Download Topic Outline for Calculus BC

Topic Outline for Calculus BC
1. Limits & continuity of functions. (10 days) -----1st Six Weeks
· Introduction of limits intuitively.
· Estimating limits using graphical and numerical approaches.
· Limit notation. Right and left hand limits, limits fail to exist.
· Calculating limits analytically (properties of limits, dividing out and rationalizing techniques)
· Infinite limits & vertical asymptotes.
· Limits at infinity & horizontal asymptotes.
· Squeeze theorem and two special trigonometric limits.
· Continuity and graphical properties of continuous functions, Intermediate Value Theorem and Extreme Value
Theorem.
· Formal Definition of Limits.
· Definition of Indeterminate forms and calculating limits comparing relative magnitudes of functions at the
given value of x.
2. The Derivative and Applications of Differentiation. (52 days)-----1st and 2nd Six Weeks
· Secant lines & Tangent Lines (Average rate of change and the instant rate of change as the limiting value of
average rate of change)
· Derivative and the tangent line problem (how to draw the best tangent line)
· Limit Definition of the derivative (three different forms)
· Differentiability and Continuity. Investigation of functions that are everywhere continuous and differentiable
as well as those that are not differentiable at all points (both graphically and algebraically)
· Differentiation Rules.
· Power, Constant, Constant multiple. Sum/Difference, Product and Quotient
· Derivatives of Trigonometric functions.
· Derivatives of Exponential & Logarithmic Functions.
· Chain Rule
· Higher Order Derivatives.
· Position, Velocity Acceleration problems (Part 1 with differentiation only)
· Investigating increasing or decreasing behavior of f(x) using the first derivative (algebraically and graphically)
· Definition of critical points.
· The First Derivative Test and the Local Extrema.
· Absolute extrema on a closed interval (justification)
· Second derivative test, Concavity and Inflection Points (Justification)
· Applied minimum/maximum problems.
· Relationships between f (x), f ’(x), and f ”(x) using tables or graphs.
· Curve stretching.
· Implicit Differentiation (using chain rule and also partial derivatives for finding y’ only)
· Related Rates
· Logarithmic Differentiation.
· Derivatives of Inverse Trig Functions.
· Differentials
· Local Linear Approximation
· Mean Value Theorem/ Rolle’s Theorem.
· Review of Inverse Functions and Derivative of an Inverse of a Function
3. The Integral & Applications of Integration (56 days) ------3rd Six Weeks
· Antiderivatives & Indefinite integration, basic integration rules.
· Approximating the total amount of consumption using the rate vs. time data. Estimating distance traveled
using the velocity graphs.
· Approximating the area of a region bounded by continuous functions
.Numerical Integration : Reimann Sums & Trapezoidal Rule
· The definite integral as a limit of Reimann Sums.
· The First & Second Fundamental Theorem of Calculus.
· The Integral of the Rate as a Net Change over a given interval
· Functions defined by Integrals.
· The average value of a Function
· Motion problems (position, velocity, acceleration problems)-Part II –Including Integration
· Integration methods:
U-substitution, Integration by Parts, Integration by Partial Fractions, Improper Integrals, Indeterminate Forms
& L’Hopital’s Rule
· Integrals involving Inverse Trig. Functions.
· Area of a region between two curves.
· Volume of solids of Rotation (Disk & Washer Methods)
· Volume (Shell Method) – (I always teach this even though it is not in BC course description. Some problems
are easier to solve like 2006 AP exam question)- This year I will probably teach this after the AP exam
· Volume of solids with other known cross sections
· Arc Length
4. Differential Equations (14 days) ------------ 4th Six Weeks
· Separable Differential equations (General & Particular solutions. Initial value problems including exponential
& growth and decay models)
· Higher order Differential Equations.
· Logistic Growth
· Slope Fields (Graphical Solution to Differential equations)
· Euler’s Method (Numerical solutions of Differential equations)
5. Calculus of Parametric, Polar and Vector Valued Functions: (15 days)
---------4th Six Weeks
· Conics (Review)
· Parametric equations (Review)
· Calculus of Parametric Functions (Slope & tangent lines. Differentiation in Parametric Form ; 1st and 2nd
derivatives. Arc length in parametric form) .
· Polar Functions (Review)
· Calculus of Polar Functions (Slope & tangent lines, Area of a polar region- Including area of the region
between two polar curves, Arc length in polar form).
· Vectors in a Plane (Review)
· Vector Valued Functions
· Differentiation & Integration of Vector Valued Functions
· Motion Problems in a Plane (Velocity, Acceleration and Position Vector Problems)
6. Polynomial Approximations & Series (20 days) -------------------- 5th Six Weeks
a. Pre Calculus Review: Overview of sequences as functions, series as a sequence of partial sums, and
convergence as the limit of the sequence of partial sums.
b. Geometric Series
c. Test for Convergence: nth term test, harmonic series, p-series, Integral test, Comparison tests, Ratio test,
Alternating Series test
d. Taylor& Maclaurin Polynomials (higher order Taylor polynomial approximation of transcendental functions)
e. Taylor& Maclaurin Series centered at x=a
f. Maclaurin Series for the exponential function, sin(x), cos(x), 1/(1x), e^ x
g. Manipulations of Taylor Series to form new series from known series (including substitution, differentiation,
integration)
h. Functions Defined by Power Series
i. Radius and Integral of Convergence of Power Series