PreCalculus and Algebra Standards
P1
Add, subtract, multiply and divide without the aid of a calculator, including fractions
and mixed numbers
P2
Factor and/or solve algebraic equations including trigonometric, exponential and
logarithms with or without the use of a calculator
P3
Sketch the graph of basic functions without the aid of a calculator.
P4
Evaluate trigonometric functions at common radian values without the aid of a
𝜋 𝜋 𝜋 𝜋 2𝜋 3𝜋 5𝜋
calculator. (0, 6 , 4 , 3 , 2 , 3 , 4 , 6 , 𝜋)
P5
Use basic trigonometric identities, including the Pythagorean identities, double angle
and power-reducing identities
P6
Use comparative relative magnitudes of functions to analyze rates of change
BC Specific Standards
P7
Analysis of planar curves, including sketches and points of intersection for:
P7.1
Parametric curves
P7.2
Polar curves
P7.3
Vector-defined functions
Limits Standards
L1
Evaluate limits graphically or from a table of values, including the limits of a constant,
one-sided limits, limits at infinity, infinite limits and non-existent limits.
L2
Evaluate limits analytically, including:
L3
L2.1
Limits at a "hole"
L2.2
One sided limits
L2.3
Limits at infinity
L2.4
Infinite limits
L2.5
Applying l'hopitals rule to evaluate limits
Use limits to define continuity
L3.1
Prove continuity in terms of limits
L3.2
Define continuity at a point and over an interval
L4
Apply the Intermediate Value Theorem
L5
Use continuity and limits as they relate to differentiability
Calculating Derivatives
D1
Use the limit definition of a derivative
D2
Find the derivative of a functions including
D3
D2.1
Sums
D2.2
Products
D2.3
Quotients
D2.4
Composites (chain rule)
Find the derivative of:
D3.1
Algebraic functions
D3.2
Trigonometric functions
D3.3
Natural exponential functions
D3.4
Exponential functions with bases other than e
D3.5
Natural logarithmic functions
D3.6
Other logarithmic functions
D3.7
Inverse trigonometric functions
D3.8
Implicitly defined functions
D3.9
Complex functions using logarithmic differentiation
D3.10 The inverse of a given function (without explicitly defining the inverse)
D4
Find higher-order derivatives.
D5
Calculate the value of a derivative at a point
D5.1
By first calculating the derivative of the function
D5.2
Using a graphing calculator
BC Specific Standards
D6
Find derivatives of:
D6.1
parametric functions
D6.2
polar functions
D6.3
vector functions, including velocity and acceleration vectors
Applying Derivatives
AD1
Find the derivative of a curve at a point, including points where the tangent is vertical
and points at which there is no tangent
AD2
Use local linear approximation to:
AD2.1 Find the equation of a line tangent to a curve
AD2.2 Estimate nearby values of a function
AD2.3 Analyze the curve to decide if an estimate is an over- or under-estimate
AD3
Approximate rate of change from graphs and tables of values
AD4
Compare the corresponding characteristics of the graphs of f, f' and f''.
AD4.1 Recognize the relationship between the sign of f' and the behavior of f
AD4.2 Recognize the relationship between the sign of f'' and the behavior of f
AD4.3 Identify relative extrema using only the first derivative
AD4.4 Identify relative extrema using the first and second derivatives
AD4.5 Identify relative and absolute extrema
AD4.6 Identify points of inflection
AD5
Use the Mean Value Theorem in predictions and proofs.
