Ch 12 introduces calculus — the mathematics of change. Students develop an intuitive understanding of limits, then use limits to define derivatives (instantaneous rate of change) and learn differentiation rules.
A limit describes the value a function approaches as the input approaches a specific value. Laws: limit of sum/difference/product/quotient = sum/difference/product/quotient of limits. Special limit: lim(x→0) sin x/x = 1.
The derivative of f(x) from first principle: f'(x) = lim(h→0) [f(x+h) − f(x)]/h. This gives the slope of the tangent at any point and the instantaneous rate of change.
Power rule: d/dx(xⁿ) = nxⁿ⁻¹. Sum rule: (f±g)' = f'±g'. Product rule: (fg)' = f'g + fg'. Quotient rule: (f/g)' = (f'g − fg')/g². d/dx(sin x) = cos x. d/dx(cos x) = −sin x.
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The derivative f'(x) measures the instantaneous rate of change of f at x. Geometrically, it's the slope of the tangent line to the curve y = f(x). It's defined as the limit of the difference quotient as the interval approaches zero.
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