Implicit Differentiation, Parametric Curves, and Related Rates

Understand derivatives as limits of difference quotients, differentiate fundamental functions from first principles, and interpret first and second derivatives geometrically.

Differentiate power, exponential, logarithmic, and trigonometric functions using product, quotient, chain, and inverse differentiation rules.

Differentiate implicit relations and parametric equations, and solve connected rates of change problems.

Determine tangent and normal equations, find extrema, analyze curve behavior, and solve real-world optimization problems.

Understand integration as the limit of Riemann sums, apply the Fundamental Theorem of Calculus, and integrate standard function families.

Master advanced integration methods including u-substitution, integration by parts, and partial fractions for definite and indefinite integrals.

Evaluate definite integrals to calculate the total geometric area under a curve and enclosed between intersecting curves.

Formulate first-order differential equations from contextual scenarios, find general and particular solutions via separation of variables, and analyze model behavior.