Arkansas

LC Limits and Continuity

• LC.1.C Students will determine the limit of a function at a value numerically, graphically, and analytically

  • LC.1.C.1 Identify vertical asymptotes in rational and logarithmic functions by identifying locations where the function value approaches infinity; estimate limits numerically and graphically; calculate limits analytically: algebraic simplification, direct substitution, one-sided limits, rationalization.

    • Find limits using graphs (C-A.1)
    • Find one-sided limits using graphs (C-A.2)
    • Determine if a limit exists (C-A.3)
    • Find limits using addition, subtraction, and multiplication laws (C-B.1)
    • Find limits using the division law (C-B.2)
    • Find limits using power and root laws (C-B.3)
    • 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)
    • Find limits involving trigonometric functions (C-B.8)

  • LC.1.C.2 Calculate infinite limits and use the result to identity vertical asymptotes in rational 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)

  • LC.1.C.3 Calculate limits at infinity and use the result to identify horizontal asymptotes in rational and exponential functions.

  • LC.1.C.4 Calculate limits at infinity and use the result to identify unbounded behavior in rational, exponential, and logarithmic functions.

    • Determine end behavior using graphs (C-D.1)
    • Determine end behavior of polynomial and rational functions (C-D.2)

  • LC.1.C.5 Identify and classify graphically, algebraically, and numerically if a discontinuity is removable or non-removable; identify the three conditions that must exist in order for a function to be continuous at x = a: f(a) is defined, the limit as x approaches a of f(x) equals f(a), the limit as x approaches a of f(x) exists.

    • Identify graphs of continuous functions (C-E.1)
    • 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 continuity on an interval using graphs (C-E.5)
    • Determine the continuity of a piecewise function at a point (C-E.6)
    • Make a piecewise function continuous (C-E.7)

  • LC.1.C.6 Apply the Intermediate Value Theorem for continuous functions.

    • Intermediate Value Theorem (C-E.8)

D Derivatives

• D.2.C Students will use derivatives to solve problems both theoretically and in real-world context

  • D.2.C.1 Approximate the derivative: graphically by finding the slope of a tangent line drawn to a curve at a given point, numerically by using the difference quotient.

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

  • D.2.C.2 Find the equation of the tangent line using the definition of derivative.

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

  • D.2.C.3 Establish and apply the relationship between differentiability and continuity.

  • D.2.C.4 Compare the characteristic of graphs of ƒ and f': generate the graph of f given the graph of f' and vice versa, establish the relationship between the increasing and decreasing behavior of ƒ and the sign of f', identify maxima and minima as points where increasing and decreasing behavior change.

  • D.2.C.5 Apply the Mean Value Theorem on a given interval.

  • D.2.C.6 Compare the characteristic of graphs of ƒ, f', and f": generate the graphs of f and f' given the graph of f" and vice versa, establish the relationship between the concavity of ƒ and the sign of f", identify points of inflection as points where concavity changes.

  • D.2.C.7 Find derivatives of functions using: power rule, product rule, quotient rule.

    • Product rule (C-G.2)
    • Quotient rule (C-G.3)
    • Power rule I (C-G.4)
    • Power rule II (C-G.5)
    • Find derivatives of polynomials (C-H.1)
    • Find derivatives of other trigonometric functions (C-H.4)
    • 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 higher derivatives of polynomials (C-J.1)
    • Find higher derivatives of rational and radical functions (C-J.2)

  • D.2.C.8 Find derivatives of: an implicitly defined equation, composite functions using chain rule, exponential and logarithmic functions, functions requiring the use of more than one differentiation rule.

    • Chain rule (C-G.6)
    • Find derivatives of exponential functions (C-H.5)
    • Find derivatives of logarithmic functions (C-H.6)
    • Find derivatives of radical functions (C-H.8)
    • 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 second derivatives of trigonometric, exponential, and logarithmic functions (C-J.3)

  • D.2.C.9 Find the equation of: a line tangent to the graph of a function at a point, a normal line 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)

  • D.2.C.10 Solve application problems involving: optimization, related rates.

  • D.2.C.11 Interpret the derivative as a rate of change and varied applied contexts, including velocity, speed, and acceleration.

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

I Integrals

• I.3.C Students will apply the techniques of integration to solve problems both theoretically and in contextual models that represent real-world phenomena

  • I.3.C.1 Define the definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval the integral from a tob of f'(x)dx = f(b) - f(a).

  • I.3.C.2 Determine the area between two curves, and identify the definite integral as the area of the region bounded by two curves.

  • I.3.C.3 Apply the Fundamental Theorem of Calculus to solve contextual models that represent real-world phenomena.

  • I.3.C.4 Compute indefinite integrals.

  • I.3.C.5 Determine the antiderivative of a function using rules of basic differentiation, and solve problems using the techniques of antidifferentiation.

  • I.3.C.6 Estimate definite integrals by using Riemann sums and trapezoidal sums, and identify the definite integral as a limit of Riemann sums.

  • I.3.C.7 Explore and apply different integration techniques.