### Making waves in STEM

• Date created

July 9, 2020

• Last updated

June 17, 2021

# Course Overview

Calculus is the mathematics of change. It is used to solve complex problems that are
continuously evolving and would otherwise be unsolvable with only algebra and
geometry. This online advanced placement course is designed to prepare students to
become deep mathematical thinkers. You will explore the calculus concepts of limits,
differentiation, and integration and apply those concepts in meaningful ways.

The course is split into two semesters. The first semester focuses on the concepts of
functions, limits, and differentiation and their applications. The second semester builds
off the first semester to focus on integrations. It will cover topics such as the definite and
indefinite integral and their applications, inverse function, and techniques for integrating.

# Course Goals

By the end of the course the student will be able to:

Work with functions represented in a variety of ways: graphical, numerical, analytical, or
verbal, and understand the connections among these representations.
Understand the meaning of the derivative in terms of a rate of change and local linear
approximation and use derivatives to solve a variety of problems.
Understand the meaning of the definite integral both as a limit of Riemann sums and as the
net accumulation of change and use integrals to solve a variety of problems.
Understand the relationship between the derivative and the definite integral as expressed in
both parts of the fundamental theorem of calculus.
Communicate mathematics both orally and in well-written sentences and explain solutions to problems.
Model a written description of a physical situation with a function, a differential equation, or an integral.
Use technology to help solve problems, experiment, interpret results, and verify conclusions.
Determine the reasonableness of solutions, including sign, size, relative accuracy, and units of measurement.
Develop an appreciation of calculus as a coherent body of knowledge and as a human accomplishment.

# Math Skills

Successful completion of four years of high school mathematics provides the
mathematical skills students need for Advanced Calculus. Typically, this includes
Algebra 1, Algebra 2, Geometry, and Precalculus (including elements of Trigonometry)
or four years of integrated high school mathematics.

# General Skills

To participate in this course, you should be able to do the following:

Complete basic operations with word processing software, such as Microsoft Word or Google Docs.
Understand the basics of spreadsheet software, such as Microsoft Excel or Google
Spreadsheets, but having prior computing experience is not necessary.
Perform online research using various search engines and library databases.
Communicate through email and participate in discussion boards.

For a complete list of general skills that are required for participation in online courses,
refer to the Prerequisites section of the Plato Student Orientation document, found at
the beginning of this course.

# Credit Value

Advanced Calculus is a 1.0-credit course. It includes semesters A and B.

# Course Materials

notebook
graphing calculator, recommend TI-83 or equivalent
computer with Internet connection and speakers or headphones
Microsoft Word or equivalent

# Teaching Strategies

### Structure

This online course is organized into units and lessons. Each lesson incorporates
multiple learning activities designed to develop, apply, and assess specific learning
objectives. (See Course Outline, below.)

### Concept Development Activities

In order to generate skills for lifelong learning and to employ the most appropriate
learning approach for each topic, twenty-five percent of the lessons will use studentdriven, constructivist approaches for concept development. The remaining lessons will
employ direct instruction approaches. In either case, students will take full advantage of
the online learning environment, linking to rich online, multimedia, and interactive
resources. Developing critical 21st Century skills is an important secondary goal of this course.

### Application

Application and inquiry will be an integrated part of the lessons, requiring higher-level
cognitive work. Students will submit written work online for review, comment, and

### Discussions/Group Work

Students will also have the opportunity to engage in online (asynchronous) discussions
during this course. Discussion topics provide the chance to dig deeper into specific
calculus and STEM concepts and applications.

Throughout AP Calculus, students will also have the opportunity to work synchronously
in groups. Group work will allow students a chance to communicate orally about the
mathematical concepts in the course and to provide explanations on how they derived
solutions to the problems presented in the course. Much of the oral discussions will
emphasize real world application/explanation of mathematical concepts.

For example, in Semester A, Unit 2, students will participate in a discussion activity
about the mathematical meaning of the word “limit” and how that compares to the more
commonly used meaning that they normally hear and use. They will communicate both
orally and in writing for this and other discussions in the course.

# Pedagogy

STEM: This course will make special effort to integrate the study of calculus in a broader
STEM perspective. In today’s global economy, employers are looking for skilled workers
who are innovative problem solvers and critical thinkers. Such skills encompass what
STEM education is. It fuses science, technology, engineering, and mathematics into one
cohesive discipline. STEM education is designed to open students’ minds to the exciting
and fulfilling opportunities in STEM-related careers.
Many guidelines have been integrated into the course design and lessons that embody
the goals of a STEM education. Lessons will use real-world scenarios that that will
illustrate to students how math, science, engineering, and technology are applied in
professional and everyday lives. Students will leverage technologies such as graphing
calculators, spreadsheets, and the Internet to help them solve difficult and complex
problems. Students will also be given culminating activities that will require them to solve
open-ended, complex problems.

