Learn advanced calculus techniques for computing limits, derivatives, and integrals, and apply calculus to solve problems in the context of related rates, optimization, particle motion, and differential equations. Dive deeper into complex numbers, vectors, matrices, parametric and polar curves, probability, and statistics.
Leverage factoring and graphing as a strategy to solve polynomial inequalities.
Visualize two-variable linear inequalities in the coordinate plane.
Apply trigonometric identities to simplify trigonometric expressions.
Solve trigonometric equations by extending previous knowledge of algebra and leveraging properties of trigonometric functions.
Limits and Continuity
Compute limits algebraically using advanced techniques such as L'Hopital's rule and conjugate multiplication.
Understand the relationship between limits and asymptotes.
Explain the behavior of a function on an interval using the Intermediate Value Theorem.
Sequences and Series
Determine whether a sequence is convergent or divergent, and compute the limit of a convergent sequence.
Express an arithmetic or geometric series using sigma notation.
Compute the sum of an arithmetic or geometric series.
Relate the graph of a function to properties of its derivative and understand the relationship between differentiability and continuity.
Understand that the derivative of position is velocity, and the derivative of velocity is acceleration.
Calculate derivatives of implicit and inverse functions and use implicit differentiation to solve related rates problems.
Estimate derivatives numerically and approximate values of functions using linearization.
Solve optimization problems by using the derivative to find extrema of functions.
Approximate areas using Riemann sums and compute the definite integral of a function as the limit of a Riemann sum.
Relate integrals and antiderivatives through the fundamental theorem of calculus.
Use integration to compute arc length and area and translate between position, velocity, and acceleration.
Interpret accumulation functions geometrically and compute their derivatives.
Solve advanced integrals using techniques like substitution, long division, completing the square, partial fractions, and integration by parts.
Evaluate improper integrals.
Verify solutions of differential equations and solve elementary differential equations using direct integration.
Solve differential equations using separation of variables and fit unknown coefficients to initial conditions.
Estimate solutions to initial value problems using Euler’s method.
Conic Sections, Parametric Curves, and Polar Curves
Translate between graphs and equations of circles, parabolas, ellipses, and hyperbolas.
Extend prior knowledge of rectangular curves to parametric and polar curves.
Extend differentiation techniques to parametric and polar curves.
Use integration to calculate the area of a polar region or the arc length of a polar curve.
Complex Numbers, Vectors, and Matrices
Connect the algebraic and geometric interpretations of complex numbers through Euler’s formula.
Generalize prior intuitions about arithmetic to vectors and matrices in higher-dimensional space.
Compute dot products and cross products of vectors and interpret them geometrically.
Interpret matrices as linear transformations of points in the coordinate plane.
Compute inverses and determinants of matrices and interpret them geometrically.
Probability and Statistics
Understand independent events both conceptually and quantitatively from the perspective of conditional probability.
Define and perform computations with probability mass functions, cumulative distributions, and expected values of discrete random variables.
Apply binomial, geometric, and normal distributions in modeling contexts.
1.1. Arithmetic Series
Expressing an Arithmetic Series in Sigma Notation
Finding the Sum of an Arithmetic Series
Finding the First Term of an Arithmetic Series
Calculating the Number of Terms in an Arithmetic Series
1.2. Finite Geometric Series
The Sum of a Finite Geometric Series
The Sum of the First N Terms of a Geometric Series
Writing Geometric Series in Sigma Notation
Finding the Sum of a Geometric Series Given in Sigma Notation
1.3. The Binomial Theorem
Pascal's Triangle and the Binomial Coefficients
Expanding a Binomial Using Binomial Coefficients
The Special Case of the Binomial Theorem
Approximating Values Using the Binomial Theorem
2.4. Single-Variable Inequalities
Solving Elementary Quadratic Inequalities
Solving Quadratic Inequalities From Graphs
Solving Quadratic Inequalities Using the Graphical Method
Solving Quadratic Inequalities Using the Sign Table Method
Inequalities Involving Powers of Binomials
Solving Polynomial Inequalities Using a Graphical Method
Solving Inequalities Involving Exponential Functions and Polynomials