Mathematics (Advance Placement Calculus)

PLAINFIELD PUBLIC SCHOOLS

Mathematics (Advance Placement Calculus)

CURRICULUM GUIDESGrade 12

INTEGRATION: `Students will learn how to integrate a function, find definite and indefinite integrals, recognize patterns in a function and substitute variables to solve complex integration problems. They will also use First and Second Fundamental Theorem of Calculus to solve problems and find the area of a given curve through different methods.
CPI / Areas of Focus / Instructional Activities
(Students should know / Students should be able to do) / MODIFICATIONS/EXTENSION / Assessment / Resources / Materials
4.2.12 E1
4.2.12 E2
4.3.12 A3
4.3.12 B1
4.3.12 B2
4.3.12 B4
4.5 A1
4.5 A2
4.5 A3
4.5 A4 /

Anti-derivatives
Students will define anti-derivatives.
They will also define differential equations, and use them to solve problems and graph them.
Basic integration rules
Solve problems by using basic integration rules.
Process of integration: Rewrite, integrate and simplify polynomial algebraic and trigonometric functions.
Indefinite integrals.

/ MODIFICATION: Students are to be given functions to differentiate. Then they will be asked to integrate to get back the original functions.
EXTENSIONS: An evergreen nursery usually sells a certain shrub after 6 years of growth and shaping. The growth rate during these 6 years is approximately by dh/dt =1.5t+5, where t is the time in years and h is the height in centimeters. The seedlings are 12 centimeters tall when planted (t=0).

Find the length after t years.
How tall are the shrubs when they are sold?

Teacher Observation
NOTE BOOK CHECK
QUIZZES
CHAPTER TESTS
CLASS WORK
HOMEWORK
PROJECTS
GROUP PRESENTATION

Q1. A ball is thrown upward with an initial velocity of 64 ft per second from an initial height of 80 ft.

Find the position function giving the height s as a function of time t.
When does the ball hit the ground?

Calculus book “CALCULUS of a SINGLE VARIABLE” 8th Edition by LARSON, HOSTETLER and EDWARDS.
Teacher’s Resource Guide
College Board, Advance Placement Program
Fast Track to Preparing For the AP Placement Calculus AB and BC Examinations.
Test Item File by Larson, et al

CPI / Areas of Focus / Instructional Activities
(Students should know / Students should be able to do) / Modifications/
EXTENSION / Assessment / Resources / Materials
4.2.12 E1
4.2.12 E2
4.3.12 A3
4.3.12 B1
4.3.12 B2
4.3.12 B4
4.5 A1
4.5 A2
4.5 A3
4.5 A4 /

Review of Sigma notation and summation formulas.
Understand and use the principles of approximation to find area of a plane region.
Upper and lower sum and their limits.
Define the area of a region in the plane.
Find area by Limit definition.
Define Riemann Sums
Definite integrals and their properties.
Evaluate definite integrals.

/ MODIFICATION: Find the area of the region bounded by the graph of
f(X)=4-X, the X-axis, and the vertical lines X=1 and X=2 by limit definition.
EXTENSION: Determine which value best approximates the area of the region between the X-axis and the graph of the function f(X)=4 - X over the interval [0, 2]. / 1.TEACHER OBSERVATION
2.NOTE BOOK CHECK
3.QUIZZES
4.CHAPTER TESTS
5.CLASS WORK
6.HOMEWORK
7.PROJECTS
GROUP PRESENTATION /

Calculus book “CALCULUS of a SINGLE VARIABLE” 8th Edition by LARSON, HOSTETLER and EDWARDS.
Teacher’s Resource Guide
College Board, Advance Placement Program
Fast Track to Preparing For the AP Placement Calculus AB and BC Examinations.
Test Item File by Larson, et al

Students will define the First Fundamental Theorem of Calculus.
Guidelines for using the First Fundamental Theorem of Calculus to evaluate the definite integrals.
The Mean Value Theorem for Integrals.
Average value of a function.
The Second Fundamental Theorem of Calculus.