Constructivism: Twenty-five percent of the course will use student-driven, constructivist
approaches in order to generate skills for lifelong learning. The remaining portion of the
course will be direct instruction on topics of importance to the subject matter.

Context: All too often, concepts are brought to the abstract algebraic level before
students have a good understanding of what the concept is. Lessons in this course will
attempt to explore the concept in an easy to understand, real-world manner. That will
help learners grasp the concept within a familiar context so that they can make a clear
connection to the abstract.

Reasoning and Sense-Making: It’s important to allow students to develop their
reasoning and sense-making skills. Students need to be able to think about mathematics
in a way that is meaningful to them. In these lessons, students will be asked to generate
logical conclusions and make meaningful connections to both the known and unknown.

Practice: As important as it is for students to be able to understand the concepts in their
own way, they still have to be able to perform and master the algebraic manipulation
required to complete this class. Therefore, practice will be integrated into the lessons.

Technology: Technology is changing at an exponential rate. It is important that students
are familiar with technology and how to use it appropriately. This course will make use of
current technologies, such as the Internet, graphing calculators, and spreadsheets to
help students solidify their understanding of the concepts that are presented in this
course.

Graphing calculators – students will need a graphing calculator in this
course to explore, experiment, and investigate calculus concepts. The course
will provide an embedded graphing tool to use within lessons, but the student
will be expected to have a handheld version as well. Graphing calculator
activities require the student to explore what happens when you change
points on the graph and require the student to draw conclusions based on
those changes. Specific graphing calculator activities are detailed in the unit
outline below. Students can either purchase the calculators for themselves or
they will be provided for them by the school or teacher.

# Student Evaluation

Multiple evaluation tools will be used to assess understanding at all appropriate
cognitive levels and to reflect AP assessment methodology:

Lesson-Level Mastery Tests: Each lesson will be accompanied by an AP-style
multiple-choice mastery test to assess mastery of the basic lesson concepts.
Self-Assessment Lesson Activities: Especially useful in constructivist-leaning lessons,
self-assessment activities will provide sample responses against which learners can
assess their own learning.
Teacher-Graded Lesson Activities: These lesson activities will require teacher
assessment, employing AP-style objective rubrics. Students will be provided with the
rubrics for each assignment.
Unit-Level Posttests: Each unit will have a multiple-choice assessment to confirm that
all the material within the unit has been retained and can be applied in a larger context
than a single-lesson format.
Unit-Level Culminating Activities: Learners will have the chance to apply their
knowledge of the concepts across the lessons within a unit. Most of the units will include
this teacher-graded activity for evaluation of higher order thinking skills.
End-of-Semester Tests: At the end of each of the two semesters, learners will take a
multiple-choice test to assess mastery of concepts and provide additional practice for a
long-form exam like the AP exam.

# Course Outline

This course will be structured in two 18-week semesters. Semester A will be divided into
four units and semester B will be divided into five units, as follows.

## Units

Semester A
Functions/Prerequisites for Calculus
Limits
Derivatives
Applications of Derivatives

Semester B
The Definite Integral
Applications of Integration
Inverse Functions
Techniques of Integration
Further Applications of Integration

This course will employ the following textbook as a resource for deep research and
learning:

Edwards, C. H, and Penney, D.E. Calculus, 6th Edition. Upper Saddle River, NJ:
Prentice Hall, Inc. 2002 Schedule, Topics, and Objectives.

# AP Topic Outline

Detailed objectives for all course lessons are documented by unit in the course outline
below. This scope and sequence is designed to introduce basic concepts and
applications first, then to spiral back to more sophisticated understandings and
applications. In the process, the AP topic outline is fully met and built up in a logical and
natural way. The following guide concisely maps the AP topic outline to the detailed
course outline below:

I. Functions, Graphs, and Limits

The following AP topics are covered in semester A, units 1 and 2. Higher-level functions
are investigated in semester B, unit 3:

analysis of graphs
limits of functions (including one-sided limits)
asymptotic and unbounded behavior
continuity as a property of functions

II. Derivatives

The following AP topics are covered in semester A, units 3 and 4. Derivatives for higherlevel functions are investigated in semester B, unit 3:

concept of the derivative
derivative at a point
derivative as a function
second derivatives
applications of derivatives
computation of derivatives

