CALCULUS SYLLABUS GRADE 12
Course Description
Grade12 Calculus has been developed taking into consideration the five standards (strands) of
Mathematics defined by NCTM: the course develops student understanding and skills in Calculus,
building on the skills learned in Pre-Calculus in Grade 11. Calculus will include numbers,
operations, geometry, measurement, data analysis, probability and problem solving. The Calculus
course will focus on Limits and continuity, Derivatives, Rules of Differentiation, integrals,
application of integration, logarithmic, exponential functions. In addition students learn different
ways to represent math, to communicate their math understanding and to make connections to
real world situations and other subject areas. Students are assessed in a variety of ways
continuously throughout the program of study to ensure they are understanding content and are
able to use and apply the knowledge and skills they are developing.
Curricular content
APPENDIX A contains the standards. APPENDIX B contains the performance indicators for each standard
for High School and APPENDIX C contains the Mathematics Focal Points for Grade 12. APPENDIX D
contains the Secondary Assessment Policy. Students use the ‘CALCULUS OF A SINGLE VARIABLE –
LARSON EDWARDS’ Textbook.
Quarter 1
Unit 0: Review of Algebra – Chapter P
 Graphing linear equations (P.1)
 Sketching the graph of a function (P.1)
 Find the intercepts of a graph (P.1)
 Test a graph for symmetry (P.1)
 Find the points of intersection of 2 graphs (P.1)
 Writing equations of lines. (From a graph, given two points, given slope and a point, given point
and parallel line, given point and perpendicular line) (P.2)
 Determining domain and range of functions (P.3)
 Determining if relations are functions(From an equation or from a graph) (P.3)
 Finding zeros of functions (P.3)
 Predicting transformations of graphs from changes in the equation of the function (P.3)
 Writing equations for combinations and compositions of functions (P.3)
Unit 1: Limits and Continuity
 Understanding the Tangent Line Problem (1.1)
 Estimate a limit using a numerical or graphical approach (1.2 E.g. 1 - 2)
 Explore limits that fail to exist (1.2 E.g. 3 - 5)
 Evaluate basic limits (1.3)
 Evaluating limits of a polynomial, of a rational function, of a composite function, and of
trigonometric functions (1.3)
 Find limits of a function analytically (1.3 E.g. 6 – 10)
 Determine continuity at a point and on an open interval (1.4)
 Evaluating one-sided limits and continuity on a closed interval (1.4)
 Evaluating infinite limits from the left and from the right (1.5)
 Determining removable and non-removable discontinuities of functions (1.5)
 Find and sketch vertical asymptotes of functions (1.5)
Unit 2: Introduction to Derivatives
 Find the slope of a tangent line to a curve at a point (2.1)
 Use the limit definition to find the derivative of a function (2.1)
 Use the derivative to find the slope at a point (2.1)
 Understand differentiability and Continuity (graphs with sharp turns & graphs with vertical
tangent lines) (2.1 E.g. 3 - 4)
 Find the derivative of a function using the power rule, therefore find the slope and the equation
of tangent lines (2.2)
 Find the derivative of sums and differences (2.2)
 Find the derivative of sine and cosine (2.2)
 Find velocity of falling objects (2.2)
 Know the position function and use the derivative to find velocity (2.2)
 Find the derivative using the Product Rule (2.3)
 Find the derivative using the Quotient Rule (2.3)
 Find the derivative of trigonometric functions (2.3)
 Find higher order derivatives (2.3)
 Find acceleration due to gravity using derivatives (2.3)
 Find the derivative of a composite function using the chain rule (2.4)
 Simplify the derivative of a function using algebra (2.4)
 Find the derivative of trig functions using the chain rule (2.4)
 Distinguish between functions written in implicit form and explicit form (2.5)
 Use implicit differentiation (2.5)
 Find a related rate and use them to solve real life problems, involving length, area, volume,
velocity, angles of elevation and angular speed verses Linear speed (2.6)
Quarter 2
Unit 3: Applications of Differentiation (Part 1)
 Understand the definition of extrema of a function on an interval (3.1)
 Understand the definition of relative extrema of a function on an open interval (3.1)
 Find extrema on a closed interval (3.1)
 Understand and use Rolle’s Theorem (3.2)
 Understand and use the Mean Value Theorem (3.2)
 Determine intervals on which a function is increasing or decreasing (3.3)
 Apply the first derivative test to find relative extrema of a function (3.3)
 Determine intervals on which a function is concave upward or concave downward (3.4)
