Student “Essential Elements” for
Math Standards
AP-Calculus
Essential Elements
Students should be able to work with functions represented in a variety of ways: graphical,
numerical, analytical or verbal. They should understand the connections among these
representations, and develop facility with the graphical and numerical processes using a graphing
utility.
Students should understand the meaning of the derivative in terms of a rate of change and local
linear approximation and should be able to use derivatives to solve a variety of problems.
Students should be able to apply the concept of limits graphically, numerically, and analytically,
and apply these concepts to the idea of taking the derivative.
Students should be able to recognize the definition of continuity as it applies to various
functions. Especially the properties that cause discontinuity (no limit, no function value, limit
and function value not agreeing).
Students should be able to apply the consequences of continuity: Intermediate value theorem
Students will be able to solve applied problems involving derivatives which require modeling
skills: Identify the question (use a variable when appropriate)
Problems types to be mastered include optimization and related rates.
Students will use approximation techniques such as tangent line approximations (linearization)
and calculating rates of change using tabular data.
Students will be able to approximate the area under a curve by using Riemann
Sums
Students will be able to use the First and Second Fundamental Theorems of
Calculus to evaluate definite and indefinite integrals.
Students will be able to solve problems involving exponential
Students will be able to solve differential equations numerically (Euler’s method),
graphically (slope field sketches), and analytically (separation of variable).
Students will be able to solve problems involving applications of integration
Students will be able to use advanced integration techniques including integration
by parts, partial fractions, and u-substitution.
Students will be able to evaluate limits using L’Hopital’s Rule.
Students will be able to evaluate improper integrals using proper notation.
Students will be able to determine the convergence or divergence of a given infinite
series using partial sums, p-series, geometric series, integral test, alternating series
test, ratio and root tests, and direct or limit comparison tests.
Students will be able to approximate a function value by creating a Taylor
polynomial.
Students will be able to determine the interval of convergence for a given power
series using the ratio test and checking of endpoints.
Students will be able to determine error approximations using the LaGrange form
of the remainder, alternating series remainder, and integral test remainder.
Students will be able to recognize the basic Taylor series for sin, cos, ln, and
exponential functions, and manipulate these with substitution of variables,
integration or differentiation.
Students will be able to determine derivatives, velocity vectors, acceleration vectors,
and speed using vector value functions.
Students will be able to find derivatives, area bounded by curves, and arc length for
functions given in polar coordinates.

I can identify a vertical translation, horizontal translation, or reflection over the x or y-axis for a
given function based on its equation, and I can rewrite the equation for a given function using
given transformations.

I can sketch a graph for a translated function with or without the use of a graphing utility.
Determine how translations affect the symbolic and graphical forms of a function, and use graphing
technology to examine translations.
 I can sketch a graph for any of the functions stated in essential outcome #6 with or without the
use of a graphing utility.
Sketch the graphs of common non linear functions (parent functions), such as square roots, absolute
value, and quadratic functions.
 I can add, subtract, and multiply given polynomials and recognize the degree of the result.

I can divide polynomials using long division, or synthetic division if appropriate.

I can express the quotient for a division of polynomials.

I can rewrite the result of a polynomial division using the form: P( x)  D( x)Q( x)  R( x) or as a
polynomial with a rational function remainder.
Perform basic algebraic operations with polynomial functions, including division and synthetic division.
 I can add, subtract, multiply and divide complex numbers and express the results in standard
form.

I can solve a quadratic equation that has non-real solutions.
Find complex solutions to quadratic equations, and perform algebraic operations with complex
numbers.
 I can use radian and/or degree measure to represent angles.

I can evaluate trig function using the unit circle.

I can use domain and period to evaluate sine and cosine functions.

I can apply the formula for arc length (s = r  ), and use the concepts of linear and angular speed
to solve applied problems.
Translate problem statements into mathematical models (equations and inequalities) including
quadratics and solve the problems using appropriate methods.
Develop skills with graphing calculator operations including producing a graph in a given window, find
function values graphically (tracing) or numerically (use a table), solve an equation numerically or
graphically.
Define and use the properties of the unit circle.
 I can use reference angles to evaluate trig functions.

I can use trigonometric functions to model and solve real-life problems.

I can use a calculator to evaluate trig functions, including the reciprocal functions vis-à-vis the
inverse functions.

I can graph sine and cosine utilizing the properties of amplitude, period, horizontal and vertical
shift.
Find values of all trig functions, their reciprocals and inverse functions.
 I can find the sum of n terms of an arithmetic or geometric series, and can determine the sum of
a convergent infinite geometric series.

I can solve applied problems involving sequences and series.

I can prove summation formulas using the Principle of Mathematical Induction.

I can expand binomials using the Binomial Theorem

I can use factorial notation to calculate binomial coefficients, and apply counting principles,
including permutations and combinations.
Apply properties of sequences and series, especially arithmetic and geometric sequences.