III. Integrals

The following AP topics are covered in semester B, units 1, 2, 3, 4, and 5:

interpretations and properties of definite integrals
fundamental theorem of calculus
techniques of antidifferentiation
applications of antidifferentiation
numerical approximations to definite integrals

# Semester A

## Unit 1 – Functions/Prerequisites for Calculus (2-3 Weeks, 10 Lessons)

Functions – Define relationships called functions and learn about the parts
of a function.

define functions
explore the meaning of function inputs
explore the meaning of function outputs
understand the function rule
understand how the different parts of a function are related

Describing Functions with Equations, Tables, and Graphs – Learn about
how equations, tables, and graphs can all represent the same function.

explore different ways to represent a function

Defining a Function with Its Rule – Learn how to determine whether a
relation represents a function.

review the meaning of a relation
define what makes a relation a function
determine whether a relation is a function

Identifying Graphs from Their Equations – Identify the graph of an equation.

understand, identify, and calculate slope
understand, identify, and calculate y-intercepts
graph an equation in slope-intercept form

Parallel Lines and Their Slopes – Identify parallel lines and their slopes.

calculate the slope of a line
recognize parallel lines
identify the slopes of parallel lines

Perpendicular Lines and Their Slopes – Identify perpendicular lines and their slopes.

calculate the slope of a line
recognize parallel lines
identify the slopes of parallel lines

Equations of Parallel or Perpendicular Lines – Identify equations of parallel
or perpendicular lines.

determine when the equations of two lines represent parallel lines
determine when the equations of two lines represent perpendicular lines
determine when the equations of two lines represent neither parallel nor
perpendicular lines

Review: Graphs – Graph linear equations and identify parallel and
perpendicular lines.

Identify the slope and points for a line on the coordinate plane
determine if two lines are parallel
determine if two lines are perpendicular

Solving Systems of Linear Equations – Use the graphing method to solve a
system of two linear equations.

graph two linear equations to find the solution to the system of equations
determine the number of solutions that exist for a system of two linear equations

Review: Linear Systems – Solve systems of linear equations or inequalities
using a variety of methods.

solve systems of linear equations and inequalities by graphing
solve systems of linear equations using substitution
solve systems of linear equations using addition
solve systems of linear equations using matrices

Solving Problems with Linear Systems – Use systems of linear equations
or inequalities to solve real-world problems.

model real-world problems using systems of linear equations or inequalities
solve systems of linear equations and inequalities using a variety of methods

Types of Functions – Categorize and describe functions.

explore categories of functions while noting the format of the equation as well as
its graph: linear, polynomial, power, rational, algebraic, inverse, exponential, logarithmic
explore piecewise functions and a specific example of a piecewise function, the
absolute value function
explore descriptors of functions such as even, odd, increasing, decreasing,
symmetry, polynomial, degree, quadratic, and cubic
review trigonometric angle measurement
calculate missing measurements given lengths and angles of a triangle
graph trigonometric functions
use trigonometric identities to solve and simplify trigonometric expressions
review trigonometric identities, functions and their graphs.
review trigonometric angle measurement
calculate missing measurements given lengths and angles of a triangle

Graphing Solution Sets of Associated Inequalities – Solve and graph the

solve inequalities that require unions of solution sets
graph solution sets for quadratic inequalities

Equations of Ellipses and Hyperbolas – Recognize the equations of ellipses
and hyperbolas and graph them.

identify equations of ellipses and hyperbolas
calculate features of ellipses and hyperbolas such as centers and axes

Graphing Linear Inequalities in 1 Variable – Learn to graph the solution
sets for single-variable inequalities.

solve and graph linear inequalities using interval notation
solve and graph linear inequalities using a number line

Translations and Transformations – Apply translations and transformations
to function graphs and their equations.

translate, rotate, and reflect function graphs
use function notation to represent function transformations
rewrite functions to represent a transformation

Functional Values – Compute functional values by translating and
transforming a function.

Find points on a transformed graph based on function notation and a point on the
original graph
Use function transformations to solve real-world problems

Composite Functions – Learn to find composite functions and compute the
points on the resulting function graph.

Use composition to combine two functions
Find points that satisfy a composite function
Determine the simpler functions that make up a composite function

## Unit 2 – Limits (3-4 Weeks, 8 Lessons)

Tangent Lines and Velocity – Examine the tangent problem and relate it to
instantaneous velocity.