/ MODIFICATION: Write in your own words what do you think are the uses of First and the Second Fundamental theorem of Calculus, the Mean Value Theorem and the Average Value of a function on an interval. Give some examples.
EXTENSION:
The velocity v of the flow of blood at a distance r from the central axis of an artery of radius R is
V= k(R- r) when k is constant of proportionality. Find the average rate of flow of blood along a radius of the artery. (Use 0 and R as the limit of integration.) / 1.TEACHER OBSERVATION
2.NOTE BOOK CHECK
3.QUIZZES
4.CHAPTER TESTS
5.CLASS WORK
6.HOMEWORK
7.PROJECTS
GROUP PRESENTATION.
1. The position function of a moving particle is given by S(t) = t - 6 t +9t -2, when 0 t 5. Find the total distance the particle travels in 5 units of time. /

Calculus book “CALCULUS of a SINGLE VARIABLE” 8th Edition by LARSON, HOSTETLER and EDWARDS.
Teacher’s Resource Guide
College Board, Advance Placement Program
Fast Track to Preparing For the AP Placement Calculus AB and BC Examinations.
Test Item File by Larson, et al

Students will recognize a PATTERN in a problem and use u-substitution to solve a problem.
Recognize f(g(x))g”(X) pattern
Anti-derivative of a composite function.
Change of variables for DEFINITE INTEGRALS
The General Power Rule for integration.
Integration by SUBSTITUTION.

/ MODIFICATION: Students will be given a few algebraic functions and will be asked to differentiate them. Then they will be asked to integrate them by finding “U”. Then they will write in their own words how they are able to observe the pattern: the connection between differentiation and integration.
EXTENSION: The rate of disbursement dQ/dt of a 2 million dollar federal grant is proportional to the square of (100-t). Time t is measured in days (0t100), and Q is the amount that remains to be disbursed after 50 days. Assume that all the money must be disbursed in 100 days. / 1.TEACHER OBSERVATION
2.NOTE BOOK CHECK
3.QUIZZES
4.CHAPTER TESTS
5.CLASS WORK
6.HOMEWORK
7.PROJECTS
GROUP PRESENTATION.
1.The normal monthly rainfall at the Seattle-Tacoma airport can be approximated by the model
R= 3.121 + 2.399 Sin (0.524t+1.377)
Where R is measured in inches and t is the time in months, with t =1, corresponding to January. (Source: US National Oceanic and Atmospheric Administration).

Determine the extrema of the function over a one –year period.
Use integration to estimate the normal annual rainfall.

Calculus book “CALCULUS of a SINGLE VARIABLE” 8th Edition by LARSON, HOSTETLER and EDWARDS.
Teacher’s Resource Guide
College Board, Advance Placement Program
Fast Track to Preparing For the AP Placement Calculus AB and BC Examinations.
Test Item File by Larson, et al

CPI / Areas of Focus / Instructional Activities
(Students should know / Students should be able to do) / Modifications/
EXTENSION / Assessment / Resources / Materials
4.2.12 E1
4.2.12 E2
4.3.12 A3
4.3.12 B1
4.3.12 B2
4.3.12 B4
4.5 A1
4.5 A2
4.5 A3
4.5 A4 /

The TRAPEZOIDAL RULE
SIMPSON’S RULE
Approximation with Simpson’s Rule
Error Analysis: Error in Trapezoid Rule and Simpson’s Rule.

/ MODIFICATION: Explain Trapezoidal rule, the Simpson’s Rule, and the error analysis in your own words. Differentiate between error and propagated error.
EXTENSION: Use Simpson’s Rule with n=14 to approximate the area of the region bounded by the graph of
Y= X CosX, Y = 0, X=0 and X = /2. / 1.TEACHER OBSERVATION
2.NOTE BOOK CHECK
3.QUIZZES
4.CHAPTER TESTS
5.CLASS WORK
6.HOMEWORK
7.PROJECTS
GROUP PRESENTATION.
1. Prove that the Simpson’s Rule is exact when approximating the integral of a cubic polynomial function and demonstrate the result for integration of f(X) = X for X= 0, X=2 and n = 2. /

Calculus book “CALCULUS of a SINGLE VARIABLE” 8th Edition by LARSON, HOSTETLER and EDWARDS.
Teacher’s Resource Guide
College Board, Advance Placement Program
Fast Track to Preparing For the AP Placement Calculus AB and BC Examinations.
Test Item File by Larson, et al

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