Limits & Continuity
Intuitive understanding of the limit process
Using graphs and tables of data to determine limits
Algebraic techniques for evaluating limits
Continuity and one-sided limits
Geometric understanding of graphs of continuous functions
Intermediate Value Theorem
Infinite limits and using limits to find asymptotes
Differentiation
Local linearity (zooming in on a curve)
Understanding the derivative graphically, numerically, and analytically
Approximating rates of change from graphs and tables
The derivative as a limit of the difference quotient and slope of a curve
Translating verbal descriptions into equations and vice versa
Relationship between differentiability and continuity
Functions with a vertical tangent or no tangent at a point
Differentiation rules: power, trig, sums, differences, products, quotients
Natural logarithm: differentiation and integration
Exponential functions: differentiation and integration
Inverse functions and their derivatives
Inverse trig functions: differentiation and integration
Applications of Derivatives
Extrema on an interval and the Extreme Value Theorem
Rolle's Theorem and Mean Value Theorem and their geometric consequences
Increasing/decreasing functions and the First Derivative Test
Optimization: relative and absolute extrema
Tangent lines and linear approximations
Position, velocity, acceleration, and rectilinear motion
Concavity and its relationship to first and second derivatives
Limits at infinity and curve sketching
Relating the graphs of f, f′, and f′′
L'Hôpital's Rule and its use in determining limits
Integration
Antiderivatives and indefinite integration
Basic properties of the definite integral
Definite integral as a limit of Riemann sums
Left, right, and midpoint Riemann sums
Fundamental Theorem of Calculus Part 1
Fundamental Theorem of Calculus Part 2
Functions defined by integrals (accumulation functions)
Integration by partial fractions
Mean Value Theorem for Integrals and average value
Improper integrals: convergence and divergence
Differential Equations
Solving separable differential equations
Exponential growth and decay models
Logistic differential equations and modeling
Slope fields: interpreting and drawing
Euler's method as a numerical solution
Applications of Integration
The integral as an accumulator of rates of change
Area of a region between two curves
Volume of a solid with known cross sections
Volume of solids of revolution (discs and washers)
Motion along a line using definite integrals with initial conditions
Distance traveled by a particle along a line
Sequences & Series
Convergence and divergence of sequences
Series as a sequence of partial sums
Geometric series and applications
nth-Term Test for Divergence
Integral Test and p-series
Comparison Test and Limit Comparison Test
Alternating Series Test and Alternating Series Remainder
Absolute vs. conditional convergence
Power series: radius and interval of convergence
Taylor and Maclaurin polynomials and approximations
Maclaurin series for sin x, cos x, eˣ, ln x, 1/(1−x), arctan x
Manipulation of series: substitution, addition, differentiation, integration
Lagrange Error Bound (Taylor's Theorem Remainder)
Parametric, Polar & Vector
Parametric equations and calculus (derivatives, arc length)
Motion along a curve: position, velocity, acceleration, speed, distance
Analysis of curves given in parametric form
Polar coordinates and polar graphs
Analysis of curves given in polar form
Area of a region bounded by polar curves
Derivatives and integrals of vector-valued functions
Position, velocity, and acceleration vectors from derivatives and integrals