"What is basic calculus" is asked by students who are about to start it and have heard that it is the hard one. It is the mathematics of change, it rests on three ideas, and each of the three can be drawn as a picture. This page gives the three ideas in plain words, a worked example of each, the first rules, and what you need to know before you begin.
The one-sentence answer
Algebra describes things that stay the same or change at a steady rate. Calculus describes things whose rate of change is itself changing: a ball speeding up as it falls, a population growing faster as it gets bigger, a curve that gets steeper as it rises. It does this with two tools, the derivative and the integral, and one idea underneath both, the limit.
| Idea | The question it answers | The picture |
|---|---|---|
| Limit | What value is this getting close to? | A point a graph approaches |
| Derivative | How fast is it changing right now? | The slope of the curve at one point |
| Integral | How much has built up in total? | The area under the curve |
Idea 1: the limit
A limit is the value a function gets close to as the input gets close to some number, whether or not the function ever reaches it.
Example. Look at f(x) = (x² − 4) / (x − 2). At x = 2 it gives 0 ÷ 0, which has no meaning. But near 2 it behaves perfectly well:
| x | 1.9 | 1.99 | 1.999 | 2.001 | 2.01 | 2.1 |
|---|---|---|---|---|---|---|
| f(x) | 3.9 | 3.99 | 3.999 | 4.001 | 4.01 | 4.1 |
The values close in on 4 from both sides. We say the limit of f(x) as x approaches 2 is 4. The algebra agrees: x² − 4 factors as (x − 2)(x + 2), the (x − 2) cancels, and what is left, x + 2, is 4 when x is 2.
Limits matter because both of the big ideas need them. A slope at a single point looks like 0 ÷ 0 too.
Idea 2: the derivative
The slope of a straight line is rise over run, and it is the same everywhere. A curve has a different slope at every point. The derivative is the slope at one point: the limit of rise over run as the run shrinks to nothing.
Example. Take y = x². Between x = 3 and x = 3.1 the rise is 9.61 − 9 = 0.61 and the run is 0.1, so the slope is 6.1. Between 3 and 3.01 it is 6.01. Between 3 and 3.001 it is 6.001. The slope at exactly x = 3 is the limit: 6.
Do that at any x and the answer is always twice x. So the derivative of x² is 2x. That is one rule that replaces all the arithmetic.
Why it matters. If a dropped ball has fallen s = 5t² metres after t seconds, its speed is the derivative: 10t metres per second. After 2 seconds it is moving at 20 m/s. Speed is the derivative of distance. The derivative also finds the highest and lowest points of any curve, because the slope there is zero, and that is how maximum profit, minimum cost and the best shape for a can are found.
Idea 3: the integral
The integral runs the other way: from a rate to a total. As a picture, it is the area under a curve, found by cutting the area into thin strips and taking the limit as the strips get thinner.
Example. The area under the line y = 2x from x = 0 to x = 3. It is a triangle with base 3 and height 6, so the area is ½ × 3 × 6 = 9. Calculus gets the same answer without the triangle: the function whose derivative is 2x is x², and 3² − 0² = 9.
Why it matters. If you know the speed of a car at every moment, the integral gives the distance it has travelled. It also gives the areas and volumes of curved shapes, the work done by a changing force, and totals of anything that arrives at a changing rate.
The theorem that joins them
Look again at the last example. To find an area, we used a derivative backwards. That is the fundamental theorem of calculus: differentiation and integration undo each other, the way multiplication and division do. It is the reason calculus is one subject and not two.
The first rules, in one table
| Rule | Derivative | Example |
|---|---|---|
| Constant | The derivative of a number is 0 | The derivative of 7 is 0 |
| Power rule | The derivative of xⁿ is n·xⁿ⁻¹ | x³ gives 3x² |
| Constant multiple | The constant stays | 5x² gives 10x |
| Sum | Term by term | x³ + 5x² gives 3x² + 10x |
| Sine and cosine | sin x gives cos x; cos x gives −sin x | |
| Exponential | eˣ gives eˣ |
| Rule | Integral | Example |
|---|---|---|
| Power rule backwards | The integral of xⁿ is xⁿ⁺¹ ÷ (n + 1), plus C | x² gives x³ ÷ 3 + C |
| Definite integral | Put in the top value, subtract the bottom | 2x from 0 to 3 gives 9 |
The "+ C" is a constant: many functions have the same derivative, and C stands for the one you cannot see. The product rule, quotient rule and chain rule come next and finish a first course.
What you need before you start
Calculus has few ideas. Almost every lost mark is algebra. Before the first lesson you should be able to do these without stopping to think:
- Functions: read f(x), substitute, and sketch a line, a parabola and a simple curve.
- Exponents: including negative and fractional ones. 1/x² is x⁻², and √x is x to the power ½.
- Factoring and expanding: quadratics, difference of squares.
- Fractions with letters: adding, simplifying, cancelling correctly.
- Trigonometry: sine, cosine and tangent, the unit circle, radians.
- Slope of a line: rise over run, and the equation y = mx + c.
A check for each of these, and a plan for the gaps, is in can I learn calculus at 14.
Where it is taught
| Country | Where basic calculus sits |
|---|---|
| United States | Grade 12 for most students (AP Calculus AB or BC, or a school calculus course); Grade 11 on the accelerated route |
| India (CBSE) | Class 11: limits and derivatives. Class 12: continuity, differentiation, applications, integrals, differential equations |
| United Kingdom | Year 12 of A-level maths: differentiation and integration of powers of x; Year 13 takes it further |
| Philippines | Basic Calculus, a Grade 11 subject in the STEM strand: limits and continuity, derivatives, integration |
The US route in detail is in what grade do you take calculus.
How to start
- Check the algebra first. A week on exponents, factoring and functions saves a term.
- Learn each idea as a picture before a formula. Draw the slope. Shade the area.
- Do the limit by hand a few times, with a table of values like the one above, so the rules mean something.
- Then learn the rules and practise them daily. Twenty minutes a day for six weeks covers derivatives.
- Check every answer. A derivative can be checked by estimating the slope; an integral can be checked by differentiating it.
How long the whole subject takes is set out in how long does it take to learn calculus.
Where to get help
A 1-on-1 online calculus class with a tutor who holds a master's in mathematics: the algebra check in the first class, each idea as a picture and then as a rule, and every line of your working read aloud, for AP Calculus, CBSE Class 11 and 12, A-level and first-year university courses. The free 30-minute first class is a real lesson on the topic you are stuck on.
Questions parents ask
1What is basic calculus?
2What is basic calculus all about?
3What are the 4 concepts of calculus?
4Is basic calculus hard?
5What grade is basic calculus taught in?
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