Georgia

MM Mathematical Modeling

• C.MM.1 Apply mathematics to real-life situations; model real-life phenomena using mathematics.

  • C.MM.1.1 Explain contextual, mathematical problems using a mathematical model.

  • C.MM.1.2 Create mathematical models to explain phenomena that exist in the natural sciences, social sciences, liberal arts, fine and performing arts, and/or humanities contexts.

  • C.MM.1.3 Using abstract and quantitative reasoning, make decisions about information and data from a contextual situation.

  • C.MM.1.4 Use various mathematical representations and structures with this information to represent and solve real-life problems.

    • Velocity as a rate of change (C-F.4)
    • Find the slope of a tangent line using limits (C-F.6)
    • Find equations of tangent lines using limits (C-F.7)

FGR Functional & Graphical Reasoning

• Relationship Between Limits and Continuity

  • C.FGR.2 Apply limit notation and characteristics of continuity to analyze behaviors of functions.

    • Determine limits graphically, numerically, and analytically.

      • C.FGR.2.1 Estimate limits from graphs and tables of values.

        • Find limits using graphs (C-A.1)
        • Find one-sided limits using graphs (C-A.2)
        • Determine if a limit exists (C-A.3)

      • C.FGR.2.2 Find limits of sums, differences, products, and quotients using substitution.

        • Find limits using addition, subtraction, and multiplication laws (C-B.1)
        • Find limits using the division law (C-B.2)

      • C.FGR.2.3 Represent asymptotic behavior using limits.

        • Find limits at vertical asymptotes using graphs (C-C.1)
        • Determine end behavior using graphs (C-D.1)

      • C.FGR.2.4 Find limits of rational functions using algebraic techniques.

        • Find the limit at a vertical asymptote of a rational function I (C-C.2)
        • Find the limit at a vertical asymptote of a rational function II (C-C.3)

      • C.FGR.2.5 Demonstrate continuity at a point using the definition and limit notation.

        • Determine continuity using graphs (C-E.2)
        • Determine one-sided continuity using graphs (C-E.3)
        • Find and analyze points of discontinuity using graphs (C-E.4)
        • Determine the continuity of a piecewise function at a point (C-E.6)

      • C.FGR.2.6 Apply the Intermediate Value Theorem to a function over a closed interval.

        • Intermediate Value Theorem (C-E.8)

• Differentiation

  • C.FGR.3 Relate limits and continuity to the derivative as a rate of change and apply it to a variety of situations including modeling contexts.

    • Apply the definition of the derivative

      • C.FGR.3.1 Interpret the derivative as an instantaneous rate of change that is a two-sided limit of an average rate of change.

        • Average rate of change I (C-F.1)
        • Average rate of change II (C-F.2)
        • Find instantaneous rates of change (C-F.3)

      • C.FGR.3.2 Demonstrate and apply the relationship between differentiability and continuity.

      • C.FGR.3.3 Apply the concept of derivative geometrically, numerically, and analytically.

        • Find values of derivatives using limits (C-F.5)
        • Find the slope of a tangent line using limits (C-F.6)

    • Apply rules of differentiation.

      • C.FGR.3.4 Find the derivatives of sums, products, quotients, and composite functions.

        • Sum and difference rules (C-G.1)
        • Product rule (C-G.2)
        • Quotient rule (C-G.3)
        • Power rule I (C-G.4)
        • Power rule II (C-G.5)
        • Chain rule (C-G.6)
        • Inverse function rule (C-G.7)

      • C.FGR.3.5 Find the derivatives of a variety of relations.