• Understand a tangent line graphically and how it relates to a rate of change
• view a tangent line approaching from the left and right sides
• relate a tangent problem to instantaneous velocity
• Instantaneous Velocity Activity
o calculate the slope and instantaneous velocity using data from a table or
a function without the use of a calculator
o given a function and table of ranges, calculate the average velocity for
each interval by hand
o explain the methods used to find answers
o determine the instantaneous velocity at a given point
• graphing calculator activity
o Use the graphing tool (or your personal calculator) to see what happens
to the secant when you reduce the interval between points P and Q:
o Enter the function f(x) = 0.25*(x^2).
o Enter 2 (the x-coordinate of P) as the value of the tangent at x. Enter 4
(the x-coordinate of Q) as the value of the secant at x. Then click New
Function to generate the graph.
o Reduce the value of the x-coordinate of Q so it is closer to the value of
the x-coordinate of P. Note how the graph changes.
o What can you conclude from this activity?

What Are Limits? – Examine limits using both a numerical and graphical
approach.

• understand the concept of limits as seen graphed or stated as a function
• use the definition of a limit to find the limit of a function
• evaluate limits from the left side, right side, and both sides

Laws of Limits? – Calculate limits using the limit laws.

• review and apply laws for constant, sum, product, quotient and exponential
functions

Continuous and Discontinuous Limits – Evaluate limits that are continuous
and discontinuous.

• examine different types of discontinuities (removable, jump) on a graph
• identify functions that have infinite discontinuities
• use the theorems of continuity to show a function is continuous

Evaluation of Limits – Evaluate limits algebraically.

• identify limits that are indeterminate at some value “c”
• develop strategies for evaluating limits algebraically

The Squeeze Theorem – Evaluate trigonometric limits.

• review the squeeze theorem graphically and algebraically
• apply the squeeze theorem to trigonometric limits
• graphing calculator activity
o Use the graphing calculator to graph the functions f, g, and h (based
on the notation of the squeeze theorem).

The Intermediate Value Theorem – Apply the Intermediate Value Theorem to
find at least one solution.

use the intermediate value theorem to find a solution to a given problem
• lesson activity
o ensure that the hypotheses of the intermediate value theorem are
satisfied in a given situation
o apply the intermediate value theorem once the hypotheses are verified
• use the bisection method to estimate f(x)= 0 on an interval
• graphing calculator activity
o Roots and Intervals Using a Graphing Calculator
o Use a graphing calculator to prove that the equation cos x = x2 has at
least one root. Note the interval in which the root lies. Apply the
Bisection Method to find a more accurate interval of length 0.01. Hint:
Use f(x) = cos x – x2.
o Use a graphing calculator to prove that the equation x4 + x3 – 3 = 0
has at least one root. Note the interval in which the root lies. Apply
the Bisection Method to find the root, rounding up to 2 decimal
places.

The Formal Definition of a Limit – Evaluate a limit using the precise
definition of a limit.

• examine the precise definition of a limit graphically
• apply the precise definition of a limit to a function on a continuous interval

## Unit 3 – Derivatives (4-5 Weeks, 11 Lessons)

Definition of the Derivative – Explore the relationship between the derivative
and a tangent line.

• analyze the geometric (or graphic) interpretation of the derivative
• find the derivative of a function at a number “a” using the definition of a derivative
• use the definition of a derivative to find the slope of a tangent line or curve at
a given point
• find the equation of a tangent line at a given point
• graphing calculator activity
o In this secant tool, you will see the graph of the function cosh(x) and
two sliders below the graph. Note that the top slider changes the
point [𝑓(𝑥 + ℎ), (𝑥 + ℎ)] while the bottom slider changes the
point[𝑓(𝑥), 𝑥]. Experiment with the two sliders to see how the slope of
the secant line changes, and then answer the following questions.
▪ Now that you’ve experimented with the sliders, use the
bottom slider to fix the value of x to x = 1. Use the top
slider to change the values of x + h to those given in the
following table. Record the slope of the secant for each case.
▪ What happens to the secant as h approaches 0?
▪ Can you guess the slope of the tangent at x + h = 1

o For the function f(x) = 2×2 + x, use the difference quotient to
find the slope of the secant joining the points at x = 1 and a = 2.
o For the function f(x) = 2×2 + x, complete the table with the values of
the difference quotient to estimate the slope at the point x = 1.
▪ Based on what you have learned in this lesson activity,
what limit formula can be used to find the tangent line of a
function at a specific point?

Instantaneous Rate of Change – Use the derivative to represent an
instantaneous rate of change.

• calculate the rate of change from function
• interpret the rate of change within the context of a problem
• approximate the rate of change from graphs and tables

Graphing the Derivative – Use the derivative as a function.

