Kentucky

Limits

• Understand the concept of the limit of a function.

  • KY.HS.C.1 Understand limits.

    • KY.HS.C.1.a Apply limits to a variety of functions, including piecewise functions.

    • KY.HS.C.1.b Prove that the limit of a function exists, based upon the definition of a limit.

      • Determine if a limit exists (C-A.3)

  • KY.HS.C.2 Demonstrate an understanding of limits by estimating and finding the limit of a function at a point graphically, numerically and algebraically.

    • Find limits using graphs (C-A.1)
    • Find one-sided limits using graphs (C-A.2)

  • KY.HS.C.3 Apply properties and theorems of limits, including limits of indeterminate forms.

    • 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)

  • KY.HS.C.4 Communicate understanding of limits using precise mathematical symbols and language.

Function Behavior

• Describe the asymptotic and unbounded behavior of functions.

  • KY.HS.C.5 Describe asymptotic behavior (analytically and graphically) in terms of infinite limits and limits at infinity.

    • 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)

  • KY.HS.C.6 Discuss the end behavior of functions; identify representative functions for each type of end behavior using precise mathematical symbols and language.

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

Continuity

• Develop an understanding of continuity as a property of functions.

  • KY.HS.C.7 Understand and use the limit definition of continuity to determine whether a given function is continuous at a specific point.

    • 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 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)

  • KY.HS.C.8 Define and identify different types of discontinuity–removable (hole) or non-removable (jump, asymptote)–in terms of limits.

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

  • KY.HS.C.9 Understand and apply continuous function theorems.

    • KY.HS.C.9.a Apply the Intermediate Value Theorem to continuous functions.

      • Intermediate Value Theorem (C-E.8)

    • KY.HS.C.9.b Apply the Extreme Value Theorem to continuous functions.

  • KY.HS.C.10 Communicate an understanding of continuity using precise mathematical symbols and language.

Understanding the Derivative

• Demonstrate an understanding of the derivative.

  • KY.HS.C.11 Define derivatives.

    • KY.HS.C.11.a Define the derivative of a function as the limit of the difference quotient.

    • KY.HS.C.11.b Understand this limit of the difference quotient can be interpreted as an instantaneous rate of change or the slope of a tangent line.

      • 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)

  • KY.HS.C.12 Use average rate of change to estimate the derivative from a table of values or a graph.

    • Average rate of change I (C-F.1)
    • Average rate of change II (C-F.2)

  • KY.HS.C.13 Understand the derivative as a function.

  • KY.HS.C.14 Apply the definition of derivative to find derivative values and derivative functions.

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

  • KY.HS.C.15 Explain why differentiability implies continuity yet continuity does not imply differentiability.

  • KY.HS.C.16 Understand and apply the Mean Value Theorem, including numerical, graphical and algebraic representations.

  • KY.HS.C.17 Understand the relationship between the concavity of a function and the sign of the second derivative.

  • KY.HS.C.18 Understand Rolle's Theorem as a special case of the Mean Value Theorem.

Applications of Derivatives

• Apply differentiation techniques.

  • KY.HS.C.19 Efficiently find derivatives of functions with and without technology.

    • Find derivatives of polynomials (C-H.1)
    • 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)

  • KY.HS.C.20 Understand and use derivative rules for sums, differences, products and quotients of two functions and calculate the derivative of a composite function using the chain rule.

    • 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)
    • 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)

  • KY.HS.C.21 Use implicit differentiation to find a derivative in an equation of two variables.

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

  • KY.HS.C.22 Use implicit differentiation to find the derivative of the inverse of a function.

  • KY.HS.C.23 Understand the relationship between the increasing and decreasing behavior of a function and the sign of the first derivative of the function.

  • KY.HS.C.24 Use the first derivative to analyze curves and identify relative extrema.

  • KY.HS.C.25 Understand the relationship of concavity to the second derivative.

  • KY.HS.C.26 Use the second derivative to find points of inflection.

    • Find second derivatives of trigonometric, exponential, and logarithmic functions (C-J.3)

  • KY.HS.C.27 Use the second derivative to analytically locate intervals on which a function is concave up, concave down or neither.

  • KY.HS.C.28 Describe how graphical characteristics of a given function, the first derivative of that function and the second derivative of that function interrelate.

  • KY.HS.C.29 Use derivatives to express rate of change in a variety of contexts.

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

  • KY.HS.C.30 Use derivatives to solve a variety of problems including related rates, optimization, tangent line approximations and growth and decay models.

    • Find tangent lines using implicit differentiation (C-I.2)

  • KY.HS.C.31 Use differentiation to solve problems involving velocity, speed and acceleration.

  • KY.HS.C.32 Understand and apply differential equations.

    • KY.HS.C.32.a Verify solutions to differential equations and use them to model real-world problems with and without technology.

    • KY.HS.C.32.b Solve separable differential equations and use them in modeling real-world problems with and without technology.

Understanding Integration

• Demonstrate understanding of a definite integral.

  • KY.HS.C.33 Understand the definite integral of a function over an interval. Interpret a definite integral as a limit of Riemann Sums and as net accumulation of change.

  • KY.HS.C.34 Write a Riemann sum that represents the definition of a definite integral.

  • KY.HS.C.35 Calculate the values of Riemann Sums over equal subdivisions to approximate definite integrals of functions represented graphically and numerically (using tables). Use left-hand sums, right-hand sums, midpoint sums and trapezoidal sums.

  • KY.HS.C.36 Recognize differentiation and integration as inverse operations.

  • KY.HS.C.37 Understand how the Fundamental Theorem of Calculus connects differentiation and integration and use it to evaluate definite and indefinite integrals and to represent particular antiderivatives.

  • KY.HS.C.38 Perform analytical and graphical analysis of functions using the Fundamental Theorem of Calculus.

  • KY.HS.C.39 Understand and use the definite integral of a function over an interval and understand its use as a mathematical tool.

Applications of Integration

• Apply techniques of integration.

  • KY.HS.C.40 Find antiderivatives of a variety of basic functions including power, exponential, logarithmic and trigonometric and apply basic properties of definite integrals.

  • KY.HS.C.41 Use substitution techniques and change of limits of integration to find antiderivatives.

  • KY.HS.C.42 Find particular antiderivatives given initial conditions.

• Define trigonometric ratios and solve problems involving right triangles.

  • KY.HS.C.43 Model, solve and interpret applications of antiderivatives including finding area, velocity, acceleration and volume of a solid.

  • KY.HS.C.44 Apply integration to solve problems including particle motion and exponential growth and decay.

  • KY.HS.C.45 Describe the application of integration to a variety of problems using precise mathematical language and notation.