La Habra High School
Mathematics Department

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Calculus Standards

LIMITS, GRAPHS, AND FUNCTIONS

• calculate limits using algebraic techniques.
• estimate limits from graphs.
• use a table of values to find limits.
• use limit theorems to evaluate limits.
• understand continuity in terms of the limit definition.
• recognize discontinuities from the examination of a graph.
• discriminate between functions that do and do not satisfy the conditions of the theorems.
• understand asymptotes in terms of graphical behavior.
• compare relative magnitudes of functions and their rates of change.
• identify and analyze the domains and ranges of functions and composite functions.
• know that a domain and a range of a composite function must be determined from the original functions.
• calculate limits using algebraic techniques.
• estimate limits from graphs.
• use a table of values to find limits.
• use limit theorems to evaluate limits.
• understand continuity in terms of the limit definition.
• recognize discontinuities from the examination of a graph.
• discriminate between functions that do and do not satisfy the conditions of the theorems.
• understand asymptotes in terms of graphical behavior.
• compare relative magnitudes of functions and their rates of change.
• identify and analyze the domains and ranges of functions and composite functions.
• know that a domain and a range of a composite function must be determined from the original functions.

DERIVATIVES

• use the limit definition of the derivative to write the equation of a line tangent to a curve.
• use derivatives to solve a variety of problems coming from physics, chemistry, economics, etc., that involve the rate of change of a function.
• analyze a graph to determine where it is continuous and/or differentiable.
• draw a graph satisfying given continuity and differentiability constraints.
• use the limit definition to establish the derivative formulas.
• calculate the slope of a curve at a point.
• use local linear approximation to solve problems.
• recognize instantaneous rate of change as the limit of average rate of change.
• find the rate of change from graphs and tables of values.
• recognize characteristics of graphs of f and f'.
• distinguish between the increasing and decreasing behavior of f and the sign of f'.
• understand the Mean Value Theorem and its geometric consequences.
• identify corresponding characteristics of the graphs of f, f', and f".
• distinguish between the concavity of f and the sign of f".
• recognize that points of inflection are places where concavity changes.
• analyze and sketch curves, including monotonicity and concavity.
• optimize functions, finding both absolute (global) and relative (local) extrema.
• model rates of change, including related rates problems.
• use implicit differentiation to find the derivative of an inverse function.
• interpret the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration.
• apply knowledge of derivatives to basic functions, including xr , exponential, logarithmic, trigonometric, and inverse trigonometric functions.
• use basic rules for the derivative of sums, products, and quotients of functions.
• use the Chain Rule to compute the derivative of a composite function.
• apply implicit differentiation to solve a wide variety of problems coming from physics, chemistry, economics, etc.
• recognize Rolle's Theorem, the Mean Value Theorem, and L'Hopitals Rule and apply them to designated functions.
• approximate the zeros of a function using Newton-Raphson's method.

INTEGRALS

• understand the basic properties of integrals.
• use the integral as a rate of change to give an accumulated change.
• setting up a Reimann sum and representing its limit as a definite integral.
• find the area of a region.
• find the volume of a solid with known cross section.
• find the distance traveled by a particle along a line.
• use the Riemann sums and the Trapezoidal Rule to approximate definite integrals of functions represented algebraically, geometrically, and by tables of values.
• use calculators and/or computers to approximate integrals numerically.
• use the Fundamental Theorem of Calculus to evaluate definite integrals.
• use the Fundamental Theorem of Calculus to represent a particular antiderivative.
• find the antiderivative from a known derivative
• integrate by substitution of variables.
• find specific integrals using initial conditions, including applications to motion along a line.
• solve separable differential equations and use them in modeling.
• know the definitions and properties of inverse trigonometric functions and their appearance as indefinite integrals.