### Course Outline

Calculus 12 is an advanced high school mathematics course. Calculus 12 is organized into four units:

- Limits and Continuity
- Derivatives
- Applications of Derivatives
- The Definite Integral and Its Applications

Students in Calculus 12 will explore the following topics:

#### Limits and Continuity

*Introduction to Limits**Rates of Change and Limits**Continuity and Piecewise functions**Rates of Change, Secant and Tangent Lines*

- Calculate and interpret average and instantaneous rate of change.
- Calculate limits for function values and apply limit properties with and without technology
- identify the intervals upon which a given function is continuous and understanding the meaning of a continuous function
- Remove removable discontinuities by extending or modifying a function
- Apply the properties of algebraic combinations and composites of continuous functions
- Apply, understand and explain average and instantaneous rates of change an extend these concepts to secant lines and tangent line slopes
- Understand the development of the slope of a tangent line from the slope of a secant line
- Find the equations of the tangent and normal lines at a given point

#### Derivatives:

*Building a conceptual understanding of the derivative including the definition of the derivative**Differentiability**Introduction to Limits**Rules for Differentiation**The Chain Rule**Implicit Differentiation**Graphs of trig functions and reciprocal trig functions**Derivatives of trig functions**Graphs and derivatives of trig functions**Determine, describe, and apply the value for ‘e’**Derivatives of Exponential and Logarithmic Functions**Velocity and Other Rates of Change*

- Demonstrate an understanding of the definition of the derivative.
- Demonstrate an understanding of the connection between the graphs of f(x) and f’(x)
- Find where a function is not differentiable and distinguish between corners, cusps, discontinuities, and vertical tangents.
- Derive, apply, and explain power, sum, difference, product and quotient rules.
- Apply the chain rule to composite functions
- Use derivatives to analyze and solve problems involving rates of change.
- Apply the rules for differentiating the six trigonometric functions
- Demonstrate understanding of implicit differentiation and identify situations that require implicit differentiation
- Apply the rules for differentiating the six inverse trigonometric functions (optional topic)
- Calculate and apply derivatives of exponential and logarithmic functions
- Apply Newton’s method to approximate zeros of a function (optional topic)
- Estimate the change in a function using differentials and apply them to real world situations
- Solve and interpret related rate problems

#### Applications of Derivatives

*Critical points and Absolute Extreme Values of Functions**Mean Value Theorem**Curve Sketching**Connecting f’(x) and f’’(x) with the graph of f(x)**Modeling and Optimization**Related Rates*

- Find the intervals on which a function is increasing or decreasing
- Apply the First and Second Derivative Tests to determine the local extreme values of a function
- Determine the concavity of a function and locate the points of inflection by analyzing the second derivative
- Solve application problems involving maximum or minimum values of a function
- Demonstrate an understanding of critical points and absolute extreme values of a function

#### Definite Integral and Its Applications

*Sigma Notation**Estimating the area under a curve with Finite Sums (Rectangular Approximation Method)**Definite Integrals and Riemann Sums**Definite Integrals and Antiderivatives**Fundamental Theorem of Calculus**Antidifferentiation by Substitution**Antidifferentiation by Parts (optional)**Initial Value Problems**Integral as Net Change**Areas in the Plane*

- Apply and understand how Riemann’s sum can be used to determine the area under a polynomial curve
- Demonstrate an understanding of the meaning of area under the curve
- Express the area under the curve as a definite integral
- Compute the area under a curve using a numerical integration procedure
- Solve initial value problems of the form dy/dx = f(x), y0=f(x0)
- Apply rules for definite integrals
- Apply the Fundamental Theorem of Calculus
- Understand the relationship between the derivative and the definite integral as expressed in both parts of the Fundamental Theorem of Calculus
- Construct antiderivatives using the Fundamental Theorem of Calculus
- Find antiderivatives of polynomials, ekx, and selected trigonometric functions of kx
- Compute indefinite and definite integrals by the method of substitution
- Apply integration by parts to evaluate indefinite and definite integrals (optional)
- Solve problems in which a rate is integrated to find the net change over time
- Apply integration to calculate areas of regions in a

**Updated August 24, 2021**