        • Find derivatives of polynomials (C-H.1)
        • Find derivatives of rational functions (C-H.2)
        • Find derivatives of sine and cosine functions (C-H.3)
        • Find derivatives of other trigonometric functions (C-H.4)
        • Find derivatives of exponential functions (C-H.5)
        • Find derivatives of logarithmic functions (C-H.6)
        • Find derivatives of inverse trigonometric functions (C-H.7)
        • Find derivatives of radical functions (C-H.8)
        • Find derivatives using the product rule I (C-H.9)
        • Find derivatives using the product rule II (C-H.10)
        • Find derivatives using the quotient rule I (C-H.11)
        • Find derivatives using the quotient rule II (C-H.12)
        • Find derivatives using the chain rule I (C-H.14)
        • Find derivatives using the chain rule II (C-H.15)
        • Find derivatives using implicit differentiation (C-I.1)
        • Find tangent lines using implicit differentiation (C-I.2)
        • Find derivatives using logarithmic differentiation (C-I.3)

      • C.FGR.3.6 Calculate higher order derivatives.

        • Find higher derivatives of polynomials (C-J.1)
        • Find higher derivatives of rational and radical functions (C-J.2)
        • Find second derivatives of trigonometric, exponential, and logarithmic functions (C-J.3)
        • Find higher derivatives of exponential and trigonometric functions (C-J.4)
        • Find higher derivatives of logarithmic and power functions (C-J.5)

• Applications of Differentiation

  • C.FGR.4 Apply derivatives to situations in order to draw conclusions including curve analysis and modeling rates of change in applications.

    • Analyze function behavior using the derivative.

      • C.FGR.4.1 Calculate the slope of a curve at a point.

        • Find the slope of a tangent line using limits (C-F.6)

      • C.FGR.4.2 Write the equation of the tangent line to a curve at a point and use it to obtain a local linear approximation of a value near the point of tangency.

        • Find equations of tangent lines using limits (C-F.7)

      • C.FGR.4.3 Identify intervals where functions are increasing, decreasing, and constant by using the relationship between the function and the sign of its first derivative.

      • C.FGR.4.4 Identify points of inflection and intervals of concavity of a function by using the second derivative of a function.

      • C.FGR.4.5 Compare characteristics of f, f', and f" graphically, numerically, analytically, and with technology.

      • C.FGR.4.6 Apply Mean Value Theorem.

    • Apply the derivative to real-world problems.

      • C.FGR.4.7 Apply Extreme Value Theorem.

      • C.FGR.4.8 Apply the derivative to real-world problems to find both local and absolute extrema, with and without technology.

      • C.FGR.4.9 Model rates of change in applied situations.

        • Velocity as a rate of change (C-F.4)

GSR Geometric & Spatial Reasoning

• Indefinite and Definite Integrals

  • C.GSR.5 Analyze the relationship between the derivative and the integral using the Fundamental Theorem of Calculus.

    • Evaluate and interpret definite integrals.

      • C.GSR.5.1 Use Riemann sums to approximate values of definite integrals.

      • C.GSR.5.2 Interpret a definite integral as a limit of Riemann sums.

      • C.GSR.5.3 Find the exact value of a definite integral using geometric formulas on a coordinate plane.

      • C.GSR.5.4 Demonstrate the use of properties of definite integrals.

      • C.GSR.5.5 Apply the Fundamental Theorem of Calculus as an interpretation of the accumulation in the rate of change of a function as equivalent to the change in the antiderivative over the interval.

    • Find the antiderivative of indefinite integrals.

      • C.GSR.5.6 Apply Fundamental Theorem of Calculus to indefinite integrals to represent the family of antiderivatives.

      • C.GSR.5.7 Apply integration by substitution to definite and indefinite integrals.

PAR Patterning & Algebraic Reasoning

• Applications of Integrals

  • C.PAR.6 Apply the definite integral and indefinite integral to contextual situations.

    • Apply Integration techniques to solve problems

      • C.PAR.6.1 Find a particular curve in a family of antiderivatives using an initial condition.

      • C.PAR.6.2 Solve separable differential equations and use them to model real-world problems.

      • C.PAR.6.3 Apply definite integrals to find the area between two curves.

      • C.PAR.6.4 Apply definite integrals to find the average value of a function over a closed interval.