Introduction

Calculus begins with two ordinary human questions:

How fast is something changing right now?
How much has accumulated over an interval?

These questions appear everywhere. A car’s position changes with time. A population grows or shrinks. A bank account accumulates interest. A falling object gains speed. A curve encloses an area. A medicine spreads through the bloodstream. A planet moves through space. A machine uses energy while it runs.

Before calculus, we can often describe change only over a whole interval. If a car travels 120 kilometers in 2 hours, its average speed is

\[ \frac{120\text{ km}}{2\text{ h}}=60\text{ km/h}. \]

But that does not tell us the car’s speed at one particular instant, such as exactly 9:15 a.m. The speedometer seems to answer that question, but mathematically the question is subtle: at a single instant, no time has passed, so the ordinary ratio

\[ \frac{\text{distance traveled}}{\text{time elapsed}} \]

appears to become

\[ \frac{0}{0}, \]

which is not a number. Calculus gives a precise way to approach this problem without pretending that \(0/0\) is meaningful.

The key idea is the limit. A limit describes what value a quantity approaches as another quantity gets closer and closer to some target. Instead of asking for speed during exactly zero seconds, we ask for average speed over smaller and smaller time intervals and see whether those average speeds approach a stable value. If they do, that value is the instantaneous rate of change. This is one of the central ideas behind the derivative.

There is a second central idea. Suppose a machine uses energy at a changing rate. During one minute it may use little energy; during the next it may use more. If we know the rate at every moment, how do we find the total energy used? Calculus answers this with the integral, which combines many tiny contributions into a total amount. In one common interpretation, an integral measures accumulated change or signed area under a curve. The rigorous development of derivatives and integrals through limits is a standard foundation of calculus, as treated in classical texts such as Apostol’s Calculus and Spivak’s Calculus (Apostol, 1967; Spivak, 2008).

This book is called Calculus from First Principles because we will not begin by memorizing formulas. We will begin by understanding what the formulas mean.

What “from first principles” means

A principle is a basic idea from which other ideas can be built. To learn from first principles is to start with definitions, examples, and reasoning instead of starting with rules alone.

For example, many students first meet the derivative through formulas such as

\[ \frac{d}{dx}x^2=2x. \]

This formula is correct, but by itself it hides the main idea. Why should the derivative of \(x^2\) be \(2x\)? What does the symbol \(\frac{d}{dx}\) mean? Why does a tangent line have anything to do with change?

From first principles, we begin with a function such as

\[ f(x)=x^2. \]

A function is a rule that assigns each allowed input exactly one output. Here the input is \(x\), and the output is \(x^2\). If \(x=3\), then \(f(3)=9\). If \(x=3.1\), then \(f(3.1)=9.61\).

To measure how \(f(x)\) changes near \(x=3\), compare two nearby values:

\[ \frac{f(3+h)-f(3)}{h}. \]

This expression is called a difference quotient. It measures average change in the output divided by change in the input. For \(f(x)=x^2\),

\[ \frac{f(3+h)-f(3)}{h} = \frac{(3+h)^2-9}{h}. \]

Expanding,

\[ (3+h)^2=9+6h+h^2, \]

so

\[ \frac{(3+h)^2-9}{h} = \frac{6h+h^2}{h} = 6+h, \]

as long as \(h\neq 0\). Now let \(h\) get closer and closer to \(0\). The expression \(6+h\) gets closer and closer to \(6\). So the derivative of \(x^2\) at \(x=3\) is \(6\).

The formula \(2x\) now has meaning: at the input \(x\), the function \(x^2\) is changing at an instantaneous rate of \(2x\). At \(x=3\), that rate is \(6\). At \(x=10\), it is \(20\). At \(x=-4\), it is \(-8\).

This is the style of the book. We will compute, but computation will grow out of meaning.

The two great movements of calculus

Most of single-variable calculus can be organized around two movements.

The first movement is local. It asks: what is happening very near one point?

The derivative belongs to this local movement. It studies a function by zooming in. If the graph of a function becomes almost straight when viewed very close to a point, the slope of that best local straight-line approximation is the derivative. This idea connects calculus to geometry, physics, economics, biology, and any field where change matters.