 Find any point of inflection of a graph of a function (3.4)
 Apply the second derivative test to find the relative extrema of a function (3.4)
SEMESTER EXAM
Quarter 3
Unit 3: Applications of Differentiation (Part 2)
 Limits at infinity (3.5)
 Analyze and sketch the graph of a function (3.6)
 Solve applied minimum and maximum (optimization) problems (3.7)
Unit 4: Integration
 Write the general solution of a differential equation (4.1)
 Use indefinite integral notation for antiderivatives (4.1)
 Use basic integration rules to find aniterivatives (4.1)
 Find a particular solution of a differential equation (4.1)
 Use sigma notation to write and evaluate a sum (4.2)
 Evaluate a definite integral using limits (4.3)
 Evaluate a definite integral using properties of definite integrals (4.3)
 Evaluate a definite integral using the Fundamental Theorem of Calculus (4.4)
 Understand the Mean Value Theorem for Integrals (4.4)
 Find the average value for a function over a closed interval (4.4)
 Understand and use the Second Fundamental Theorem of Calculus (4.4)
 Understand and use the Net Change Theorem (4.4)
 Use pattern recognition and change of variables to find an indefinite integral (4.5)
 Use the General Power Rule for integration to find an indefinite integral (4.5)
 Use a change of variables to find an definite integral (4.5)
 Evaluate a definite integral involving an even or odd function (4.5)
Unit 5 – Applications of Integration
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Find the area of a region between two curves using integration (7.1).
Find the area of a region between intersecting curves using integration (7.1).
Describe integrations as an accumulation process (7.1)
Find the volume of a solid of revolution using the disk method (7.2).
Find the volume of a solid of revolution using the washer method (7.2).
Find the volume of a solid with known cross sections (7.2).
Quarter 4
Unit 6 – Logarithmic, Exponential and Transcendental Functions
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Develop and use properties of the natural logarithmic function (5.1).
Understand the definition of the number e (5.1).
Find derivatives of functions involving the natural logarithmic function (5.1).
Use the Log Rule for Integration to integrate a rational function (5.2).
Integrate trigonometric functions (5.2).
Verify that one function is the inverse function of another function (5.3).
Determine whether a function has an inverse function (5.3).
Find the derivative of an inverse function (5.3).
Develop properties of the natural exponential function (5.4).
Differentiate natural exponential functions (5.4).
Integrate natural exponential functions (5.4).
Define exponential functions that have bases other than e (5.5).
Differentiate and integrate exponential functions that have bases other than e (5.5).
Use exponential functions to model compound interest and exponential growth (5.5).
Develop properties of the six inverse trigonometric functions (5.6).
Differentiate an inverse trigonometric function (5.6).
Review the basic differentiation rules for elementary functions (5.6).
Unit 6 – Differential Equations
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Use initial conditions to find particular solutions of differential equations (6.1).
Use slope fields to approximate solutions of differential equations.
Use Euler’s Method to approximate solutions of differential equations.
Use separation of variables to solve a simple differential equation(6.2).
Use exponential functions to model growth and decay in applied problems.
Recognize and solve differential equations that can be solved by separation of variables.
Recognize and solve homogeneous differential equations.
Use differential equations to model and solve applied problems(6.3).
Solve and analyse logistic differential equations.
Unit 8 – Integration Techniques
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Review procedures for fitting an integrand to one of the basic integration rules (8.1).
Find an antiderivative using integration by parts (8.2).
Use a tabular method to perform integration by parts.
FINAL EXAM
Summative Assessments: Weight: 40%
There will be 2-3 major assessments in each quarter. These will be in the form of full period tests.
Each test in this category will be worth 100 points.
Formative Assessments: Weight: 45%
This category is for all the graded work that prepares you for the summative assessments. Graded
work includes quizzes, problem sets, and projects. There will be approximately 5-8 grades in this
category in each quarter. Each item in this category will be worth approximately 10 – 25 points
depending on the nature of the assignment. The point values/rubrics for each of these assessments
will be given to you in advance.
