Advanced Calculus II (2020 - 2021)

Process Skills

Integrals

Functions

Geometry

Sequences

Series

Vectors

Vector Calculus

use appropriate technology to solve mathematical problems

build new mathematical knowledge through problem-solving

solve problems that arise in mathematics and in other areas

apply and adapt a variety of appropriate strategies to solve problems

monitor and reflect on the process of mathematical problem-solving

recognize reasoning and proof (evidence) as fundamental aspects of mathematics

make and investigate mathematical conjectures

investigate, develop, and evaluate mathematical arguments and proofs

select and use various types of reasoning and methods of proof

organize and consolidate mathematics thinking

communicate mathematical thinking coherently to peers, teachers, and others

analyze and evaluate the mathematical thinking and strategies of others

use the terminology and language of mathematics to express mathematical ideas precisely

recognize and use connections among mathematical ideas

explain how mathematical ideas interconnect and build on one another to produce a coherent whole

recognize and apply mathematics in context outside of mathematics

create and use pictures, manipulatives, models, and symbols to organize, record, and communicate mathematical ideas

select, apply, and translate among mathematical representations to solve problems

use representations to model and interpret physical, social, and mathematical phenomena

evaluate integrals using integration by parts

evaluate integrals of rational functions using partial fractions

evaluate improper integrals

apply concepts of functions including domain, range, intercepts, and symmetry

define parabolas, ellipses, and hyperbolas geometrically

explain the role of conic sections in the reflection of light and sound

sketch curves in polar coordinates

calculate the area of a region with boundary given in polar coordinates

express curves in parametric form

find equations of tangents to curves given parametrically

calculate arc length using integrals

calculate the area of a surface generated by revolution

use Pappus's Theorem on Surface Area to find area of a surface

find least upper bounds and greatest lower bounds

determine the limit of a sequence

state the convergence or divergence of a sequence

calculate the limit of a sequence using L'Hospital's Rule

calculate limits of sequences with indeterminate forms

determine the convergence of improper integrals

write a given sequence in sigma notation

evaluate the sum of series

determine if a series converges or diverges using the integral test, basic comparison test, limit comparison test, root test, and the ratio test

test for absolute and conditional converges of alternating series

test for convergence of MacLaurin series

find a Taylor polynomial for given functions and use them to estimate function values

find the Lagrange form of the remainder of a series and use it to test for accuracy of the polynomial

apply Taylor's Theorem to find the Taylor polynomial of a given function for a given value

find the interval of convergence for a power series

differentiate and integrate power series

calculate distance between points in 3-space

write an equation for spheres with given conditions

calculate the norm of a vector

find a unit vector for a given vector

calculate dot products

find projection vectors

find direction angles of a vector

calculate cross products

calculate the angle between two vectors

find the volume of parallelepipeds

find vector parameterizations for lines

find equations for a plane

calculate distance from a point to a plane

determine whether vectors are parallel, skew, or intersecting

find points of intersections of intersecting vectors

find the angle between two planes

determine the co-planarity of vectors

find unit normal vectors for a plane

find a set of scalar parametric equations for lines formed by the intersection of planes

integrate vectors

apply the rules of differentiation to find the derivative of vectors

find the tangent vector at a given point

sketch curves defined by vectors

calculate the arc length of a curve defined in vector form

find the angular speed and the magnitude of the acceleration of a particle moving along a curve

calculate acceleration vectors