Calculus core ideas: limits, derivatives, integrals and the rules
Three ideas carry most of first-year calculus: limits, derivatives and integrals. These pages take each one from the definition to a worked example, then show how the rules and the Fundamental Theorem connect them.
7 pages in this section
- ReferenceDerivative rules cheat sheetDerivative rules of a first calculus course on one page: all six trig functions, a^x, logs, inverse trig, with examples.
- GuideWhat Is a Derivative in Calculus? Simple ExplanationA derivative is the instantaneous rate of change of a function: the slope of its tangent line. Plain meaning, worked examples, notation and when it fails.
- GuideWhat Is a Limit in Calculus? Plain Meaning, ExamplesA limit is the value a function approaches as its input nears a point. Plain meaning, worked examples, when limits fail to exist, and the formal definition.
- GuideWhat Is an Integral in Calculus? Area, Sums, ExamplesAn integral adds up a changing quantity: for a positive function it is the area under the curve. Plain meaning, worked examples, the Fundamental Theorem.
- StoryFundamental Theorem of Calculus Explained: Parts 1 and 2The Fundamental Theorem of Calculus links derivatives and integrals. Part 1, Part 2, worked examples, when to use each, and when it fails.
- GuideChain Rule Explained: Differentiating Composite FunctionsThe chain rule in plain words: inside and outside functions, rates that multiply, step-by-step examples, real-life uses and the mistakes to avoid.
- GuideOptimization Problems in Calculus: How to Solve ThemHow to solve optimization problems with derivatives: the steps, 4 worked examples with answers (fence, box, revenue, can), checks and common mistakes.