Effort: Weight: 15%
This category is based on two things. Each week, you will receive a classwork grade and a
homework grade. The classwork grade is based on preparedness for class, participation and
behavior. Each week you can earn 5 possible points on classwork.
APPENDIX A: Mathematics Standards
1. Number and Operations
Instructional programs from prekindergarten through grade 12 should enable all students to- understand numbers, ways of representing numbers, relationships among numbers,
and number systems;
 understand meanings of operations and how they relate to one another;
 compute fluently and make reasonable estimates
2. Algebra
Instructional programs from prekindergarten through grade 12 should enable all students to- understand patterns, relations, and functions;
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represent and analyze mathematical situations and structures using algebraic symbols;
use mathematical models to represent and understand quantitative relationships;
analyze change in various contexts
3. Geometry
Instructional programs from prekindergarten through grade 12 should enable all students
to- analyze characteristics and properties of two- and three-dimensional geometric
shapes and develop mathematical arguments about geometric relationships;
 specify locations and describe spatial relationships using coordinate geometry and
other representational systems;
 apply transformations and use symmetry to analyze mathematical situations;
 use visualization, spatial reasoning, and geometric modeling to solve problems.
4. Measurement
Instructional programs from prekindergarten through grade 12 should enable all students to- understand measurable attributes of objects and the units, systems, and processes of
measurement;
 apply appropriate techniques, tools, and formulas to determine measurements.
5. Data Analysis and Probability
Instructional programs from prekindergarten through grade 12 should enable all students to- formulate questions that can be addressed with data and collect, organize, and
display relevant data to answer them;
 select and use appropriate statistical methods to analyze data;
 develop and evaluate inferences and predictions that are based on data;
 understand and apply basic concepts of probability
6. Problem Solving
Instructional programs from prekindergarten through grade 12 should enable all students
to- build new mathematical knowledge through problem solving;
 solve problems that arise in mathematics and in other contexts;
 apply and adapt a variety of appropriate strategies to solve problems;
 monitor and reflect on the process of mathematical problem solving.
7. Reasoning and Proof
Instructional programs from prekindergarten through grade 12 should enable all students
to- recognize reasoning and proof as fundamental aspects of mathematics;
 make and investigate mathematical conjectures;
 develop and evaluate mathematical arguments and proofs;
 select and use various types of reasoning and methods of proof.
8. Communication
Instructional programs from prekindergarten through grade 12 should enable all students
to- organize and consolidate their mathematical thinking through communication;
 communicate their mathematical thinking coherently and clearly to peers, teachers,
and others;
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analyze and evaluate the mathematical thinking and strategies of others;
use the language of mathematics to express mathematical ideas precisely.
9. Connections
Instructional programs from prekindergarten through grade 12 should enable all students
to- recognize and use connections among mathematical ideas;
 understand how mathematical ideas interconnect and build on one another to
produce a coherent whole;
 recognize and apply mathematics in contexts outside of mathematics.
10. Representation
Instructional programs from prekindergarten through grade 12 should enable all students
to- create and use representations to organize, record, and communicate mathematical
ideas;
 select, apply, and translate among mathematical representations to solve problems;
 use representations to model and interpret physical, social, and mathematical
phenomena.
APPENDIX B: Performance Indicators Grades 9-12
1. Numbers and Operations Standard
Instructional
programs from
prekindergarten
through grade 12
should enable all
students to
In grades 9 through 12 all students should—
1.1 Understand
numbers, ways of
 develop a deeper understanding of very large and very
small numbers and of various representations of them;
 compare and contrast the properties of numbers and
representing
numbers,
relationships among
numbers, and
number systems
1.2 Understand
meanings of
operations and how
they relate to one
another
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1.3 Compute fluently
and make reasonable
estimates
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number systems, including the rational and real
numbers, and understand complex numbers as solutions
to quadratic equations that do not have real solutions;
understand vectors and matrices as systems that have
some of the properties of the real-number system;
use number-theory arguments to justify relationships
involving whole numbers.
judge the effects of such operations as multiplication,
division, and computing powers and roots on the
magnitudes of quantities;
develop an understanding of properties of, and
representations for, the addition and multiplication of
vectors and matrices;
develop an understanding of permutations and
combinations as counting techniques.
develop fluency in operations with real numbers,
vectors, and matrices, using mental computation or
paper-and-pencil calculations for simple cases and
technology for more-complicated cases.
judge the reasonableness of numerical computations and
their results.
