Calculus: Integration and Differentiation

Define the derivative from first principles (limit definition) and interpret it as instantaneous rate of change and slope of a tangent.

Explain the relationship between differentiability and continuity, identifying points where functions are continuous but not differentiable.

Differentiate polynomial, exponential, and logarithmic functions using standard formulas.

Differentiate trigonometric and inverse trigonometric functions using standard formulas.

Apply the product rule and quotient rule to differentiate combinations of functions.

Apply the chain rule to differentiate composite functions.

Perform implicit differentiation on equations not explicitly solved for y.

Perform logarithmic differentiation for functions of the form [f(x)]^[g(x)].

Find equations of tangents and normals to curves at given points.

Solve related rates problems in geometric and physical contexts (e.g., kinematics, fluid flow).

Differentiate parametrically defined functions and apply to physical contexts such as projectile motion.

Compute higher-order derivatives and interpret the second derivative physically as acceleration.

Analyze the angle between two intersecting curves.

Determine the shortest distance between two curves or a point and a curve.

State, prove, and apply Rolle's Theorem to verify the existence of stationary points.

State, prove, and apply the Lagrange Mean Value Theorem (LMVT) and understand its geometric interpretation.

Identify the seven indeterminate forms (0/0, infinity/infinity, etc.) and convert them into a form suitable for L'Hopital's rule.

Apply L'Hopital's rule to evaluate limits involving indeterminate forms.

Determine intervals of increasing and decreasing behavior (monotonicity) using the first derivative.

Find local maxima and minima using the first derivative test.

Determine concavity and locate points of inflection using the second derivative.

Find local maxima and minima using the second derivative test.

Identify horizontal, vertical, and slant asymptotes of a function using limits.

Combine derivative analysis, asymptotes, and key features to construct an accurate sketch of a function's graph.

Formulate a mathematical function and constraints to model a given optimization word problem.

Find the absolute maximum and minimum of a continuous function on a closed interval.

Apply calculus techniques to solve optimization problems in geometric contexts (e.g., maximizing area, minimizing surface area).

Apply optimization techniques to physics problems such as maximizing range in projectile motion or applying Fermat's principle of least time.

Understand integration as the process of anti-differentiation and use it to integrate standard functions (polynomial, exponential, trig).

Perform integration by substitution (change of variable) for a wide variety of functions.

Simplify and evaluate integrals using trigonometric identities.

Perform integration by parts and apply the ILATE/LIATE priority rule for choosing u and dv.

Apply integration by parts multiple times or in combination with substitution to solve complex integrals.

Integrate rational functions by decomposing them into partial fractions (including linear, repeated linear, and irreducible quadratic factors).

Evaluate integrals that result in inverse trigonometric forms.

Evaluate integrals involving sqrt(a^2-x^2), sqrt(a^2+x^2), and sqrt(x^2-a^2) using trigonometric substitutions.

Evaluate integrals of the form ∫ dx/(ax^2+bx+c) and ∫ dx/sqrt(ax^2+bx+c) by completing the square.

Evaluate integrals of the form ∫ (px+q)/(ax^2+bx+c) dx and its variations.

Derive and apply reduction formulas for integrals of powers of trigonometric functions (e.g., sin^n(x), sec^n(x)).

Define the definite integral as the limit of a Riemann sum and evaluate simple integrals from first principles.

State and apply both parts of the Fundamental Theorem of Calculus to evaluate definite integrals.

Apply properties of definite integrals: linearity, interval splitting, and comparison.

Evaluate definite integrals using substitution, ensuring correct transformation of limits.

Use symmetry properties (even/odd functions) to simplify definite integrals.

Apply the King, Queen, and other standard properties (e.g., ∫f(x)dx = ∫f(a+b-x)dx) to solve challenging definite integrals.

Apply the periodicity property for integrals of periodic functions.

Differentiate integrals with variable limits using the Leibniz integral rule.

Calculate the area under a curve and the area enclosed between two or more curves.

Calculate areas for regions defined by functions involving modulus or piecewise definitions.

Calculate the volume of solids of revolution using the disk and washer methods.

Calculate the volume of solids of revolution using the cylindrical shells method.

Calculate the arc length of a plane curve defined in Cartesian, parametric, or polar forms.

Compute displacement from a velocity function and work done by a variable force.

Calculate the center of mass of a one-dimensional rod with variable density.

Calculate the center of mass of a two-dimensional lamina of uniform density (e.g., semicircle, triangle).

Calculate the moment of inertia for simple objects (rod, ring, disk) about a given axis.

Model and solve problems related to fluid pressure and hydrostatic force on a vertical surface.

Form a differential equation from a family of curves by eliminating arbitrary constants.

Determine the order and degree of a given differential equation.

Solve differential equations using the method of separation of variables.

Solve homogeneous differential equations using the substitution y = vx.

Solve first-order linear differential equations using the integrating factor method.

Apply differential equations to solve geometric problems (e.g., finding orthogonal trajectories).

Model and solve physical problems involving exponential growth/decay (e.g., radioactive decay, Newton's cooling).

Model and solve problems involving first-order chemical reaction rates.