### 1: Precalculus and Limits

#### 1.A: General Learner Expectations

1.A.1: Students are expected to understand that functions, as well as variables, can be combined, using operations, such as addition and multiplication, and demonstrate this, by:

1.A.1.1: describing the relationship among functions after performing translations, reflections, stretches and compositions on a variety of functions

1.A.1.2: drawing the graphs of functions by applying transformations to the graphs of known functions

1.A.1.3: expressing final algebraic and trigonometric answers in a variety of equivalent forms, with the form chosen to be the most suitable form for the task at hand

1.A.2: Students are expected to understand that functions can be transformed, and these transformations can be represented algebraically and geometrically, and demonstrate this, by:

1.A.2.1: describing the relationship among functions after performing translations, reflections, stretches and compositions on a variety of functions

1.A.2.2: drawing the graphs of functions by applying transformations to the graphs of known functions

1.A.2.3: expressing final algebraic and trigonometric answers in a variety of equivalent forms, with the form chosen to be the most suitable form for the task at hand

#### 1.B: Conceptual Understanding

1.B.1: Students will demonstrate conceptual understanding of the algebra of functions, by:

1.B.1.3: expressing the sum, product, difference and quotient, algebraically and graphically, given any two functions

1.B.1.5: illustrating the difference between the concepts of equation and identity in trigonometric contexts.

1.B.2: Students will demonstrate conceptual understanding of the transformation of functions, by:

1.B.2.1: describing the similarities and differences between the graphs of y = f(x) and y = af [k(x+c)]+d, where a, k, c and d are real numbers

1.B.2.2: describing the effects of the reflection of the graphs of algebraic and trigonometric functions across any of the lines y = x, y = 0, or x = 0

1.B.3: Students will demonstrate conceptual understanding of equivalent forms, by:

1.B.3.1: describing what it means for two algebraic or trigonometric expressions to be equivalent.

#### 1.C: Procedural Knowledge

1.C.1: Students will demonstrate competence in the procedures associated with the algebra of functions, by:

1.C.1.2: finding the sum, difference, product, quotient and composition of functions

1.C.1.2.a: primary and reciprocal ratio

1.C.1.2.c: sum and difference sin (A±B) cos (A±B)

1.C.1.2.d: Pythagorean

1.C.2: Students will demonstrate competence in the procedures associated with the transformation of functions, by:

1.C.2.1: sketching the graph of, and describing algebraically, the effects of any translation, reflection or dilatation on any of the following functions or their inverses:

1.C.2.1.a: linear, quadratic or cubic polynomial

1.C.2.1.b: absolute value

1.C.2.1.d: exponential

1.C.3: Students will demonstrate competence in the procedures associated with the construction of equivalent forms, by:

1.C.3.2: rationalizing expressions containing a numerator or a denominator that contains a radical

#### 1.D: Problem-Solving Contexts

1.D.1: Students will demonstrate problem-solving skills, by:

1.D.1.3: translating problem conditions into equation or inequality form.

### 2: Derivatives and Derivative Theorems

#### 2.B: Conceptual Understanding

2.B.2: Students will demonstrate conceptual understanding of derivative theorems, by:

2.B.2.7: describing the second derivative geometrically.

2.B.3: Students will demonstrate conceptual understanding of the derivatives of trigonometric functions, by:

2.B.3.1: demonstrating that the three primary trigonometric functions have derivatives at all points where the functions are defined

#### 2.C: Procedural Knowledge

2.C.1: Students will demonstrate competence in the procedures associated with derivatives, by:

2.C.1.1: finding the slopes and equations of tangent lines at given points on a curve, using the definition of the derivative

2.C.2: Students will demonstrate competence in the procedures associated with derivative theorems, by:

2.C.2.1: finding the derivative of a polynomial, power, product or quotient function

2.C.2.5: finding the slope and equations of tangent lines at given points on a curve

2.C.2.6: finding the second and third derivatives of functions.

2.C.3: Students will demonstrate competence in the procedures associated with derivatives of trigonometric functions, by:

2.C.3.1: calculating the derivatives of the three primary and three reciprocal trigonometric functions

2.C.3.3: using the power, chain, product and quotient rules to find the derivatives of more complicated trigonometric functions

### 3: Applications of Derivatives

#### 3.A: General Learner Expectations

3.A.1: Students are expected to understand that calculus is a powerful tool in determining maximum and minimum points and in sketching of curves, and demonstrate this, by:

3.A.1.6: fitting mathematical models to situations described by data sets.

#### 3.C: Procedural Knowledge

3.C.1: Students will demonstrate competence in the procedures associated with maxima and minima, by:

3.C.1.4: determining vertical, horizontal and oblique asymptotes, and domains and ranges of a function

### 4: Integrals, Integral Theorems and Integral Applications

#### 4.B: Conceptual Understanding

4.B.1: Students will demonstrate conceptual understanding of antiderivatives, by:

4.B.1.2: showing that many different functions can have the same derivative

4.B.2: Students will demonstrate conceptual understanding of area limits, by:

4.B.2.2: establishing the existence of upper and lower bounds for the area under a curve.

#### 4.C: Procedural Knowledge

4.C.1: Students will demonstrate competence in the procedures associated with antiderivatives, by:

4.C.1.2: finding the family of curves whose first derivative has been given

### 5: Calculus of Exponential and Logarithmic Functions

#### 5.A: General Learner Expectations

5.A.1: Students are expected to understand that exponential and logarithmic functions have limits, derivatives and integrals that obey the same theorems as do algebraic and trigonometric functions, and demonstrate this, by:

5.A.1.5: fitting mathematical models to situations described by data sets.

#### 5.B: Conceptual Understanding

5.B.1: Students will demonstrate conceptual understanding of the calculus of exponential and logarithmic functions, by:

5.B.1.1: defining exponential and logarithmic functions as inverse functions

### 6: Numerical Methods

#### 6.B: Conceptual Understanding

6.B.1: Students will demonstrate conceptual understanding of the principles of numerical analysis, by:

6.B.1.2: identifying when a particular numerical method is likely to give poor results

6.B.1.5: describing the basis of a limit, derivative, equation root or integral procedure in geometric terms

### 7: Volumes of Revolution

#### 7.D: Problem-Solving Contexts

7.D.1: Students will demonstrate problem-solving skills, in one or both of the following, by:

7.D.1.1: deriving formulas for the volume of a cylinder, cone and sphere

### 9: Applications of Calculus to Biological Sciences

#### 9.B: Conceptual Understanding

9.B.1: Students will demonstrate conceptual understanding of the links between calculus and the biological sciences, by:

9.B.1.1: defining exponential and logarithmic functions as inverse functions

Correlation last revised: 9/24/2019

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