Introduction to Calculus
Calculus is the mathematics of change and accumulation. Two big ideas: the derivative (rate of change) and the integral (accumulated area).
Limits
A limit describes what a function approaches as x gets close to a value — even if the function is undefined there.
lim(x→2) of (x² − 4)/(x − 2) = lim(x→2)(x+2) = 4
Derivatives
The derivative measures instantaneous rate of change — the slope of the tangent line at any point.
If f(x) = x², then f'(x) = 2x (power rule)
At x = 3: slope = 2(3) = 6
Power rule: d/dx(xⁿ) = nxⁿ⁻¹
Common Derivative Rules
| Function | Derivative |
|---|---|
| c (constant) | 0 |
| xⁿ | nxⁿ⁻¹ |
| sin x | cos x |
| eˣ | eˣ |
Integrals
The integral accumulates values — geometrically, the area under a curve. It is the inverse of differentiation (Fundamental Theorem of Calculus).
∫x² dx = x³/3 + C (reverse power rule: add 1, divide by new power)
What a Derivative Really Measures
Average speed over a whole trip is easy: distance divided by time. A derivative answers the harder question — how fast were you going at one exact instant?
From average to instantaneous
Pick two points on a curve and the line between them gives an average rate of change. Slide the second point closer and closer to the first, and that line settles onto the tangent at a single point. The limit of those slopes is the derivative.
That is the entire idea. The rules are just shortcuts that skip the sliding.
Using the power rule
If f(x) = 3x⁴, bring the exponent down and reduce it by one: f'(x) = 12x³.
If f(x) = 5x, the derivative is 5 — a straight line has the same slope everywhere.
If f(x) = 7, the derivative is 0 — a constant never changes.
Reading a Derivative
The sign of a derivative describes the shape of the original function, which is why calculus is so useful for finding maximums and minimums.
- Where
f'(x) > 0, the function is increasing. - Where
f'(x) < 0, the function is decreasing. - Where
f'(x) = 0, the curve has levelled off — a peak, a valley, or a flat spot.
Finding a maximum
For f(x) = −x² + 6x, the derivative is −2x + 6. Setting it to zero gives x = 3. Since the parabola opens downward, that is the maximum, and f(3) = 9.
Integrals as Accumulation
If a derivative takes a total and reports its rate, an integral takes a rate and rebuilds the total.
Speed and distance
Plot speed against time. The area under that graph is distance travelled. Driving at a steady 60 km/h for 2 hours makes a rectangle of area 120 — and that is 120 km.
When speed varies, the graph is a curve rather than a rectangle, and integration is how you measure the area anyway.
+ C stands for every constant that would have produced the same derivative.
Common Mistakes
- Forgetting
+ Con an indefinite integral. It is a real part of the answer, not decoration. - Assuming a limit is just substitution. Often it is — but when substitution gives
0/0, you must factor or simplify first. - Applying the power rule to
1/x. That case is the exception, and its integral involves a logarithm. - Reading
f'(x) = 0as automatically a maximum. It could be a minimum or neither; you have to check.
Where You Will Use This
- Physics, where velocity is the derivative of position and acceleration is the derivative of velocity.
- Engineering, for finding the design that minimizes material or maximizes strength.
- Economics, where marginal cost is the derivative of total cost.
- Medicine and biology, for modelling how drug concentration or a population changes over time.
Summary
- A limit describes what a function approaches, even where it is undefined.
- A derivative is an instantaneous rate of change, and the slope of the tangent line.
- The power rule handles most early derivatives: multiply by the exponent, then subtract one from it.
- An integral accumulates — geometrically, the area under a curve.
- Differentiation and integration are inverse operations.
FAQ
What is C in an integral? The constant of integration — any constant disappears when you differentiate, so we add C back.
What is calculus used for? Physics (motion), engineering (optimization), economics (marginal cost), biology (population growth).
Do I need to be good at algebra first? Yes, and it is the single best predictor of how calculus goes. Most calculus errors are algebra errors wearing a disguise.
What is the difference between a definite and an indefinite integral? An indefinite integral returns a family of functions and needs + C. A definite integral has limits attached and returns a single number.
What does dy/dx mean? It is another way of writing the derivative — the change in y with respect to the change in x. It means the same thing as f'(x).
Who invented calculus? Isaac Newton and Gottfried Leibniz developed it independently in the late 1600s. The notation most classes use today is Leibniz's.
Quick Quiz
Test what you just learned. Choose the best answer for each question.