West Virginia

Algebra

• Understand the key concepts, connections and applications of functions, limits, continuity, derivatives, and integrals represented in multiple ways.

  • M.C.1 Use abstract notation to apply properties of algebraic, trigonometric, exponential, logarithmic, and composite functions, as well as their inverses, represented graphically, numerically, analytically, and verbally; and demonstrate an understanding of the connections among these representations.

    • Identify functions (PC-A.2)
    • Evaluate functions (PC-A.3)
    • Find values using function graphs (PC-A.4)
    • Composition of functions (PC-A.8)
    • Identify inverse functions (PC-B.1)
    • Find values of inverse functions from tables (PC-B.2)
    • Find values of inverse functions from graphs (PC-B.3)
    • Find inverse functions and relations (PC-B.4)
    • Characteristics of quadratic functions (PC-D.2)
    • Match quadratic functions and graphs (PC-D.4)
    • Match polynomials and graphs using end behavior (PC-F.8)
    • Match polynomials and graphs using zeros (PC-F.9)
    • Match polynomials and graphs (PC-F.10)
    • Domain and range of polynomials (PC-F.11)
    • Rational functions: asymptotes and excluded values (PC-G.1)
    • Check whether two rational functions are inverses (PC-G.3)
    • Match exponential functions and graphs (PC-H.2)
    • Domain and range of logarithmic functions (PC-J.1)
    • Graph logarithmic functions (PC-J.2)
    • Find properties of sine functions (PC-S.1)
    • Graph sine functions (PC-S.4)
    • Graph translations of sine functions (PC-S.5)
    • Find properties of cosine functions (PC-S.6)
    • Graph cosine functions (PC-S.9)
    • Graph translations of cosine functions (PC-S.10)
    • Graph sine and cosine functions (PC-S.14)
    • Graph translations of sine and cosine functions (PC-S.15)

  • M.C.2 Demonstrate a conceptual understanding of the definition of a limit via the analysis of continuous and discontinuous functions represented using multiple representations (e.g., graphs and tables).

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

  • M.C.3 Use the properties of limits including addition, product, quotient, composition, and squeeze/sandwich theorem to calculate the various forms of limits: one-sided limits, limits at infinity, infinite limits, limits that do not exist, and special limits such as the limit as x approaches zero of ^{sin x}/_x = 1 and the limit as x approaches zero of ^{1 – cos x}/_x = 0.

    • 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)
    • 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 of polynomial and rational functions (C-D.2)

  • M.C.4 Apply the definition of continuity to determine where a function is continuous or discontinuous including continuity at a point, continuity over an interval, application of the Intermediate Value Theorem, and graphical interpretation of continuity and discontinuity.

    • 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)
    • Intermediate Value Theorem (C-E.8)

  • M.C.5 Investigate and apply the definition of the derivative graphically, numerically, and analytically at a point; conceptually interpret the derivative as an instantaneous rate of change and the slope of the tangent line.

    • Find values of derivatives using limits (C-F.5)
    • Find the slope of a tangent line using limits (C-F.6)
    • Find equations of tangent lines using limits (C-F.7)

  • M.C.6 Discriminate between the average rate of change and the instantaneous rate of change using real-world problems.

    • 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)

  • M.C.7 Recognize when the Extreme Value Theorem indicates that function extrema exist.

    • Find critical points of functions (C-M.2)

  • M.C.8 Quickly recall and apply rules of differentiation including the constant multiple rule, the sum rule, the difference rule, the product rule, the quotient rule, the power rule, and the chain rule as applied to algebraic, trigonometric, exponential, logarithmic, and inverse trigonometric functions using techniques of both explicit and implicit differentiation.

    • 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 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 of trigonometric functions using the product and quotient rules (C-H.13)
    • 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)
    • 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)

  • M.C.9 Apply Rolle's Theorem and the Mean Value Theorem to real-world problems.

    • Mean Value Theorem (C-M.1)

  • M.C.10 Construct and use mathematical models to solve optimization, related-rates, velocity, and acceleration problems.