2. Algebra Standard
Instructional
programs from
prekindergarten
through grade 12
should enable all
students to—
2.1 Understand
patterns, relations,
and functions
In grades 9 through 12 all students should—
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generalize patterns using explicitly defined and
recursively defined functions;
understand relations and functions and select, convert
flexibly among, and use various representations for
them;
analyze functions of one variable by investigating rates
of change, intercepts, zeros, asymptotes, and local and
global behavior;
understand and perform transformations such as
arithmetically combining, composing, and inverting
commonly used functions, using technology to perform
such operations on more-complicated symbolic
expressions;
understand and compare the properties of classes of
functions, including exponential, polynomial, rational,
logarithmic, and periodic functions;
interpret representations of functions of two variables
2.2 Represent and
analyze
mathematical
situations and
structures using
algebraic symbols
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2.3 Use
mathematical models
to represent and
understand
quantitative
relationships
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2.4 Analyze change in
various contexts
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understand the meaning of equivalent forms of
expressions, equations, inequalities, and relations;
write equivalent forms of equations, inequalities, and
systems of equations and solve them with fluency—
mentally or with paper and pencil in simple cases and
using technology in all cases;
use symbolic algebra to represent and explain
mathematical relationships;
use a variety of symbolic representations, including
recursive and parametric equations, for functions and
relations;
judge the meaning, utility, and reasonableness of the
results of symbol manipulations, including those carried
out by technology.
identify essential quantitative relationships in a
situation and determine the class or classes of functions
that might model the relationships;
use symbolic expressions, including iterative and
recursive forms, to represent relationships arising from
various contexts;
draw reasonable conclusions about a situation being
modeled.
approximate and interpret rates of change from
graphical and numerical data.
3. Geometry Standard
Instructional
programs from
prekindergarten
through grade 12
should enable all
students to—
3.1 Analyze
characteristics and
properties of twoand threedimensional
geometric shapes and
develop
mathematical
arguments about
geometric
relationships
In grades 9 through 12 all students should—
analyze properties and determine attributes of two- and
three-dimensional objects;
 explore relationships (including congruence and
similarity) among classes of two- and three-dimensional
geometric objects, make and test conjectures about
them, and solve problems involving them;
 establish the validity of geometric conjectures using
deduction, prove theorems, and critique arguments
made by others;
 use trigonometric relationships to determine lengths
and angle measures.
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3.2 Specify locations
and describe spatial
relationships using
coordinate geometry
and other
representational
systems
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3.3 Apply
transformations and
use symmetry to
analyze
mathematical
situations
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3.4 Use visualization,
spatial reasoning,
and geometric
modeling to solve
problems
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use Cartesian coordinates and other coordinate systems,
such as navigational, polar, or spherical systems, to
analyze geometric situations;
 investigate conjectures and solve problems involving
two- and three-dimensional objects represented with
Cartesian coordinates.
understand and represent translations, reflections,
rotations, and dilations of objects in the plane by using
sketches, coordinates, vectors, function notation, and
matrices;
 use various representations to help understand the
effects of simple transformations and their
compositions.
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draw and construct representations of two- and threedimensional geometric objects using a variety of tools;
visualize three-dimensional objects and spaces from
different perspectives and analyze their cross sections;
use vertex-edge graphs to model and solve problems;
use geometric models to gain insights into, and answer
questions in, other areas of mathematics;
use geometric ideas to solve problems in, and gain
insights into, other disciplines and other areas of
interest such as art and architecture.
