## Calculus 12

### 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