• explore the relationship between the graph of a function and its derivative
• match the graph of a function with the graph of its derivative function
• understand the various notations for the derivative
• graphing calculator activity
o First play around with the Calculus Grapher tool to explore the
relationship between the graphs of a function and its derivative. You
can create a function graph by dragging the line below and above the
x-axis. As you create a graph, its derivative graph is automatically
created. Here are some things you can do to explore different graphs
in the tool:
▪ Create graphs of different shapes using the buttons on the right.
▪ Combine different shapes and observe the corresponding
▪ Derivatives.

▪ Use the slider to stretch your graphs along the x-axis.
o For every change you make in the graph, watch carefully how the
derivative changes as you distort the function.
o Activate the Grid and Cursor from the panel on the right and
compare the values of the function and its derivative. Once you’ve
familiarized yourself with the tool, go to the Calculus Grapher section
of your Lesson Activities document, where you’ll find a list of
instructions to perform with the Calculus Grapher and questions to

Differentiability and Continuity – Determine when a function can fail to be
differentiable.

• understand the difference between differentiability and continuity
• identify instances when the derivative does not exist (corner, discontinuity,
vertical tangent)

The Formal Definition of the Derivative – Differentiate a function using the
basic rules of differentiation.

• develop the rules for finding derivatives by using the formal definition of the derivative

Differentiation Rules – Learn and apply rules for differentiating functions.

• find the derivative using the constant function, power function and constant multiple rules
• find the derivative of a sum or difference function using the rules of differentiation
• find the derivative using the product rule
• find the derivative using the quotient rule
• find the derivative using the general power rule

Higher Order Derivatives – Explore higher order derivatives.

• find higher-order derivatives (i.e. second, third, etc.)
• identify the curves on a graph that represent the original function as well as
the first, second and third derivatives
• use higher order derivatives to determine rates of change (velocity,
acceleration, jerk)

Differentiation of Trigonometric Functions – Differentiate trigonometric functions.

• develop rules for finding derivatives of trigonometric functions by using the
definition of the derivative
• review limits of trigonometric functions
• find the derivative of a trigonometric function
• explore the use of simplification using the trigonometric identities prior to
finding the derivative of a function involving multiple trigonometric functions

The Chain Rule – Differentiate using the chain rule.

• demonstrate differentiation on composite functions by using the chain rule
• demonstrate the use the power rule combined with the chain rule to find the
derivative of composite power function

Differentiation of Implicit Functions – Perform implicit differentiation.

• describe how and when implicit differentiation is used
• perform implicit differentiation
• use technology to explore the graphs of an equation and its derivative

Differentiation in the Real World – Apply differentiation techniques in rates
of change problems.

• explain the difference between the average rate of change and the
instantaneous rate of change
• setup and calculate problems to find the rate of change

## Unit 4 – Application of Derivatives (4-5 Weeks, 10 Lessons)

Linear Approximations – Use linear approximation and graphs to solve
problems requiring linearization.

• define how “differentials” can be used as independent and dependent variables
• perform linearization of a function
• investigate approximation error and accuracy

Fermat’s Theorem and Maximum and Minimum Values – Identify extrema on
an interval from a graph.

• find maximum and minimum values on open and closed intervals (identify as
absolute or local)
• describe the difference between absolute and local maximum (and minimum) values
• identify extreme values on continuous and non-continuous intervals
• use Fermat’s theorem to find local maximum and minimum values of a function
• identify critical numbers of a function on a closed interval

The Mean Value Theorem – Explore the mean value theorem.

• explore a graphical representation of the mean value theorem
• use Rolle’s theorem to show the proof of the mean value theorem
• use the mean value theorem to find a number “c” that satisfies the conclusion
• understand the mean value theorem from a geometrical perspective
• explore the three hypotheses of Rolle’s theorem
• use Rolle’s theorem to find a number where Rolle’s theorem is satisfied

The First Derivative Test – Examine the first derivative test and use it to aid
in graphing.

• explore the use of the first derivative test graphically
• use the first derivative test to identify the interval on which a function is
increasing and decreasing
• apply the first derivative test to identify local maximum or local minimum
values at a critical number “c”
• graph a function while using the results of the first derivative test

The Second Derivative Test – Examine the second derivative test and use it
to aid in graphing.

• explore the use of the second derivative test graphically
• use the second derivative test to identify intervals on which a function is
concave upward or downward
• use the second derivative test to identify any points of inflection
• use the second derivative test to identify local maximum and minimum values
• graph a function while using the results of the second derivative test

Limits at Infinity and Asymptote – Identify limits at infinity.

