COURSE OBJECTIVES:

Students will . . .

• understand and interpret the behavior of functions using a variety of techniques. These techniques include limits, derivatives, and integrals.
• apply their knowledge to functions that are described as equations, as graphs, as verbal/written descriptions, and as numerical data.
• use numerical, analytical, and graphical techniques to arrive at solutions. Students will understand when and how technology (specifically a graphing calculator) can be used to arrive at a solution and when the technology can be misleading.
• communicate their knowledge and solutions verbally, graphically (using the graphing calculator), and in written responses.

Throughout the curriculum, as skills are acquired, they are applied to solving problems involving related rates, position, velocity and acceleration, tangents, curve analysis, optimization, area between curves, average value of a function, growth and decay, separation of variables, volume of solids and slope fields.

CONTENT

• Review of precalculus topics

a. graphs and models

b. linear models and rates of change

c. functions and their graphs

d. trigonometry

e. parametric and polar functions

f. graphing calculator features

• Limits and their properties

a. definition of limit

i. graphical representation

ii. tables values through graphing calculator

iii. no limit

iv. properties of limits

v. formal definition of limit

1. delta-epsilon proof

• Evaluation of limits

a. numerical substitution

b. algebraic manipulation

c. infinite

d. one-sided limits

• Graphical interpretation of

a. discontinuities

i. vertical asymptotes

ii. holes

iii. removable versus essential

b. definition of continuity

i. understand continuity with respect to limits

ii. intermediate value theorem

c. special limits

i. squeeze Theorem

ii. apply to trig limits ex., f(x) = sin(x) / x as x approaches 0

d. limits as x approaches infinity

i. horizontal asymptotes

ii. algebraic evaluation

iii. end behavior models

CONTENT

• Derivative as a limit

a. Graphical interpretation

b. Algebraic interpretation

c. ‘Delta Process’

d. Differentiation and continuity

• Basic

a. Power rule and polynomials

b. Product and quotient rules

c. Chain rule

d. Trigonometric derivatives

e. Higher order derivatives

f. Derivative of absolute value functions

g. Derivatives of polar functions

h. Derivatives of parametric and vector functions

i. Evaluate derivative at a point

i. numerically with nderiv() , and CALC feature of graphing calculator

ii. graph tangent line to curve

• Applications of basic derivatives

a. Derivative as a rate of change

b. Relation between a function and its derivative

c. Local linearity and linear approximations

d. Instantaneous versus average rate of change

e. Rectilinear motion

i. position, velocity and acceleration realtionships

f. L’Hopital’s Rule for limits and indeterminant forms

• Implicit differentiation

a. Technique

b. Applications

i. tangents

ii. orthogonal curves

iii. related rates

• Further application of the derivative

a. Extrema

ii. extreme value theorem

b. Existence theorems

i. Rolle’s theorem

ii. mean value theorem

iii. geometric interpretation

c. Connecting ^{f ′} and ^{f ′′} to the graph of ^f

i. first derivative test

ii. second derivative test

d. Interval testing for

i. increasing /decreasing

ii. concavity

e. Curve analysis

i. numerical/analytical

ii. graphical approach

iii. reconstructing ^f from ^{f ′}

iv. applications to polar functions

f. Curve analysis of parametric and polar functions

i. modeling projectile motion

ii. modeling motion in the xy-plane

iii. use graphing calculator to model motion

g. Analysis of vector functions

i. notation for writing velocity and acceleration vectors

h. Modeling and optimization problems

i. minimum / maximum value

ii. feasible domain

iii. interpretation in context of question

i. Differentials

i. local linearization

ii. estimating change

1. absolute, relative and percentage change

iii. applications

CONTENT

• Antidifferentiation

a. History – geometric versus algebraic development of integral calculus

b. Antiderivative

i. Polynomial functions

ii. Trigonometric functions

c. General versus specific solutions

d. Notation

• Area under a Curve

a. Approximation techniques - geometry

i. upper and lower sums

ii. midpoint rule

iii. trapezoid rule

iv. error analysis

b. Reimann sums and the definite integral

i. notation

ii. limit of Reimann sum

iii. definite integration

iv. areas of common geometric figures

v. simple properties of integrals

• Fundamental theorem of calculus

a. Antiderivative as an integral of f(t)dt, from a to x

b. Mean value theorem for definite integrals

c. First fundamental theorem of calculus

d. Second fundamental theorem of calculus

e. Evaluate the integral

i. Numeric with the graphing calculator

1. fint() and Calc feature

2. analytic

f. Average value of a function

g. Total versus net area

• Integration by substitution

a. Pattern recognition

b. Change of variables in evaluating definite forms

c. Applications

CONTENT

• Review of exponential and logarithmic functions
• Derivative and integral involving

a. Natural logarithmic functions

b. Logarithmic functions base b

c. Trigonometric functions as ratios: tan(x), sec(x) , etc.