4. Measurement Standard
Instructional
programs from
prekindergarten
through grade 12
should enable all
students to—
In grades 9 though 12 all students should—
4.1 Understand
measurable
attributes of objects
and the units,
systems, and
processes of
measurement
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4.2 Apply
appropriate
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make decisions about units and scales that are
appropriate for problem situations involving
measurement.
analyze precision, accuracy, and approximate error in
measurement situations;
 understand and use formulas for the area, surface area,
techniques, tools, and
formulas to
determine
measurements
and volume of geometric figures, including cones,
spheres, and cylinders;
 apply informal concepts of successive approximation,
upper and lower bounds, and limit in measurement
situations;
 use unit analysis to check measurement computations
5. Data Analysis and Probability Standard
Instructional
programs from
prekindergarten
through grade 12
should enable all
students to—
5.1 Formulate
questions that can be
addressed with data
and collect, organize,
and display relevant
data to answer them
In grades 9 through 12 all students should—
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5.2 Select and use
appropriate
statistical methods to
analyze data
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5.3 Develop and
evaluate inferences
and predictions that
are based on data
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understand the differences among various kinds of
studies and which types of inferences can legitimately be
drawn from each;
know the characteristics of well-designed studies,
including the role of randomization in surveys and
experiments;
understand the meaning of measurement data and
categorical data, of univariate and bivariate data, and of
the term variable;
understand histograms, parallel box plots, and
scatterplots and use them to display data;
compute basic statistics and understand the distinction
between a statistic and a parameter.
for univariate measurement data, be able to display the
distribution, describe its shape, and select and calculate
summary statistics;
for bivariate measurement data, be able to display a
scatterplot, describe its shape, and determine regression
coefficients, regression equations, and correlation
coefficients using technological tools;
display and discuss bivariate data where at least one
variable is categorical;
recognize how linear transformations of univariate data
affect shape, center, and spread;
identify trends in bivariate data and find functions that
model the data or transform the data so that they can be
modeled.
use simulations to explore the variability of sample
statistics from a known population and to construct
sampling distributions;
understand how sample statistics reflect the values of
population parameters and use sampling distributions
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5.4 Understand and
apply basic concepts
of probability
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as the basis for informal inference;
evaluate published reports that are based on data by
examining the design of the study, the appropriateness
of the data analysis, and the validity of conclusions;
understand how basic statistical techniques are used to
monitor process characteristics in the workplace.
understand the concepts of sample space and probability
distribution and construct sample spaces and
distributions in simple cases
use simulations to construct empirical probability
distributions;
compute and interpret the expected value of random
variables in simple cases;
understand the concepts of conditional probability and
independent events;
understand how to compute the probability of a
compound event.
APPENDIX C: Focal Points Calculus
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Students demonstrate knowledge of both the formal definition and the graphical
interpretation of limit of values of functions. This knowledge includes one-sided limits,
infinite limits, and limits at infinity.
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Students know the definition of convergence and divergence of a function as the domain
variable approaches either a number or infinity.
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Students prove and use theorems evaluating the limits of sums, products, quotients, and
composition of functions.
Students use graphical calculators to verify and estimate limits.
Students prove and use special limits, such as the limits of (sin(x))/x and (1-cos(x))/x as x
tends to 0.
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Students demonstrate knowledge of both the formal definition and the graphical
interpretation of continuity of a function.
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Students demonstrate an understanding and the application of the intermediate value
theorem and the extreme value theorem.
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Students demonstrate an understanding of the formal definition of the derivative of a
function at a point and the notion of differentiability:
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Students demonstrate an understanding of the derivative of a function as the slope of the
tangent line to the graph of the function.
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Students demonstrate an understanding of the interpretation of the derivative as an
instantaneous rate of change. Students can use derivatives to solve a variety of problems
from physics, chemistry, economics, and so forth that involve the rate of change of a
function.
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Students understand the relation between differentiability and continuity.
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Students derive derivative formulas and use them to find the derivatives of algebraic,
trigonometric, inverse trigonometric, exponential, and logarithmic functions.
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Students know the chain rule and its proof and applications to the calculation of the
derivative of a variety of
composite functions.
Students find the derivatives of parametrically defined functions and use implicit
differentiation in a wide variety of problems in physics, chemistry, economics, and so forth.
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Students compute derivatives of higher orders.
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Students know and can apply Rolle's theorem, the mean value theorem, and L'Hôpital's
rule.
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Students use differentiation to sketch, by hand, graphs of functions. They can identify
maxima, minima, inflection points, and intervals in which the function is increasing and
decreasing.
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Students know Newton's method for approximating the zeros of a function.
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Students use differentiation to solve optimization (maximum-minimum problems) in a
variety of pure and applied contexts.
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Students use differentiation to solve related rate problems in a variety of pure and applied
contexts.
