Quick check · Calculus I · Change and local behavior
Level II · Optional · Open anytime
Calculus I & II
Start with a changing quantity, then build the mathematics to describe its slope, accumulation, and approximation error. Calculus I develops limits and derivatives; Calculus II develops integrals, differential equations, and series. Four closing sections build fast mental arithmetic: two-digit products, division, roots and logarithms, and checks. Each short section has a worked example and two explained checks. Familiarity with algebra, functions, logarithms, and basic trigonometry is helpful; you can revisit Level I mathematics at any time.
Study any section in any order. These checks do not affect your core course or assessments. Choice answers and elective progress are tracked in this browser.
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Calculus I · Change and local behavior · Section 1 of 2
Functions, domains, and the language of change
A function assigns one output to each allowed input. Write y = f(x), read it as ‘y depends on x,’ and state the domain before calculating. For f(x) = √x the real domain is x ≥ 0; for ln(x) it is x > 0. A formula outside its domain does not produce a meaningful real answer.
A graph puts inputs horizontally and outputs vertically. An average rate of change over [a, b] is [f(b) − f(a)]/(b − a). Its units are output units per input unit. This is a secant slope between two points, not necessarily the slope anywhere else on the curve.
Composition means feeding one output into another function: f(g(x)). An inverse reverses an operation on a domain where the original function is one-to-one. The functions exp and ln reverse one another for positive outputs. Keep the pure mathematical operation separate from any economic interpretation of the input.
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Keep in mind: Always identify the domain, the input units, and the output units.
Pause and try two quick checks.
You can move on at any time. Each answer explains the reasoning.
Quick check · Calculus I · Change and local behavior
What is the real domain of ln(x − 2)?
Further reading and documentation
- OpenStax: Calculus Volume 1 — functions, limits, derivatives, and integration ↗
- OpenStax: Calculus Volume 2 — integration techniques, differential equations, and series ↗
- Optional external course: Krista King, Become a Calculus 1 Master (Udemy; independent provider) ↗
- Optional external course: Krista King, Become a Calculus 2 Master (Udemy; independent provider) ↗
- Arthur Benjamin: A performance of mathemagic (TED talk); the techniques are developed in Benjamin & Shermer, Secrets of Mental Math ↗
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