AD6
Use calculus to find the maxima or minima in "real life" situations
AD7
Use calculus in modeling rates of change and related rates
AD7
Interpret the derivative as a rate of change in applied contexts, including velocity, speed,
and acceleration
BC specific standards
AD8
Numerically estimate the value of differential equations using Euler’s method
AD9
l’Hopital’s Rule to test the convergence of improper integrals and series
Calculating Integrals
I1
Find antiderivatives directly from known derivatives of basic functions including:
I1.1
Power rule
I1.2
Trigonometric forms
I1.3
Exponentials
I1.4
Exponential functions with bases other than e
I1.5
Integrals resulting in a logarithm
I1.6
Using substitution (including change of limits for definite integrals)
I2
Find a particular solution given an integral and initial condition
I3
Evaluate definite integrals
I3.1
By first evaluating the indefinite integral
I3.2
Using a graphing calculator
BC Specific Topics
I4
Find antiderivatives using:
I4.1
Integration by parts
I4.2
Partial fractions
I4.3
Using limits to evaluate improper integrals
I4.4
Trigonometric integrals using power-reducing formulas
Applications of the Integral
AI1
Interpret the definite integral as a limit of a sum
AI2
Interpret the definite integral as an area
AI3
Approximate integrals of function represented algebraically, graphically and by a table
of values using:
AI3.1
Riemann sums
AI3.2
Trapezoidal sums
AI3.3
Analyze the relative accuracy of Riemann and trapezoids sums
𝑏
1
AI4
Find the average value of a function using 𝑏−𝑎 ∫𝑎 𝑓(𝑥) 𝑑𝑥
AI5
Associate the average value of a rate as the average rate over the interval
AI6
Interpret the integral of a rate as the accumulation of an amount
AI7
Analyze an integral with a variable as one of the bounds
AI8
Evaluate the derivative of an integral using 𝑑𝑥 ∫𝑎 𝑓(𝑡)𝑑𝑡 = 𝑓(𝑢)𝑢′ .
AI9
Use the integration techniques calculate the area of a region bound by one or more
functions, including:
AI10
𝑑
𝑢
AI9.1
Integrating with respect to x or with respect to y
AI9.2
Evaluating areas with multiple regions, including both positive and negative
regions
Use integration techniques to calculate the volume of a solid formed by:
AI10.1 A region bound by one or more functions being rotated around a coordinate axis
AI10.2 A region bound by one or more functions being rotated around a line parallel to a
coordinate axis
AI10.3 A region that is rotated around a line removed from the region such that the solid
formed is hollow
AI10.4 Cross-sections of consistent shape with a base defined by one or more functions
AI11
Calculate the distance traveled by a particle along a line given the position, velocity or
acceleration equation
AI12
Sketch a slope field and particular solution to a differential equation
AI13
Associate given slope fields and differential equations
AI14
Solve differential equations including:
AI14.1 Writing a differential equation from a "real life" situation
AI14.2 Separate variables and integrate
AI14.3 Find particular solutions given an initial value
AI14.4 Comfortably use the form y'=ky and exponential growth
BC Specific Standards
AI15
Calculate the area of a region bound by polar curves
AI16
Calculate the length of a curve defined by:
AI16.1 An algebraic, trigonometric or exponential function
AI16.2 A parametric function
AI17
Work logistical differential equations
BC Specific Standard: Infinite Series
S1
Define a series as a sequence of partial sums
S2
Test for convergence of a series. This will include:
S3
S2.1
Decimal expansion
S2.2
Geometric series
S2.3
Harmonic series
S2.4
Telescoping series
S2.5
P-series
S2.6
Alternating series with analysis of error bound
S2.7
Ratio test
S2.8
Integral test
S2.9
Direct comparison test
S2.10
Limit comparison test
Use of Taylor series. This will include:
S3.1
Taylor polynomial approximations with graphical demonstration of convergence
S3.2
Finding the Maclaurin series and the general Taylor series for a given function
centered at a given value by using derivatives
S3.3
Know and use Maclaurin series for the functions:
S3.3a
ex
S3.3b
sin x
S3.3c
cos x
S3.3d
S3.3e
S4
1
1−𝑥
ln x
Manipulation of Taylor series including:
S4.1
Substitution
S4.2
Differentiation
S4.3
Antidifferentiation
S4.4
The formation of new series from known series
S5
Use of functions defined by power series to approximate values
S6
Determining the radius and interval of convergence of power series
S7
Calculating Lagrange error bound of a Taylor polynomial