• determine limits at infinity
• determine horizontal asymptotes of a graph or function (if they exist)
• use the precise definition of a limit to find the corresponding number N
• graphing calculator activity
o Using a graphing calculator, find the vertical asymptote for the
o Using a graphing calculator, find the vertical asymptote for the

Curve Sketching – Apply the techniques from limits and derivatives to
curve sketching.

• demonstrate how the use function basics as well as first and second
derivative tests and asymptotes to aid in sketching a curve
• demonstrate how calculators or technology can aid in curve sketching

Comparing Related Rates – Calculate related rates problems.

• use the derivative to find the solution to problems that involve the rate of
change of one quantity in terms of the rate of change of another quantity

Maximum and Minimum Applications – Perform differentiation in order to
solve applied problems.

• In this lesson, students will:
o solve applied maximum and minimum problems in the areas of physics,
o solve optimization problems using provided equations or verbal
descriptions of situations

Newton’s Method – Explore Newton’s method graphically and algebraically.

• explore Newton’s method graphically
• perform the necessary calculations involved in Newton’s method to find the
approximate value

# Semester B

## Unit 1 – The Definite Integral (4-5 Weeks, 10 Lessons)

The Antiderivative – Explore the concept of the antiderivative by using the
derivative.

• explore the definition of the antiderivative
• understand the use of “C” when finding the antiderivative
• create the graph of an antiderivative using a given function
• find the antiderivative in its general form and with given specific conditions
• graphing calculator activity
o Assume a function, F(x) = sin x + C, is an antiderivative of a function
f(x). Using a graphing calculator, substitute different values for C and
plot each function. Do you notice any similarity in the graphs of these

Area Under a Curve – Explore the area problem graphically and by using
summation notation.

• estimate the area below a curve using by splitting the region into rectangles
(2, 4, 8, etc.)
• estimate distance traveled by using the velocities at various times
• state the differences in estimating the area below a curve using the left
endpoint, right endpoint and midpoint for curves that are increasing and decreasing
• find the area below a curve by splitting the region into “n” rectangles while
using the left and right endpoints as “n” approaches infinity
• Area Under the Curve Activity
o interpret and make use of sigma notation and limit notation to solve for
the area under a curve
o make use of smaller intervals of estimation to find better estimations for
the area under a curve
o make use of limits to express the exact area under a curve as the
Reimann sum with the number of subdivisions tending toward infinity

The Definite Integral – Examine the definition of the definite integral.

• perform integration by using the definition of the definite integral
• understand how formulas for sums are used in finding the definite integral
• compare area found using sum to definition of definite integral
• explore how the area is affected by curves that are above the x-axis, below
the x-axis, and both above and below the x-axis

Properties of the Definite Integral – Demonstrate the use of the properties of
the definite integral.

• equate switching endpoints of a definite integral with the negative of a definite integral
• show that the area from a to a on a definite integral is zero
• find the value of definite integrals involving a constant, sum and difference

Comparison Properties of the Definite Integral – Explore comparison
properties of the definite integral.

• review the comparison properties of the definite integral graphically
• use the comparison properties to insure a plausible answer for a definite integral
• lesson activity
o plot the graphs of two given functions
o compare the functions and write down any observations in sentence form

The Fundamental Theorem of Calculus – Explore the fundamental theorem
of calculus graphically and by using the definite integral.

• explain the fundamental theorem of calculus graphically
• examine the proof of the fundamental theorem of calculus (parts 1 and 2)
• use the fundamental theorem of calculus to find the value of a definite integral
• graphing calculator activity
o Plot the functions in the table for the given values of a and x.
Observe the graph of each function, its integral, and the
antiderivative of the integral. Write your observations in the table.
The first one has been done for you.
o Explore the geometric aspect of the theorem. Use the same
functions you plotted in the earlier graph. Change the values of a and
b, observe the graphs, and write your observations in the table.

Indefinite Integrals – Use the tables of indefinite integrals to perform
integration.

• explain the difference between a definite and indefinite integral
• find the definite and indefinite integrals of constants, simple functions, and
trigonometric functions using the table of indefinite integrals

Applications for the Definite Integral – Explore the applications of definite
integrals in terms of distance and displacement.

• find the integral of a velocity function to determine displacement
• use the net change theorem to examine displacement vs. distance
• use the net change theorem in other practical application problems

The Substitution Rule – Perform integration by using the substitution rule.

• identify when the strategy of substitution is used to find definite and indefinite integrals
• evaluate integrals using substitution

Symmetric Functions and Integration – Perform integration of symmetric functions.

