South Carolina

C.LC Limits and Continuity

• C.LC.1 Understand the concept of a limit graphically, numerically, analytically, and contextually.

  • C.LC.1.a Estimate and verify limits using tables, graphs of functions, and technology.

    • 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.LC.1.b Calculate limits, including one-sided limits, algebraically using direct substitution, simplification, rationalization, and the limit laws for constant multiples, sums, differences, products, and quotients.

    • Find limits using addition, subtraction, and multiplication laws (C-B.1)
    • Find limits using the division law (C-B.2)
    • Find limits using limit laws (C-B.4)
    • Find limits of polynomials and rational functions (C-B.5)
    • Find limits involving factorization and rationalization (C-B.6)
    • Find limits involving absolute value functions (C-B.7)

  • C.LC.1.c Calculate infinite limits and limits at infinity. Understand that infinite limits and limits at infinity provide information regarding the asymptotes of certain functions, including rational, exponential and logarithmic functions.

    • Find limits at vertical asymptotes using graphs (C-C.1)
    • 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)
    • Determine end behavior using graphs (C-D.1)
    • Determine end behavior of polynomial and rational functions (C-D.2)

• C.LC.2 Understand the definition and graphical interpretation of continuity of a function.

  • C.LC.2.a Apply the definition of continuity of a function at a point to solve problems.

    • Identify graphs of continuous functions (C-E.1)
    • Determine continuity using graphs (C-E.2)
    • Determine one-sided continuity using graphs (C-E.3)
    • Determine the continuity of a piecewise function at a point (C-E.6)

  • C.LC.2.b Classify discontinuities as removable, jump, or infinite. Justify that classification using the definition of continuity.

    • Find and analyze points of discontinuity using graphs (C-E.4)

  • C.LC.2.c Understand the Intermediate Value Theorem and apply the theorem to prove the existence of solutions of equations arising in mathematical and real-world problems.

    • Intermediate Value Theorem (C-E.8)

C.D Derivatives

• C.D.1 Understand the concept of the derivative of a function geometrically, numerically, analytically, and verbally.

  • C.D.1.a Interpret the value of the derivative of a function as the slope of the corresponding tangent line.

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

  • C.D.1.b Interpret the value of the derivative as an instantaneous rate of change in a variety of real-world contexts such as velocity and population growth.

    • Find instantaneous rates of change (C-F.3)
    • Velocity as a rate of change (C-F.4)

  • C.D.1.c Approximate the derivative graphically by finding the slope of the tangent line drawn to a curve at a given point and numerically by using the difference quotient.

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

  • C.D.1.d Understand and explain graphically and analytically the relationship between differentiability and continuity.

  • C.D.1.e Explain graphically and analytically the relationship between the average rate of change and the instantaneous 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)
    • Velocity as a rate of change (C-F.4)

  • C.D.1.f Understand the definition of the derivative and use this definition to determine the derivatives of various functions.

    • Find values of derivatives using limits (C-F.5)

• C.D.2 Apply the rules of differentiation to functions.

  • C.D.2.a Know and apply the derivatives of constant, power, trigonometric, inverse trigonometric, exponential, and logarithmic functions.

    • 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)

  • C.D.2.b Use the constant multiple, sum, difference, product, quotient, and chain rules to find the derivatives of 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)
    • 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)

  • C.D.2.c Understand and apply the methods of implicit and logarithmic differentiation.

    • Find derivatives using implicit differentiation (C-I.1)
    • Find derivatives using logarithmic differentiation (C-I.3)

• C.D.3 Apply theorems and rules of differentiation to solve mathematical and real-world problems.

  • C.D.3.a Explain geometrically and verbally the mathematical and real-world meanings of the Extreme Value Theorem and the Mean Value Theorem.

  • C.D.3.b Write an equation of a line tangent to the graph of a function at a point.

    • Find equations of tangent lines using limits (C-F.7)
    • Find tangent lines using implicit differentiation (C-I.2)

  • C.D.3.c Explain the relationship between the increasing/decreasing behavior of f and the signs of f'. Use the relationship to generate a graph of f given the graph of f', and vice versa, and to identify relative and absolute extrema of f.

  • C.D.3.d Explain the relationships among the concavity of the graph of f, the increasing/decreasing behavior of f' and the signs of f''. Use those relationships to generate graphs of f, f', and f'' given any one of them and identify the points of inflection of f.

  • C.D.3.e Solve a variety of real-world problems involving related rates, optimization, linear approximation, and rates of change.

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

C.I Integrals

• C.I.1 Understand the concept of the integral of a function geometrically, numerically, analytically, and contextually.

  • C.I.1.a Explain how the definite integral is used to solve area problems.

  • C.I.1.b Approximate definite integrals by calculating Riemann sums using left, right, and mid-point evaluations, and using trapezoidal sums.

  • C.I.1.c Interpret the definite integral as a limit of Riemann sums.

  • C.I.1.d Explain the relationship between the integral and derivative as expressed in both parts of the Fundamental Theorem of Calculus. Interpret the relationship in terms of rates of change.

• C.I.2 Apply theorems and rules of integration to solve mathematical and real-world problems.

  • C.I.2.a Apply the Fundamental Theorems of Calculus to solve mathematical and real-world problems.

  • C.I.2.b Explain graphically and verbally the properties of the definite integral. Apply these properties to evaluate basic definite integrals.

  • C.I.2.c Evaluate integrals using substitution.