d. Applications

• Exponential and Inverse Function

a. Derivative of a functions and its inverse.

b. Derivative of natural exponential function

c. Logarithmic differentiation

d. Derivative of inverse trigonometric functions

e. Integration

i. Exponential functions base e and other bases

ii. Resulting in inverse function forms

f. Applications

• Differential Equations

a. Separation of variables

b. Solutions

i. General

ii. Particular (given initial condition )

c. Slope Fields

i. Construct by hand and with graphing calculator

ii. Use initial conditions to identify particular solution

d. Exponential growth and decay models

e. Newton’s law of cooling

f. Euler’s Method

i. numerical method of solution

ii. calculate by hand –recursively

iii. calculate with graphing calculator program

iv. graphical analysis of solution

g. Logistic Differential Equations

i. recognizing forms

ii. finding particular solution

iii. limits and long term behavior

iv. construct slope field using technology

v. logistic Regression and data sets

1. Using graphing calculator capabilities

vi. real-world applications

CONTENT

• Integral as a net change

a. Writing a Reimann sum to model a situation

b. Write as a definite integral

c. Evaluate

• Area between two curves

a. Vertical representative rectangles

b. Horizontal representative rectangles

c. Calculate with and without technology

d. Represent graphically with and without technology

e. Apply to polar functions

• Volume of a Solid

a. Disk / washer

b. Shell method

c. Known cross section

d. Choosing a method

• Arc Length

a. Develop formula

b. Vertical tangents, sharp points

c. Apply to polar curves

• Surface of Revolution

a. Vertical rotations

b. Horizontal rotations

c. Apply to polar curves

• Work

a. Develop as a limit of a Reimann sum

b. Hooke’s law

c. Other applications of work

• General Applications

a. Recognize and write Reimann sums to represent a variety of physical situations.

b. Write the limit of the Reimann sum as a definite intergral and interpret the answer in terms of the given situation.

CONTENT

• Integration Techniques

a. Fitting integral to basic rules

b. Integration by parts

i. tabular method

ii. cycles in

c. Powers of sine and cosine

d. Trigonometric substitutions

e. Partial fractions with non-repeating linear factors

• Improper integrals

a. Discontinuous functions and the integral bounds

b. Infinite upper and /or lower bounds

c. Evaluate

i. Convergence versus divergence

ii. Use technology to investigate

d. Comparison test

e. Limit comparison test

f. Applications

i. Gabriel’s horn

ii. Normal probability function

CONTENT

• Review sequences, known series and notation
• Definition of a series as a limit of partial sums
• Convergence / divergence

a. Evaluate the limit of the sequence of partial sums

b. Graphical with and without technology

• Special Series

a. Geometric

i. Recognition / representation

ii. Properties

b. p-series

i. Recognition / representation

ii. Properties

c. Harmonic

i. Recognition / representation

ii. Properties

d. Alternating

i. Recognition / representation

ii. Properties

• Tests of Convergence

a. n^th term test

b. The comparison test

c. Ratio test

d. Root test

e. Integral test

f. Limit comparison test

• Taylor and Maclaurin Series

a. Development of Taylor and Maclaurin polynomials centered at x = a

i. Use graphing calculator to interpret

b. Develop Maclaurin series for sin(x), cos(x), e^x, 1/(1-x), ln(x), etc.

c. Formal manipulation of Taylor series to form new series

i. algebraic substitution

ii. using differentiation

iii. using integration

d. Functions defined by power series

e. Taylor’s theorem

f. Proving convergence

g. Lagrange error bound

h. Taylor’s inequality (remainder estimation theorem)

i. Interval of convergence

i. determining

ii. end point analysis

iii. graphical interpretation using technology

j. Applications to functions

• The AP Calculus BC exam is given in early May.