function differ by a constant.
Think of it as "working backwards from the derivative."
Limit
The value that a function (or sequence) approaches as the input (or index) approaches some value. Limits underpin both derivatives and integrals.
In simple terms, it answers the question "what value is this heading towards?"
Continuous function
A function with no breaks, jumps or holes in its graph. Formally, f is continuous at a point c if the limit of f(x) as x approaches c equals f(c).
Think of it as a curve you can draw without lifting your pen.
Concerned with the instantaneous rate of change of a quantity
The central operation is differentiation: given a function f(x), find f'(x)
Geometrically, the derivative at a point is the slope of the tangent line to the curve at that point
Builds on the concept of a limit: f'(x) = lim(h→0) [f(x+h) - f(x)] / h
Applications include velocity, acceleration, optimisation problems, and curve sketching
Concerned with the accumulation of quantities and the total effect of a continuously changing rate
The central operation is integration: given a rate function f(x), find the total accumulated quantity
Geometrically, the definite integral represents the signed area between the curve and the x-axis
Two flavours: indefinite integrals (families of antiderivatives, written with + C) and definite integrals (a number, computed over an interval [a, b])
Applications include area, volume, displacement, work, and probability
Differentiation and integration are inverse processes. If you differentiate an integral, you recover the original function, and if you integrate a derivative, you recover the original function up to a constant. This relationship is formalised by the Fundamental Theorem of Calculus.
If f is continuous on [a, b] and F(x) = the integral from a to x of f(t) dt, then F'(x) = f(x).
This says that if you define a new function by integrating f from a fixed starting point up to a variable endpoint, then the derivative of that new function is the original f
It guarantees that every continuous function has an antiderivative
Practical upshot: differentiation undoes integration
If f is continuous on [a, b] and F is any antiderivative of f, then the integral from a to b of f(x) dx = F(b) - F(a).
This converts the problem of computing an area (a limit of Riemann sums) into a simple subtraction
You find any antiderivative F, evaluate it at the two endpoints, and subtract
This is the result that makes integration practical for most textbook problems
Part 1 establishes the theoretical link: integration and differentiation are inverses. Part 2 gives you the computational shortcut: instead of summing infinitely many rectangles, you evaluate a single antiderivative at two points. Nearly every integration problem in MATH 104 relies on Part 2.
Definition of the derivative (limit definition)
f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}Fundamental Theorem of Calculus, Part 1
\frac{d}{dx} \int_a^x f(t)\,dt = f(x)Fundamental Theorem of Calculus, Part 2
\int_a^b f(x)\,dx = F(b) - F(a), \quad \text{where } F'(x) = f(x)Basic power rule for differentiation
\frac{d}{dx}[x^n] = n \cdot x^{n-1}Basic power rule for integration
\int x^n\,dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1Derivatives are the basis for every rate-of-change calculation in science and engineering: velocity and acceleration in physics, marginal cost and marginal revenue in economics, and reaction rates in chemistry all come from differentiation.
Integrals show up whenever you need a total from a continuously varying quantity. The area under a velocity-time graph gives displacement. Engineers use integration to calculate the volume of irregularly shaped objects, the work done by a variable force, and the total charge flowing through a circuit over time.
Students often think "calculus" and "differential calculus" are the same thing. They are not. Calculus has two equally important halves: differential and integral.
Students sometimes believe the derivative gives the value of a function at a point. It gives the rate of change (slope) at that point, not the value itself.
A common error is forgetting the constant of integration (+ C) when computing indefinite integrals. Every indefinite integral represents a family of functions, not a single function.
Students often confuse the Fundamental Theorem Part 1 with Part 2. Part 1 is about differentiating an integral (theoretical link). Part 2 is about evaluating a definite integral using antiderivatives (the computational shortcut you will use most often).
The limit definition of the derivative is a staple early-exam question. Be able to compute f'(x) from the definition, not just by shortcut rules.
Expect at least one question requiring you to state or apply the Fundamental Theorem of Calculus. Know both parts and when each applies.
Questions asking you to distinguish between differential and integral calculus (or between definite and indefinite integrals) appear regularly on concept-check portions of exams.
The relationship between a function, its derivative and its antiderivative is foundational. If you can explain how differentiation undoes integration and vice versa, you are well placed for the rest of the course.
True or false: Integral calculus is concerned with rates of change.
Answer: False. Integral calculus is concerned with accumulation and areas. Rates of change belong to differential calculus.
Fill in the blank: The derivative of a function at a point gives the ______ of the tangent line at that point.
Answer: slope
True or false: The Fundamental Theorem of Calculus says that differentiation and integration are inverse operations.
Answer: True.
Fill in the blank: An indefinite integral always includes a ______ because the antiderivative is not unique.
Answer: constant of integration (+ C)
True or false: The word "calculus" originally referred to a small stone used for counting.
Answer: True. It comes from Latin.
Q: What are the two main branches of calculus, and what does each one study?
A: Differential calculus studies rates of change and slopes of curves. Integral calculus studies the accumulation of quantities and the areas under and between curves.
Q: State the Fundamental Theorem of Calculus, Part 2, in your own words.
A: If F is any antiderivative of a continuous function f on [a, b], then the definite integral of f from a to b equals F(b) minus F(a). In other words, you can evaluate a definite integral by finding an antiderivative and subtracting its values at the endpoints.
Q: Using the limit definition, write the expression for f'(x).
A: f'(x) = lim(h to 0) [f(x+h) - f(x)] / h.
Q: Why does an indefinite integral include "+ C"?
A: Because the antiderivative is not unique. Any two antiderivatives of the same function differ by a constant, so the general antiderivative is written with an arbitrary constant C.
Q: Explain, in one sentence, why Part 1 of the Fundamental Theorem of Calculus matters.
A: It guarantees that every continuous function has an antiderivative, establishing the theoretical foundation for integration.
This material connects directly to limits and continuity, which you will study in detail before formally defining the derivative. A solid grasp of limit laws is the prerequisite for understanding differentiation.
Later in the course, techniques of integration (substitution, integration by parts) build on the FTC by giving you methods to find the antiderivatives Part 2 requires.
In multivariable calculus (Calc 2 and 3), these same ideas extend to functions of several variables, partial derivatives and multiple integrals. The single-variable foundations covered here carry through the entire sequence.
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