For example, if \(s(t)\) gives the position of an object at time \(t\), then the derivative \(s'(t)\) gives its velocity, provided the derivative exists. If \(C(q)\) gives the cost of producing \(q\) units of a product, then \(C'(q)\) describes marginal cost: the approximate additional cost of producing one more unit when production is near \(q\). The mathematical idea is the same: the derivative measures local rate of change.

The second movement is global. It asks: what total effect has built up over an interval?

The integral belongs to this global movement. It studies a quantity by adding many small pieces. If \(v(t)\) is velocity, then integrating velocity over time gives displacement. If \(r(t)\) is the rate at which water flows into a tank, then integrating \(r(t)\) over time gives the amount of water added. If \(f(x)\) is a curve above the \(x\)-axis, then integrating \(f(x)\) over an interval gives the area under the curve.

The remarkable discovery is that these two movements are deeply connected. Differentiation breaks accumulated change into an instantaneous rate. Integration builds a total amount from instantaneous rates. The Fundamental Theorem of Calculus states this relationship precisely: under appropriate conditions, differentiation and integration are inverse processes in a specific mathematical sense. This connection is one of the central achievements of calculus and is treated as a foundational theorem in standard rigorous developments of the subject (Apostol, 1967; Courant and John, 1999).

A brief historical orientation

The ideas that became calculus developed over many centuries, with contributions from ancient geometry, medieval mathematics, and early modern science. In the seventeenth century, Isaac Newton and Gottfried Wilhelm Leibniz independently developed systematic methods for what we now call differential and integral calculus. Newton’s work was strongly tied to motion and physical problems, while Leibniz introduced notation, including forms related to \(dx\) and \(dy\), that became central to later calculus. The historical development was complex and involved many predecessors and successors, but Newton and Leibniz remain the two central names in the formation of calculus as a coherent method (Boyer, 1959).

This book is not mainly a history book, but the history teaches an important lesson: calculus was created to solve real problems that older methods could not handle efficiently. It was not invented as a collection of classroom exercises. It was invented because people needed a language for motion, change, area, approximation, and accumulation.

What you will learn

You will begin with functions, because calculus studies how quantities depend on one another. A function is the mathematical way to say: “when this input changes, that output changes.” Without functions, derivatives and integrals have no stable home.

Then you will study average rate of change. This is the bridge from algebra to calculus. If a function changes from \(f(a)\) to \(f(b)\), its average rate of change on \([a,b]\) is

\[ \frac{f(b)-f(a)}{b-a}. \]

This ratio is the slope of the secant line through two points on the graph. A secant line is a line that passes through two points of a curve. Calculus asks what happens when those two points move closer together.

Next come limits and continuity. A function is continuous at a point when, informally, its values near that point approach the function’s actual value there. More precisely, continuity will be defined using limits. Continuous functions behave without sudden jumps or holes at the point in question. This matters because many theorems of calculus require continuity or differentiability as hypotheses.

After that, you will define and use derivatives. You will learn derivative rules, not as magic shortcuts, but as consequences of the limit definition. These rules make calculus efficient. For example, once you understand the product rule, you can differentiate functions such as

\[ f(x)=x^2\sin x \]

without returning to the limit definition every time.

You will then apply derivatives to approximation, graph analysis, optimization, and related rates. These topics show the power of local information. If you know how a function behaves near a point, you can approximate values, find maximum and minimum behavior, and model changing systems.

The book then turns to antiderivatives and integrals. An antiderivative of a function \(f\) is a function \(F\) whose derivative is \(f\). For example, since

\[ \frac{d}{dx}x^3=3x^2, \]

an antiderivative of \(3x^2\) is \(x^3\). The definite integral will be built from Riemann sums, which approximate accumulation by adding many rectangular pieces. This construction is historically and mathematically central to the rigorous treatment of integration (Apostol, 1967).

Later chapters extend calculus to infinite processes. A sequence is an ordered list of numbers, such as

\[ 1,\frac12,\frac13,\frac14,\dots \]

An infinite series is a sum of infinitely many terms, such as

\[ 1+\frac12+\frac14+\frac18+\cdots. \]

Some infinite series approach finite values, and others do not. Calculus gives tools for deciding which behavior occurs. Power series and Taylor series then show how functions can sometimes be represented or approximated by infinite polynomials. This is one of the most useful bridges between calculus, computation, and advanced mathematics.

