Use first- and second-derivative sign analysis to find critical numbers, determine intervals of increase and decrease, classify local extrema, and identify concavity and points of inflection. Use derivative information to write tangent and normal line equations and sketch standard polynomial, rational, exponential, logarithmic, and trigonometric curves using intercepts, domain restrictions, end behavior, and asymptotes where appropriate. Model and solve single-variable optimization and related-rates problems by applying derivative tests, endpoint analysis, implicit differentiation, constraints, feasible domains, units, and contextual interpretation.