Calculus is the mathematics of change and of accumulation. A derivative answers how fast a quantity is changing at one moment. An integral answers how much has piled up over an interval. Limits are the shared language: they say what happens when you look closer, or when you add finer and finer pieces. This pillar is a map of that first course. It says who the pages help, what you should be able to do, and which explainers, calculators, glossary entries, and pathways already on this site to open next.

Who this pillar is for

Start here if you are beginning calculus, or if you can apply derivative rules and still cannot say what the symbols measure. You might be entering a first college course, repairing a course that moved too fast, or getting ready for series and a first differential equation. You do not need to be fast. You do need a sentence you can say before you trust a formula.

If computation is already comfortable, skim this map and follow the AP Calculus AB study pathway. If the next demand is writing arguments, this pillar is the vocabulary and intro to proofs is the writing practice.

Prerequisites

You need algebra, graphs, and function notation. You should be able to evaluate a formula, solve a linear equation, and read a point off a graph. Fractions and exponents show up constantly. Sine and cosine appear in a standard first course. You can still learn the ideas of limit, derivative, and integral on polynomials first, and add trigonometry when a problem requires it.

If a word is doing real work, use the glossary entries for limit, derivative, continuity, and definite integral. If you want a full book beside this map, OpenStax Calculus and MIT OpenCourseWare are on the sources page. Come back when you can say, in one sentence each, what those three words ask.

What you should be able to do

When you finish the core of this map, you should be able to do the following in words, not only with a memorized rule.

  • Explain a limit: nearby inputs, and a value the outputs are forced toward. The function value at the point itself may be missing.
  • Say what continuity adds. The function value exists, and it equals that limit.
  • Compute derivatives of common functions, and also say that $f'(a)$ is an instantaneous rate and the slope of the best linear approximation.
  • Set up a definite integral as net signed accumulation, and connect it to derivatives with the Fundamental Theorem of Calculus.
  • Tell a sequence (a list) from a series (an attempt to add the whole list) well enough to know which later explainer to open.

Limits and continuity

Saying $\lim_{x \to a} f(x) = L$ means you can force the outputs as close to $L$ as you like by taking $x$ close enough to $a$, and not necessarily equal to $a$. The function may be undefined at $a$, or defined as something else. The limit does not consult $f(a)$.

Continuity at $a$ adds two facts: $f(a)$ exists, and it equals the limit. A hole is a limit that exists while the function value is missing or wrong. A jump is a refusal of the story from the left and the story from the right to agree. A vertical blow up is outputs that will not stay near any finite $L$.

A numerical table can show settling. A proof still has to be written. The numerical limit estimator samples nearby values. The epsilon delta finder lets you type $\varepsilon$ and see a numerical $\delta$, with the band drawn around $L$. The precise version, tolerance on outputs first and a window on inputs second, is the epsilon delta definition. It is optional on a first pass. Epsilon delta definition and delta finder is the friendly entry if a later course will ask for proofs.

Check before you plug in. For $f(x)=(x^{2}-1)/(x-1)$ and $a=1$, the formula is undefined at the point. For every other $x$, it simplifies to $x+1$, so the limit is $2$. A table should approach $2$. Substituting $x=1$ into the original formula is not available. That is the situation a limit is built for.

Derivatives as linear approximation

The derivative $f'(a)$, when it exists, is the unique number such that near $a$,

$$f(x) \approx f(a) + f'(a)(x-a).$$

The slope of the tangent line and the instantaneous rate are the same number wearing different units. If position is in meters and time is in seconds, the derivative is in meters per second. "The derivative is $3$" is unfinished. "Volume is increasing by $3$ liters per minute at this time" is finished. What a derivative actually measures walks through that meaning. The numerical derivative calculator estimates a slope. Compare it with an algebraic derivative, and change the step size if the estimate looks wild.

Implicit differentiation is the same idea when the equation does not give $y$ as a formula in $x$. You differentiate both sides, treating $y$ as a function of $x$, and then solve for $dy/dx$. Read implicit differentiation intuition when a curve is given as a relation. For $x^{2}+y^{2}=1$, the slope is $-x/y$ when $y$ is not zero. The glossary line for implicit differentiation is the short reminder.

Integrals as accumulation

A definite integral $\int_a^b f(x)\,dx$ measures net signed accumulation of $f$ from $a$ to $b$. Where $f$ is positive, area adds. Where $f$ is negative, area subtracts. Riemann sums make the picture concrete: rectangles of height $f$ at a sample point and width $\Delta x$ approximate the total. The integral is the limit of those sums as the mesh goes to zero, when that limit exists.

