Study the expanded notes and the credited detailed textbook lessons. The current chapter index covers core topics; it is not a claim of exhaustive coverage of every examination syllabus. Check the official syllabus and primary references.
Calculus — lesson overview
Limits and continuity
A limit describes the value a function approaches near a point, where it exists; the function’s value at that point is a separate question.
Continuity at a point requires an appropriate limit equal to the function’s defined value there.
A removable hole can leave the limiting value unchanged while making the original function discontinuous at that point.
Derivatives
A derivative measures local rate of change as a limit of difference quotients, when that limit exists. Geometrically it gives tangent slope in the standard setting.
For f(x)=x², f′(x)=2x. At x=3 the tangent slope is 6; this is not the slope of every chord through the graph.
Differentiability implies continuity at a point, but continuity alone does not ensure differentiability; a sharp corner provides an introductory counterexample.
Integration
An indefinite integral represents a family of antiderivatives and includes an arbitrary constant: ∫2x dx=x²+C.
A definite integral can describe signed accumulation; regions below an axis contribute negatively under the ordinary coordinate convention.
Geometric area and signed integral need not be numerically equal unless signs and intervals are handled appropriately.
Limits describe behaviour near a point, not merely substitution
A limit asks what value f(x) approaches as x approaches a specified point through its domain. The function’s value at the point may exist, may differ from the limit, or may be undefined. For f(x)=(x²−1)/(x−1), the original expression is undefined at x=1, but for x≠1 it equals x+1. Hence the limit as x approaches 1 is 2; this does not make the original undefined value suddenly equal to 2.
A two-sided limit requires compatible approach from both sides. For a piecewise function with values approaching 1 on the left and 3 on the right, a two-sided limit at the joining point does not exist, even if a value is assigned at that point. Continuity requires the function to be defined at the point, the limit to exist, and the value to equal the limit.
Algebraic simplification is therefore justified by the behaviour near the point, not by illegally cancelling a zero denominator at the point itself. Check domains before cancelling, rationalising, or dividing. This habit prevents a valid limit computation from being misreported as a claim about a function’s value where its original formula is undefined.
Derivative from average change to instantaneous change
The derivative f′(a) is the limit of [f(a+h)−f(a)]/h as h approaches zero, when the limit exists. The fraction is an average rate over a small interval; the limit produces an instantaneous rate and the tangent slope. It is not simply the function divided by its input.
For f(x)=x², the difference quotient is [(x+h)²−x²]/h=(2xh+h²)/h=2x+h for h≠0. Letting h approach zero gives f′(x)=2x. At x=3 the tangent slope is 6. The computation explains the power rule in this example instead of treating differentiation as an unexplained change in exponents.
Differentiability implies continuity at the point, but continuity does not imply differentiability. The function |x| is continuous at zero but has left slope −1 and right slope 1, so no single derivative exists there. A graph can be unbroken and still have a corner. Check one-sided slopes when the definition or graph changes at a joining point.
Rules work because functions combine in specific ways
The derivative of a sum is the sum of derivatives. For a product, (uv)′=u′v+uv′, not u′v′. For a quotient, (u/v)′=(u′v−uv′)/v² where the expression is defined. The chain rule differentiates a composition: [f(g(x))]′=f′(g(x))g′(x). Identify the structure before choosing a rule.
Example: for y=(3x+1)², the outer square contributes 2(3x+1) and the inner expression contributes 3, so y′=6(3x+1). Expanding first gives 9x²+6x+1 and derivative 18x+6, confirming the same result. Omitting the inner derivative gives a wrong answer even though it may resemble a familiar power rule.
In an application, keep units. If s(t) is position in metres and t is seconds, s′(t) is velocity in metres per second and s″(t) is acceleration in metres per second squared. Negative velocity indicates direction under the chosen convention; it does not mean negative total distance travelled. Optimisation also requires checking the relevant domain, critical points, and endpoints rather than setting a derivative to zero and stopping.
Integration: accumulation and antiderivatives
An indefinite integral represents a family of antiderivatives. Since the derivative of x³/3 is x², ∫x² dx=x³/3+C. The constant is needed because all functions differing by a constant have the same derivative. The power rule for integrating xⁿ has the restriction n≠−1; the special form for 1/x is ln|x|+C on an interval excluding zero.
A definite integral describes signed accumulation over an interval and, for an appropriate continuous function, can be evaluated through an antiderivative using the Fundamental Theorem of Calculus: ∫ from a to b of f(x) dx=F(b)−F(a). For ∫ from 0 to 2 of x² dx, use x³/3 to obtain 8/3. There is no arbitrary +C in the final definite value.
Signed area and total geometric area differ when a function crosses the horizontal axis. The integral of velocity gives displacement; total distance requires accumulating speed, the absolute value of velocity. This parallel connects graphical interpretation, units, and mechanics. A negative definite integral is therefore not inherently an error; it can indicate net negative signed accumulation.
