A: Rate of Change

A.1: demonstrate an understanding of rate of change by making connections between average rate of change over an interval and instantaneous rate of change at a point, using the slopes of secants and tangents and the concept of the limit;

A.1.1: describe examples of real-world applications of rates of change, represented in a variety of ways (e.g., in words, numerically, graphically, algebraically)

Compound Interest

A.3: verify graphically and algebraically the rules for determining derivatives; apply these rules to determine the derivatives of polynomial, sinusoidal, exponential, rational, and radical functions, and simple combinations of functions; and solve related problems.

A.3.3: determine algebraically the derivatives of polynomial functions, and use these derivatives to determine the instantaneous rate of change at a point and to determine point(s) at which a given rate of change occurs

Graphs of Derivative Functions

B: Derivatives and Their Applications

B.1: make connections, graphically and algebraically, between the key features of a function and its first and second derivatives, and use the connections in curve sketching;

B.1.2: recognize the second derivative as the rate of change of the rate of change (i.e., the rate of change of the slope of the tangent), and sketch the graphs of the first and second derivatives, given the graph of a smooth function

Graphs of Derivative Functions

B.1.3: determine algebraically the equation of the second derivative f"(x) of a polynomial or simple rational function f(x), and make connections, through investigation using technology, between the key features of the graph of the function (e.g., increasing/ decreasing intervals, local maxima and minima, points of inflection, intervals of concavity) and corresponding features of the graphs of its first and second derivatives (e.g., for an increasing interval of the function, the first derivative is positive; for a point of inflection of the function, the slopes of tangents change their behaviour from increasing to decreasing or from decreasing to increasing, the first derivative has a maximum or minimum, and the second derivative is zero)

Graphs of Derivative Functions

B.1.5: sketch the graph of a polynomial function, given its equation, by using a variety of strategies (e.g., using the sign of the first derivative; using the sign of the second derivative; identifying even or odd functions) to determine its key features (e.g., increasing/ decreasing intervals, intercepts, local maxima and minima, points of inflection, intervals of concavity), and verify using technology

Graphs of Polynomial Functions

C: Geometry and Algebra of Vectors

C.1: demonstrate an understanding of vectors in two-space and three-space by representing them algebraically and geometrically and by recognizing their applications;

C.1.1: recognize a vector as a quantity with both magnitude and direction, and identify, gather, and interpret information about real-world applications of vectors (e.g., displacement, forces involved in structural design, simple animation of computer graphics, velocity determined using GPS)

Adding Vectors
Vectors

C.1.2: represent a vector in two-space geometrically as a directed line segment, with directions expressed in different ways (e.g., 320º; N 40º W), and algebraically (e.g., using Cartesian coordinates; using polar coordinates), and recognize vectors with the same magnitude and direction but different positions as equal vectors

Vectors

C.1.3: determine, using trigonometric relationships [e.g., x = rcos Theta, y = rsin Theta, Theta = tan to the -1 power (y / x) or tan to the -1 power (y / x) + 180º, r = square root of (x² + y²)], the Cartesian representation of a vector in two-space given as a directed line segment, or the representation as a directed line segment of a vector in two-space given in Cartesian form [e.g., representing the vector (8, 6) as a directed line segment]

Adding Vectors
Vectors

C.1.4: recognize that points and vectors in three-space can both be represented using Cartesian coordinates, and determine the distance between two points and the magnitude of a vector using their Cartesian representations

Vectors

C.2: perform operations on vectors in two-space and three-space, and use the properties of these operations to solve problems, including those arising from real-world applications;

C.2.1: perform the operations of addition, subtraction, and scalar multiplication on vectors represented as directed line segments in two-space, and on vectors represented in Cartesian form in two-space and three-space

Adding Vectors
Vectors

C.2.2: determine, through investigation with and without technology, some properties (e.g., commutative, associative, and distributive properties) of the operations of addition, subtraction, and scalar multiplication of vectors

Adding Vectors
Vectors

C.2.3: solve problems involving the addition, subtraction, and scalar multiplication of vectors, including problems arising from real-world applications

Adding Vectors
Vectors

C.2.4: perform the operation of dot product on two vectors represented as directed line segments (i.e., using vector a times vector b = (absolute value of vector a)(absolute value of vector b)(cos Theta)) and in Cartesian form (i.e., using vector a times vector b = (a base 1 of b base 1) + (a base 2 of b base 2) or (vector a times vector b) = (a base 1 of b base 1) + (a base 2 of b base 2) + (a base 3 of b base 3)) in two-space and three-space, and describe applications of the dot product (e.g., determining the angle between two vectors; determining the projection of one vector onto another)

Vectors

C.2.5: determine, through investigation, properties of the dot product (e.g., investigate whether it is commutative, distributive, or associative; investigate the dot product of a vector with itself and the dot product of orthogonal vectors)

Vectors

C.2.8: solve problems involving dot product and cross product (e.g., determining projections, the area of a parallelogram, the volume of a parallelepiped), including problems arising from real-world applications (e.g., determining work, torque, ground speed, velocity, force)

Vectors

C.3: distinguish between the geometric representations of a single linear equation or a system of two linear equations in two-space and three-space, and determine different geometric configurations of lines and planes in three-space;

C.3.1: recognize that the solution points (x, y) in two-space of a single linear equation in two variables form a line and that the solution points (x, y) in two-space of a system of two linear equations in two variables determine the point of intersection of two lines, if the lines are not coincident or parallel

Solving Equations by Graphing Each Side
Solving Linear Systems (Matrices and Special Solutions)
Solving Linear Systems (Slope-Intercept Form)
Solving Linear Systems (Standard Form)