• examine why a function must be continuous to find a definite integral
• apply the properties of symmetric functions to find during integration

## Unit 2 – Applications of Integration (2-3 Weeks, 6 Lessons)

Area Between Two Curves – Determine the area between two curves.

• graphically estimate the area between two curves
• use “n” rectangles to find the area between two curves
• use the formula for area to find the area between two curves with given boundaries
• use the formula for area to find the area between two intersecting curves
• examine area problems where areas need to be split into more than one integral
• identify and calculate areas where the functions are identified as x = f(y)
• graphing calculator activity.
o Using a graphing calculator, plot the functions y = sin x + 5 and y =
cos x + 3 on a coordinate plane. Shade the region under the curve y
= sin x + 5 over the interval [2, 5] and label it A1. On a second
coordinate plane, again plot the functions y = sin x + 5 and y = cos x
+ 3. Shade the region under the curve y = cos x + 3 over the interval
[2, 5] and label it A2.
▪ What does A1 represent?
▪ What does A2 represent?
▪ What do you get if you subtract A2 from A1?
the area of the region between two curves y = f(x) and y = g(x) over
the interval [a,b] given that 𝑓(𝑐) ≥ 𝑔(𝑥) for all x in [a, b]?

Volume (Disk Method) – Determine the volume of a solid created by a
continuous function that is rotated around a vertical or horizontal line (disk
method).

• explore graphic interpretation of finding the volume of multiple cross-sections
• use technology to examine the shapes of solids created by rotated a curve around an axis
• use the definition of volume to find the volume of a curve that is rotated around the x-axis
• use the definition of volume to find the volume of a curve that is rotated around the y-axis

Volume (Cylindrical Shells Method) – Determine the volume of a solid by
using the method of cylindrical shells.

• identify the graphs of volumes that are easier to calculate using the method of cylindrical shells
• use technology and graphing to visualize the shape created by rotating
• visualize the cylindrical shells that will be used make up the volume of the solid
• perform integration to find volume

Work – Use the definite integral to calculate work.

• examine the relationships between work, force, and distance
• relate the definite integral to work
• calculate work

Mean Value Theorem for Integrals – Use the mean value theorem for
integrals to find the average value of a function.

• illustrate the average value of a function graphically
• compute the average value of a function

## Unit 3 – Inversion Functions (4-5 Weeks, 11 Lessons)

Inverse Functions – Calculate the inverse of a function.

• explore the concept of one-to-one functions
• graph a function and its inverse
• find the inverse function of a one-to-one function

Differentiating Exponential Functions – Differentiate exponential functions.

• review the properties of exponential functions
• explore the graphs of exponential functions to determine where the function is continuous
• find the derivative of the exponential function using the definition of the derivative
• find the derivative of exponential functions that involve the product rule,
quotient rule, and chain rule, etc.

Integrating Exponential Functions – Evaluate integrals that include
exponential functions.

• evaluate definite and indefinite integrals that include the exponential function
(bases a and e)

Differentiating Logarithmic Functions – Differentiate logarithmic functions.

• review the properties of logarithmic functions
• explore the graphs of logarithmic functions to determine where the function is continuous
• find the derivative of logarithmic functions
• find the derivative of logarithmic functions that involve the product rule,
quotient rule, and chain rule, etc.
• use logarithmic differentiation to find the derivative of a complicated function
that includes products, quotients, or powers.

The Logarithm as an Integral – Evaluate integrals that include logarithmic
functions.

• evaluate definite and indefinite integrals that result in logarithmic functions (base e)

Exponential Growth and Decay – Complete problems that involve
exponential growth and decay.

• use the formulas to solve problems using laws of natural growth and decay

L’Hôpital’s Rule – Explore indeterminate forms and l’Hôpital’s rule.

• use l’Hôpital’s rule to evaluate the limit of function

Differentiating Inverse Trigonometric Functions – Differentiate inverse
trigonometric functions.

• explore the graphs of trigonometric functions and their inverses
• perform differentiation on inverse trigonometric functions

Integration Formulas for Inverse Trigonometric Functions – Evaluate
integrals of inverse trigonometric functions.

• evaluate definite and indefinite integrals that require inverse trigonometric
functions

Differentiating Hyperbolic Functions – Differentiate hyperbolic functions.

• explore the graphs of hyperbolic functions
• perform differentiation on hyperbolic functions

Integrating Hyperbolic Functions – Evaluate integrals of hyperbolic functions.

