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