Finally, you will move beyond one input variable. In multivariable calculus, a function may depend on several inputs:

\[ f(x,y)=x^2+y^2. \]

Such a function might describe temperature on a flat plate, height on a landscape, or cost depending on two different production choices. You will learn partial derivatives, gradients, multiple integrals, and vector calculus. These ideas generalize the one-variable story: local change, accumulated quantity, and geometric structure remain the central themes, but now they occur in higher-dimensional settings. Apostol’s second volume develops this broader multivariable viewpoint in detail (Apostol, 1969).

How to think while studying calculus

Calculus rewards active reading. Do not try to read it like a novel, and do not try to memorize every formula before understanding why it is true. Instead, ask three questions often.

First: What quantity is changing?

For example, if \(A(r)=\pi r^2\) is the area of a circle, then the area changes when the radius changes. The derivative

\[ A'(r)=2\pi r \]

tells us how sensitive the area is to a small change in radius.

Second: With respect to what variable is it changing?

This phrase “with respect to” is important. If position depends on time, velocity is change in position with respect to time. If area depends on radius, \(A'(r)\) is change in area with respect to radius. In multivariable calculus, this question becomes even more important because a quantity may depend on several variables at once.

Third: What are the units?

Units protect meaning. If distance is measured in meters and time in seconds, then velocity is measured in meters per second. If \(V(t)\) is volume in liters and \(t\) is time in minutes, then \(V'(t)\) has units liters per minute. The derivative is not just a number; it is a rate with meaning.

For integrals, units also guide interpretation. If a rate is measured in liters per minute and you integrate with respect to minutes, the result is measured in liters. Symbolically,

\[ \left(\frac{\text{liters}}{\text{minute}}\right)(\text{minutes})=\text{liters}. \]

This is not a proof, but it is a powerful check on whether your answer makes sense.

An example of the whole story

Suppose water flows into a tank at a rate

\[ r(t)=2t \]

liters per minute, where \(t\) is measured in minutes after the flow begins.

At \(t=3\),

\[ r(3)=6. \]

So after 3 minutes, the instantaneous flow rate is 6 liters per minute.

But how much water has entered during the first 3 minutes? The rate is changing, so we cannot simply multiply one constant rate by 3 minutes unless we choose an approximation. The exact accumulated amount is given by the definite integral

\[ \int_0^3 2t\,dt. \]

Using the antiderivative \(t^2\), we get

\[ \int_0^3 2t\,dt = 3^2-0^2=9. \]

So 9 liters of water entered during the first 3 minutes.

This example contains the central architecture of calculus:

• the function \(r(t)=2t\) describes a changing rate;
• the value \(r(3)=6\) describes the rate at one instant;
• the integral \(\int_0^3 2t\,dt\) describes total accumulation;
• the antiderivative \(t^2\) connects the rate to the accumulated amount.

The rest of the book develops these ideas carefully and completely.

The promise of calculus

Calculus is sometimes difficult because it asks you to think at the edge of the finite and the infinite: intervals shrink toward zero, sums contain more and more pieces, series continue without end, and surfaces and fields live in higher dimensions. But each idea will be built step by step.

You do not need to understand everything at once. You need to learn how to move from examples to definitions, from definitions to theorems, from theorems to methods, and from methods back to interpretation.

If you stay close to meaning, calculus becomes more than a collection of techniques. It becomes a language for describing change, accumulation, approximation, and structure.

That is the path we now begin.

References

Apostol, Tom M. Calculus, Volume 1: One-Variable Calculus, with an Introduction to Linear Algebra. 2nd ed., Wiley, 1967.

Apostol, Tom M. Calculus, Volume 2: Multi-Variable Calculus and Linear Algebra, with Applications to Differential Equations and Probability. 2nd ed., Wiley, 1969.

Boyer, Carl B. The History of the Calculus and Its Conceptual Development. Dover Publications, 1959.

Courant, Richard, and Fritz John. Introduction to Calculus and Analysis, Volume I. Springer, 1999.

Spivak, Michael. Calculus. 4th ed., Publish or Perish, 2008.