Honors Calculus

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Curriculum Standards

Pre-Calculus, Limits and Continuity

HC-1 Functions

1.1 Review of functions and their graphs Combining functions using sums, differences, products, and quotients. Composite functions. 1.2 Horizontal and vertical shifting of functions. Horizontal and vertical reflecting of functions. Horizontal and vertical scaling of functions. 1.3 Review of trigonometric functions and their graphs

HC-2 Limits and Continuity 2.1 Rates of change and tangent lines to curves 2.2 Limit of a function and limit laws 2.4 One-sided limits 2.5 Continuity. Intermediate Value Theorem for continuous functions 2.6 Limits involving infinity. Horizontal asymptotes. Limits involving Trig .

Differential Calculus and Applications

HC-3 Derivatives 3.1 Tangent lines and the derivative at a point 3.2 The definition of derivative as the limit of the difference quotient 3.3 Differentiation rules for sums and differences, products and quotients, and powers (integer exponents). 3.4 The derivative as an instantaneous rate of change 3.5 Derivatives of trigonometric functions 3.6 The Chain Rule 3.7 Implicit differentiation 3.8 Related rates problems. Differentiating with respect to time 3.9 Linearization and differentials

HC-4 Applications of Derivatives 4.1 Extreme Value Theorem. Definitions of absolute maximum and minimum, local (relative) maximum and minimum. The First Derivative Theorem for Local Extreme Values. Definition of “critical point” 4.1.a Finding absolute and local extrema on a closed or open interval. 4.2 Rolle’s Theorem. The Mean Value Theorem for Derivatives 4.2.a Applying the Mean Value Theorem 4.3 Monotonic functions. Definition of increasing and decreasing applied to functions. 4.3.a First derivative test for monotonic functions. The relationship between the sign of the first derivative and the increasing 4.3.b and decreasing intervals for a function. 4.3.c The first derivative test for local extrema. 4.4 Concavity. Definition of concave up and concave down. 4.4.a The second derivative test for concavity. 4.4.b The relationship between the sign of the second derivative and the concave up and concave down intervals for a function. 4.4.c Definition of inflection point. 4.4.d Analyzing graphed functions. The relationship between the shape of a graph and its first and second derivatives. 4.5 Applied optimization problems. 4.6 Newton’s Method and its application. 4.7 Antiderivatives following directly from the derivatives of basic functions. 4.7.a Initial value problems, particularly finding velocity from acceleration and displacement from velocity. 4.7.b Indefinite Integrals.

Integral Calculus and Applications

HC-5 Integrals and The Fundamental Theorem of Calculus 5.1 Estimating with finite sums using left endpoints, right endpoints, and midpoint values. 5.1.a Displacement versus distance traveled. 5.1.b The average value of a function. 5.2 Limits of finite sums and Riemann sums. 5.3 The definite integral as a limit of Riemann sums. 5.3.a The existence of definite integrals. 5.3.b Integrable and nonintegrable functions. 5.3.c Rules satisfied by definite integrals (order of integration, zero width integral, constant multiple, sum and difference, additivity, max/min inequality, domination). 5.3.d Area under a curve as a definite integral. 5.4 The Mean Value Theorem for Definite Integrals. 5.4.a Total Area vs Net Area 5.4.b The fundamental Theorem of Calculus Part 1 5.4.c The fundamental Theorem of Calculus Part 2 5.4.d The relationship between the graph of a function and values of its derivative and integral. 5.5 Using substitution with integration formulas (running the Chain Rule backward) 5.5.a Integration of (sinx)^2 and (cosx)^2 as well as trig identities 5.6 Finding areas between curves using vertical and horizontal rectangles. 5.6.a Area of regions bounded by polar curves

HC-6 Integral Applications to Volume, Surface Area and Length of Curves 6.1 Volumes by slicing. (Disk and Washer method) 6.2 Volumes by the cylindrical method. 6.2.a Volumes of the solid of revolution by the cylinder method

Differential and Integral Calculus – Transcendental Functions

HC-7 Transcendental Functions 7.1 One-to-one functions and the horizontal line test. 7.1.a Definition of an inverse function and finding the inverse of a function. 7.1.b Inverse functions are symmetric about the line y=x. 7.1.c The Derivative Rule for inverse functions. 7.2 The natural logarithms and the number “e”. 7.2.a Properties of logarithms (product rule, quotient rule, and power rule including reciprocals). The change of base formula for logarithms. 7.2.b Derivatives and Integrals involving logs and natural logs 7.2.c Logarithmic differentiation 7.3 The natural exponential function. The relationship between the logarithmic and exponential functions (they are inverses). 7.3.a The derivative and integral formulas for natural exponential functions. 7.3.b The number “e” expressed as a limit. 7.3.c The Power Rule with real number exponents. 7.3.d Derivatives and integrals involving exponential functions, base "e" and or base "b" 7.5 Indeterminate forms and L’Hopital’s Rule