AP Calculus BC

Learning objectives

• Define the derivative as a limit of the difference quotient and interpret it as instantaneous rate of change
• Apply power, constant, sum, and constant-multiple rules to differentiate polynomial functions
• Differentiate all six trigonometric functions and their compositions
• Differentiate exponential (e^x, a^x) and logarithmic (ln x, log_a x) functions
• Use the product and quotient rules to differentiate combinations of functions

Learning objectives

• Apply the chain rule to differentiate composite functions including multi-layer compositions
• Use implicit differentiation to find dy/dx for equations not explicitly solved for y
• Compute derivatives of inverse functions using the inverse function theorem
• Differentiate all six inverse trigonometric functions (arcsin, arccos, arctan, etc.)
• Calculate and interpret second and higher-order derivatives in context

Learning objectives

• Interpret the derivative as a rate of change in applied contexts (population growth, temperature, economics)
• Set up and solve related-rates problems using implicit differentiation
• Use linearization and differentials to approximate function values near a known point
• Apply L'Hopital's Rule to evaluate limits of indeterminate forms (0/0 and infinity/infinity)
• Analyze straight-line motion by connecting position, velocity, and acceleration functions

Learning objectives

• State and apply the Mean Value Theorem to prove the existence of specific rates of change
• Use the first derivative test to classify relative extrema on open intervals
• Use the second derivative test to determine concavity and classify critical points
• Identify inflection points where concavity changes and verify with a sign chart
• Set up and solve optimization problems including checking endpoint and boundary cases

Learning objectives

• Approximate area under a curve using left, right, midpoint Riemann sums and the trapezoidal rule
• State and apply both parts of the Fundamental Theorem of Calculus
• Evaluate definite and indefinite integrals using power, trig, exponential, and 1/x rules
• Apply u-substitution to evaluate integrals, including changing bounds for definite integrals
• Interpret the definite integral as net accumulation of a rate of change in context

Learning objectives

• Write differential equations to model real-world growth, decay, and rate problems
• Sketch and interpret slope fields to visualize families of solution curves
• Use Euler's method to generate approximate numerical solutions step by step
• Solve separable differential equations by separating variables and integrating both sides
• Apply exponential growth/decay models (dy/dt = ky) and logistic growth models

Learning objectives

• Find the average value of a function on an interval using the Mean Value Theorem for Integrals
• Set up and evaluate definite integrals for the area between two curves (vertical and horizontal slicing)
• Compute volumes of solids of revolution using the disc and washer methods around both axes
• Compute volumes of solids with known cross sections (squares, semicircles, equilateral triangles)
• Determine which integration method to use based on the geometry of the problem

Learning objectives

• Compute first and second derivatives (dy/dx and d2y/dx2) for parametric curves
• Calculate arc length for parametric and polar curves using integral formulas
• Convert between polar and Cartesian coordinates and graph polar curves (roses, cardioids, limacons)
• Find area enclosed by polar curves using the formula A = (1/2) integral r^2 d-theta
• Analyze planar motion using vector-valued position, velocity, and acceleration functions

Learning objectives

• Apply the nth-term, ratio, root, comparison, integral, and alternating series tests to determine convergence or divergence
• Construct Taylor and Maclaurin series for e^x, sin x, cos x, 1/(1-x), and ln(1+x)
• Find the radius and interval of convergence for a power series, including endpoint analysis
• Apply the Lagrange error bound to quantify the accuracy of Taylor polynomial approximations
• Represent functions as power series and perform algebraic operations (addition, substitution, differentiation, integration) on series