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Students know the definition of the definite integral by using Riemann sums. They use this
definition to approximate integrals.
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Students apply the definition of the integral to model problems in physics, economics, and
so forth, obtaining results in terms of integrals.
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Students demonstrate knowledge and proof of the fundamental theorem of calculus and
use it to interpret integrals as antiderivatives.
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Students use definite integrals in problems involving area, velocity, acceleration, volume of
a solid, area of a surface of revolution, length of a curve, and work.
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Students compute, by hand, the integrals of a wide variety of functions by using techniques
of integration, such as substitution, integration by parts, and trigonometric substitution.
They can also combine these techniques when appropriate.
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Students know the definitions and properties of inverse trigonometric functions and the
expression of these functions as indefinite integrals.
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Students compute, by hand, the integrals of rational functions by combining the techniques
in standard 17.0 with the algebraic techniques of partial fractions and completing the
square.
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Students compute the integrals of trigonometric functions by using the techniques noted
above.
Students understand the algorithms involved in Simpson's rule and Newton's method.
They use calculators or computers or both to approximate integrals numerically.
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Students understand improper integrals as limits of definite integrals.
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Students demonstrate an understanding of the definitions of convergence and divergence
of sequences and series of real numbers. By using such tests as the comparison test, ratio
test, and alternate series test, they can determine whether a series converges.
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Students understand and can compute the radius (interval) of the convergence of power
series.
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Students differentiate and integrate the terms of a power series in order to form new
series from known ones.
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Students calculate Taylor polynomials and Taylor series of basic functions, including the
remainder term.
Students know the techniques of solution of selected elementary differential equations and
their applications to a wide variety of situations, including growth-and-decay problems
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APPENDIX D: Secondary Assessment Policy
Assessment monitors the progress of student learning and produces feedback for students, teachers,
parents and external institutions. The following policy outlines the general assessment procedures for
the school. Teachers are responsible for communicating their individual assessment policies to the
students and parents at the beginning of the school year.
Teachers are expected to communicate assessment expectations and criteria, including major
assignments and projects clearly to students prior to a chunk of learning. Assessment should take into
account the ISS diverse group of learners and learning styles. Feedback on assignments should be
positive, constructive and prompt. Teachers should provide a wide variety of different assessment
opportunities which are relevant and motivational to students. Formative assessments assist student in
building understanding, knowledge and skills and summative assessments assess students’ acquired
understanding, knowledge and skills.
External
Summative
Formative
Definition
External
assessments are
assessments which
are designed and
marked externally
Summative assessments
are those assessments
given within a class at the
end of a chunk of learning
(such as a unit).
Formative assessments are those
given regularly and continuously
throughout the school year.
Primary
Purpose
To measure growth
and progress, to
inform teaching, to
identify needs, to
collect data, to
determine level of
understanding, to
determine reading
or math levels
against national
norms, assessing
student learning,
providing a
qualification for
university or
college entry.
To inform teaching, to
identify needs, to
determine level of
understanding, to measure
progress, to communicate
with parents
To determine prior knowledge, to
determine student interest, to modify
teacher practice, measure
understanding, ensuring short-term
knowledge and understanding
objectives and targets are being met,
to ensure students are progressing
Policies
Some external
assessments are
taken twice a year,
some are once and
Assessments are aligned to
curriculum, teachers model
in advance, authentic
assessments, differentiated
Assessments are aligned to
curriculum, differentiated if
necessary.
Practices
some are on-going.
if necessary.
STAR Math, NWEA,
Accelerated Math,
PSAT, SAT, AP
Essays, projects, test,
RAFTS, portfolio,
investigations, real-world
examples, exams, oral
presentation, reports,
reflections, mid-trimester
reports, mid-quarter
reports
Observation, journal, quiz, exit cards,
peer assessment or self-assessment
(not graded on Gradequick), role play,
conferencing, small group discussion,
debate, create/present, note-taking,
reflection, homework, classwork,
effort, behavior, participation,
Gradequick reporting,
Teachers will be asked to implement IEP's/ILP's in their classroom should it contain students receiving
necessary support. Teachers will be provided with the document, as well as support in how to
effectively implement the modifications in order to ensure student success. We strongly suggest that
teachers consult with the learning specialist or principal during the design and implementation of all
summative evaluations for students with IEPs.