• explore the graphs of exponential functions and hyperbolic functions
• use formulas and substitution to evaluate the integrals of hyperbolic functions

## Unit 4 – Techniques of Integration (3-4 Weeks, 8 Lessons)

Integration by Parts – Evaluate integrals by using integration by parts.

• identify integrals that cannot be completed using previously learned methods
• find an indefinite integral using integration by parts
• find a definite integral using integration by parts

Strategies for Evaluating Complex Trigonometric Functions – Evaluate
integrals that include powers of trigonometric functions.

• use trigonometric identities to form strategies for evaluating trigonometric
functions involving powers
• evaluate indefinite and definite integrals of trigonometric functions

Trigonometric Substitution – Evaluate integrals that can be solved by
trigonometric substitution.

• identify integrals that can be handled using trigonometric substitution
• use the trigonometric functions and the parts of a triangle to determine valid substitutions
• perform integration using trigonometric substitution

Integration by Partial Fractions and Long Division – Evaluate integrals of
rational functions.

• use long division of algebraic statements to rewrite a rational function
• use partial fractions to rewrite rational functions
• evaluate indefinite and definite integrals of rational functions

Using Integration Tables – Evaluate integrals using the tables of integration formulas.

• describe the categories of integrals included in tables
• use tables and substitution to evaluate integrals

Strategies for Integrating Functions – Strategize methods for integration.

• examine all methods for evaluating integrals
• determine strategy and evaluate integrals using any method
• explore technology that can be used to evaluate integrals

Methods for Approximating an Integral – Calculate approximations of integrals.

• explore the midpoint rule graphically
• calculate the area using the midpoint rule
• explore the trapezoidal rule graphically
• calculate the area using the trapezoidal rule
• explore Simpson’s rule graphically
• calculate the area using Simpson’s rule
• compare the results of calculating area using the trapezoidal rule and Simpson’s rule.

Improper Integrals – Evaluate improper integrals.

• evaluate integrals that converge and diverge
• evaluate integrals that are discontinuous

## Unit 5 – Further Applications of Integration (1-2 Weeks, 5 Lessons)

Arc Length – Calculate the arc length of a continuous curve.

• estimate the length of a curve graphically by dividing the curve into 2, 4, 6, etc. segments
• use the arc length formula to calculate the length of a curve using either dy/dx or dx/dy
• apply the use of arc length in practical setting

Surface Area – Calculate the surface area of a revolution.

• use technology and graphing to visualize the surface area of a solid
• compare techniques of finding basic solids to solids created from rotating an
• calculate the surface area of a revolution by using either dy/dx or dx/dy
• apply the use of surface area in a practical setting

Calculus in Physics, Engineering, and Biology – Explore applications to
physics, biology and engineering.

• create and evaluate integrals that represent volumes and surface areas
which may include pressure and center of mass
• create and evaluate integrals that represent blood flow, cardiac output, etc.

Calculus in Economics – Explore applications to economics.

• create and evaluate integrals that represent consumer surplus

Calculus in Probability – Explore applications to probability.

• create and evaluate integrals that represent are below the curve which
represent probabilities
• use the function for the normal distribution to evaluate areas below the curve
(compare to tables)

##### Related Classes ## Dolphin STEM

Inspiring passions in STEM.  Educating a new generation.
2017-03-13T11:54:16+00:00 Dolphin STEM Inspiring passions in STEM.  Educating a new generation. Dolphin STEM Academy 2017-03-13T11:54:16+00:00 Inspiring passions in STEM.  Educating a new generation. https://dolphinstemacademy.com/testimonials/testimonials/ ## We are super proud of him and appreciate your support!

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Erin, mom of a 5th grader This is our 1st year homeschooling and we are so glad that we chose Dolphin STEM Academy ## Thank you for always being here for me!

Big thanks to Mrs Debbie for all the support and encouragement!

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Mirabel from Malaysia Thank you for always being here for me! Big thanks to Mrs Debbie for all the support and encouragement! Mirabel from Malaysia Dolphin STEM Academy 5 2021-06-15T15:36:43+00:00 Mirabel from Malaysia Big thanks to Mrs Debbie for all the support and encouragement! https://dolphinstemacademy.com/testimonials/thank-you-for-always-being-here-for-me/ ## I believe his confidence in himself has grown!

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Keenna H. I believe his confidence in himself has grown! Mrs. Debbie and her team have been awesome through this entire school year and I appreciate your support and patience with me! Keenna H. Dolphin STEM Academy 5 2021-06-18T01:33:46+00:00 Keenna H. Mrs. Debbie and her team have
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