The Riemann sum calculator draws that picture. For an increasing positive function, a left sum undershoots and a right sum overshoots. Say which is which before you run a modest number of rectangles. The integral of $x$ from $0$ to $1$ is $1/2$. A right sum on that interval should sit above $1/2$.

The Fundamental Theorem of Calculus connects the two ideas. Differentiating an accumulation function returns the integrand, under the usual hypotheses, and a definite integral of a continuous function can be computed from an antiderivative at the ends. If you can antidifferentiate and cannot say what the number accumulates, the picture is still missing. The glossary entries for integral and definite integral keep the indefinite and definite meanings apart.

An antiderivative of a rational function often starts by splitting the fraction. The partial fraction decomposition calculator writes that algebra in exact fractions: division when the fraction is improper, then one term for each linear factor and a linear numerator for each irreducible quadratic. If a quadratic factor might still split over the rationals, the quadratic formula calculator reports the discriminant that decides. The Riemann sum calculator answers a different question. It estimates a definite integral with rectangles.

Sequences and series, the next door

Many first courses end with sequences and series. A sequence is an ordered list. A series adds that list, defined as the limit of partial sums when the limit exists. Terms going to zero are required and not enough. The harmonic series is the warning: terms shrink, and partial sums still wander off. Read sequences, series, and when infinite sums make sense, then convergence tests for series without fog. The sequence calculator adds finitely many terms. A finite total is a different question from infinite convergence. The polynomial that matches a function and its derivatives at one point is a different finite object again: the Taylor polynomial calculator builds it, with the remainder bound, for e^x, sine, cosine, and the other standard series.

Suggested learning order

Work in this order. One step per sitting is enough. Open a calculator only after you can say what the number means. The FAQ on calculators versus understanding is the rule on this site.

  1. Glossary: limit, continuity, derivative, definite integral.
  2. What a derivative actually measures, then the numerical derivative calculator on $x^{2}$ at $x=4$ (the algebraic slope is $8$).
  3. The numerical limit estimator on a hole such as $(x^{2}-1)/(x-1)$ near $1$.
  4. Optional: epsilon delta intuition and the glossary line, with the epsilon delta finder after you can state the quantifiers. Continue on intro to proofs if this is the step you care about.
  5. Implicit differentiation. On the unit circle, compare $-x/y$ with a numerical slope.
  6. The Riemann sum calculator with the definite integral entry. Predict which sum is larger.
  7. Sequences and series, then convergence tests, with the ratio test. The sequence calculator is only for a finite sum. The Taylor polynomial calculator is the local polynomial for e^x, sine, and ln(1+x).
  8. Next subject: differential equations and the separable walkthrough, or the AP Calculus AB study pathway for the full route.

Checkpoints and practice habits

Do these on paper. A calculator check comes after a sentence you can say.

Practice habit. For a derivative you already know how to compute, write one sentence naming the quantity whose rate you are computing. Distance, cost, population, or temperature is enough. Include units. If the sentence has no units and no quantity, it is not finished.

Slope check. The derivative of $x^{2}$ at $x=4$ is $8$. Ask the numerical derivative calculator for an estimate. If the estimate and $8$ disagree by a lot, change the step size before you blame the algebra.

Accumulation check. $\int_0^1 x\,dx = 1/2$. On an increasing positive function, a right Riemann sum overshoots. Say that sentence, then confirm the picture on the Riemann sum calculator.

Series check. In one sentence, state how a sequence differs from a series, and which object a convergence test is about. If you never mention adding terms, reread sequences, series, and when infinite sums make sense.

Common pitfalls

Speed with the power rule can hide a missing definition. If what a derivative actually measures feels slow, the pillar is doing its job. A calculator output is a picture or an estimate. A proof and an algebraic derivative are separate work.

Pitfall. Substituting $x=a$ is a useful check when $f$ is continuous there. It is not the definition of a limit. Limits care about nearby values. A related mistake is to treat a finite geometric sum as a verdict about infinite convergence, or to trust the ratio test when the limit of successive absolute ratios is $1$. That case is silent. Silence is neither convergence nor divergence. Switch tests, as convergence tests for series without fog explains.

Related reading

Epsilon delta leads to intro to proofs and real analysis. Several variables lead through the linear algebra first course and Jacobian matrices into multivariable calculus. Models of change stay with differential equations. Compare routes on the pathways hub. A full book is on the sources page.

If the rules feel empty, start with the derivative explainer. If nearby values are the confusion, open the limit estimator after you predict what the table should do.

Read what a derivative measures
Disclaimer. Educational only. This page is not a course, not exam preparation, and not affiliated with College Board. See the disclaimer.