Calculus 1 answers two questions: how fast is this changing, and how much has accumulated. Derivatives answer the first, integrals the second, and the fundamental theorem shows they are inverse operations.
Practise derivativesA limit is about approach, not arrival. The function need not be defined at the point, which is exactly why limits can describe instantaneous rates that direct substitution cannot.
Continuity requires three things at a point: the function is defined there, the limit exists, and the two agree. Removable discontinuities fail only the third condition.
The derivative is the limit of a difference quotient, but in practice you apply rules. Identify the outermost structure. Sum, product, quotient or composition, and apply the matching rule before touching the inner parts.
The chain rule is the one students under-apply. Any function of a function, including powers of expressions and trig of expressions, needs the derivative of the inside as a factor.
The sign of the first derivative gives increase and decrease; the sign of the second gives concavity. Together they locate maxima, minima and inflection points without plotting a single value.
Optimisation and related rates follow one recipe: write the relationship between the quantities, differentiate with respect to the right variable, then substitute the known values only at the end.
An antiderivative reverses differentiation, so every indefinite integral carries a constant. The fundamental theorem then evaluates definite integrals by differences of antiderivatives.
Substitution is the chain rule read backwards. Look for a factor that is the derivative of another part of the integrand, and the integral collapses to a table entry.