    • Relate position, velocity, speed, and acceleration using derivatives (C-K.1)
    • Introduction to related rates (C-K.2)
    • Introduction to optimization (C-O.1)

  • M.C.11 Determine antiderivatives that follow from derivatives of basic functions and apply substitution of variables.

    • Find antiderivatives (C-S.1)
    • Find indefinite integrals using the power rule (C-S.2)
    • Find indefinite integrals involving exponential and logarithmic functions (C-S.3)
    • Find indefinite integrals involving trigonometric functions (C-S.4)
    • Find indefinite integrals using substitution (C-U.1)

  • M.C.12 Evaluate definite integrals using basic integration properties such as addition, subtraction, constant multipliers, the power rule, substitution, and change of limits.

    • Properties of definite integrals (C-Q.6)
    • Evaluate definite integrals using the power rule (C-T.1)
    • Evaluate definite integrals involving exponential and logarithmic functions (C-T.2)
    • Evaluate definite integrals involving trigonometric functions (C-T.3)
    • Evaluate definite integrals using substitution (C-U.)

  • M.C.13 Characterize the definite integral as the total change of a function over an interval and use this to solve real-world problems.

  • M.C.14 Apply the Fundamental Theorem of Calculus to evaluate definite integrals and to formulate a cumulative area function and interpret the function as it relates to the integrand.

    • Fundamental Theorem of Calculus, Part 1 (C-R.1)
    • Fundamental Theorem of Calculus, Part 2 (C-R.2)

  • M.C.15 Use limits to deduce asymptotic behavior of the graph of a function.

    • 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)

  • M.C.16 Compare and contrast the limit definition (not delta epsilon) of continuity and the graphical interpretation of the continuity of a function at a point; recognize different types of discontinuities.

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

  • M.C.17 Develop tangent lines as best linear approximations to functions near specific points; explain this conceptually; construct these tangent lines; and apply this concept to Newton's Method.

    • Linear approximation (C-P.1)

  • M.C.18 Investigate and explain the relationships among the graphs of a function, its derivative and its second derivative; construct the graph of a function using the first and second derivatives including extrema, points of inflection, and asymptotic behavior.

    • Find intervals of increase and decrease (C-M.)
    • Find local extrema using the first derivative test (C-M.3)
    • Find absolute extrema on a closed interval (C-M.4)
    • Identify the graph of the derivative from the graph of the function (C-M.5)
    • Identify the graph of a function from the graph of its derivative (C-M.6)
    • Identify the graph of the second derivative from the graph of the function (C-N.1)

  • M.C.19 Approximate areas under a curve using Riemann sums by applying and comparing left, right, and midpoint methods for a finite number of subintervals.

    • Approximate the area under a curve: left and right endpoints (C-Q.1)
    • Approximate the area under a curve: midpoints (C-Q.2)
    • Approximate the area under a curve: trapezoids (C-Q.3)

Geometry

• Apply the key concepts, connections and applications of limits, continuity, derivatives, and integration for a wide variety of regions.

  • M.C.20 Justify why differentiability implies continuity and classify functional cases when continuity does not imply differentiability.

  • M.C.21 Calculate a definite integral using Riemann sums by evaluating an infinite limit of a sum using summation notation and rules for summation.

  • M.C.22 Use integration to solve problems that involve linear displacement, total distance, position, velocity, acceleration, and area between curves by looking at both functions of x and functions of y; utilize units to interpret the physical nature of the calculus process.

    • Average value of a function (C-W.1)
    • Relate position, velocity, speed, and acceleration using integrals (C-W.)
    • Areas of regions bounded by graphs and the x-axis (C-W.2)
    • Areas of regions bounded by two curves: functions of x (C-W.)

Data Analysis and Probability

• Apply the key concepts and applications of limits, continuity, derivatives, and integration to analyze functions that represent a collection of data.

  • M.C.23 Identify a real-world situation that involves quantities that change over time; pose a question; make a hypothesis as to the answer; develop, justify, and implement a method to collect, organize, and analyze related data; extend the nature of collected, discrete data to that of a continuous function that describes the known data set; generalize the results to make a conclusion; compare the hypothesis and the conclusion; present the project numerically, analytically, graphically, and verbally using the predictive and analytic tools of calculus.