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The Canterbury Episcopal School
Scope and Sequence
AP Calculus I
Text: Calculus: Early Transcendentals, Stewart, James, 5th Edition, Thompson Learning Inc., 2003.
Students understand the relationship of calculus to other areas of mathematics, realizing:
● calculus was co-invented by Isaac Newton & Gottfried Liebnitz
● calculus logically follows geometry, algebra, & trigonometry
● responsibility toward applying material from geometry, algebra, and trigonometry found on textbook reference
pages and on handouts
Students review the basic properties of functions, being reminded of:
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verbal, numerical, graphical, and algebraic representations of functions
properties of the Cartesian coordinate plane
function notation and the arithmetic of functions in that notation
domain, range, and asymptotes of functions
types, families, or groups of functions
> linear
> algebraic polynomial, power, root, reciprocal, and rational
> trigonometric
> exponential
> logarithmic
● composite functions
● laws of exponents
● laws of logarithms
Students come to realize calculus is operating upon functions with two basic operations, the derivative
and the integral.
Students understand the basics of sequences and series, knowing how to:
● generate a sequence or series from the nth term
● generate the nth term from a listed sequence or series
● use properties of series notation
Students demonstrate understanding of the concept of the limit by being able to find:\
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the limit of a continuous function at x = c
limits for functions as x → ±∞ and limits of functions whose value is ±∞
limits of the arithmetic combination of functions
limits requiring algebraic simplification and/or composite expressions
limits using the Squeeze Theorem
whether or not a function (including a piecemeal function) is continuous at x = c, employing approaching c
from the left and from the right, if necessary
● limits of rational polynomial functions as x → ± ∞ using the theorem stating the limit of (1/x n), where n is a positive
integer, under such an approach of values of x is zero
● the value of the base of the natural logarithms, e, as one of two special limits
Students understand the definition of a derivative as a limit of a function notation expression and,
using the definition demonstrate:
● the direct relation between the derivative and the slope of tangent lines to the function
● the ability to find the derivative of a function using the definition of the derivative
● the ability to use both the “prime” and the “d” notation for the derivative
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With the help of Pascal’s Triangle, and a “silly” mnemonic or two, rules for differentiation (taking the
derivative are derived, and the student learns to find derivatives using the following rules as
“shortcuts,” so that the definition of the derivative no longer needs to be employed:
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the Power Rule
rules for taking derivatives of arithmetic combinations of functions
higher order derivatives
the derivative of ex
the Product Rule
the Quotient Rule
Using primarily physics applications, students demonstrate they can apply the derivative in the context
of stated problems by:
● solving kinematical equations employing v = ds/dt and a = dv/dt
● solving for linear equations of tangent lines to functions at particular values of x employing the fact that the slope
m in the linear form y = mx + b is found by taking the derivative (df/dx) of the function f(x)
Trigonometric identities and two specially derived limits are used so that students understand and can
apply:
● the derivatives of the sine, cosine, tangent, cotangent, secant, and cosecant functions
● the cyclic derivates of sine and cosine toward finding exaggerated orders of their derivatives
Students demonstrate the ability to apply the Chain Rule by using:
● the Chain Rule with “u-substitution”
● the Chain Rule “by sight”
● the Chain Rule in combination with any function as a “u” and in any combination of the Power Rule, the Product
Rule, and the Quotient Rule
● any number of factors in an applied Chain Rule
Students demonstrate the ability to solve problems applying the derivative to sets of parametric
equations.
Implicit differentiation is demonstrated, so that the students can:
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find dy/dx of functions defined implicitly
determine if two curves are orthogonal to each other
understand and apply the derivatives of the six inverse trigonometric function
perform differentiation of the logarithmic functions of any base
perform logarithmic differentiation
Students can apply the rules of differentiation to find the differential dy of any y(x).
Students demonstrate the ability to apply derivatives in three major areas:
● Related Rates problems
● performing a full function analysis by finding for a given f(x):
> all critical numbers or critical values
> all critical points
> intervals of increasing and decreasing and regions of concavity
> all inflection points
> all relative extremes (maxima & minima) and all absolute extremes (maxima & minima) using the 2 nd derivative
test
● Optimization problems using principles of full function analysis
Students demonstrate the ability to apply L’Hospital’s Rule of Indeterminate Forms to find limits of
problematic forms using derivatives.
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Students understand the concept of the integral as an area under a function, just as they understand the
derivative to be the slope of a tangent line to the function.
● approximations of the area
> “undershooting”
> “overshooting”
> “mid-width”
● precise representations of the area using a limit of the Riemann sum
● area as the integral or the anti-derivative
● integral notation utilizing the differential
> indefinite integral
> definite integral
● The Fundamental Theorem of Calculus
The students demonstrate the ability to perform integration using the basic rules of integration (power
rule of integration and arithmetic of integration):
● find indefinite integrals of algebraic polynomials
● find definite integrals of algebraic polynomials
Using associated rules of differentiation, rules of integration are derived, allowing the students to
integrate (both indefinitely and definitely):
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exponential functions of any base
all six trigonometric functions
special trigonometric integrands
special algebraic forms to obtain inverse trig functions as integrals
Students demonstrate they can integrate using the full spectrum of integration techniques:
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by substitution of variables
by parts (using the mnemonic LIATE as a guide)
by using trig identities
by trigonometric substitution
by partial fractions
integration tables
Students apply integrals and integration techniques to find:
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areas beneath functions
areas between curves
multiple, nested integrals and differentials
volumes as integrals in:
> rectangular coordinates
> cylindrical coordinates
> spherical coordinates
● volumes of geometrical solids using the:
> disk method
> shell method
Students apply integration to solve and apply:
● simple, separable variable differential equations
● problems of exponential growth and decay
Infinite series are revisited, so that students can:
● test series for convergence or divergence with such techniques as the Ratio Test
● recognize those functions than can be represented as a converging infinite series
● generate the infinite series representation of certain functions using Taylor and Maclaurin series