# Integration Teacher Resources

Find Integration lesson plans and worksheets
Showing 1 - 200 of 270 resources
Lesson Planet

#### Introduction to Calculus

For Teachers 11th - Higher Ed Standards
This heady calculus text covers the subjects of differential and integral calculus with rigorous detail, culminating in a chapter of physics and engineering applications. A particular emphasis on classic proof meshes with modern graphs,...
4:26
Lesson Planet

#### Average Value of a Function

For Students 11th - Higher Ed
Viewers learn how to find the average of a variable density with a video made for AP Calculus. The presenter applies the formula to find the average value of a function with an example that provides a density function for a rod. The...
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#### Integral Calculus

For Teachers 10th - Higher Ed
From derivatives to antiderivatives and back again. Building on the second instructional activity of the three-part series covering functions, learners explore the concept of an antiderivative. They connect the concept to the graph of...
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#### Historical Reflections on the Fundamental Theorem Of (Integral) Calculus

For Teachers 11th - 12th
Highlight the process of mathematical discovery for your classes. This lesson examines the development of the Fundamental Theorem of Calculus over time. Understanding the history of the theorem helps create a better understanding of the...
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#### Study Guide for the Advanced Placement Calculus AB Examination

For Teachers 11th - 12th Standards
Is this going to be on the test? A calculus study guide provides an organized list of important topics and a few examples with answers. The topics include elementary functions, limits, differential calculus, and integral calculus,...
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#### The Fundamental Theorems of Calculus

For Teachers 11th - Higher Ed
Are your calculus pupils aware that they are standing on the shoulders of giants? This lesson provides a big picture view of the connection between differential and integral calculus and throws in a bit of history, as well. Note: The...
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#### Math Handbook: Calculus

For Students 10th - 12th
If only there was a reference that listed all the formulas covered in calculus. Luckily, there is. This comprehensive 197-page handbook provides formulas and explanations for all topics in AP Calculus or in a standard college calculus...
20:45
Lesson Planet

#### Integration and the Fundamental Theorem of Calculus | Essence of Calculus, Chapter 8

For Students 11th - Higher Ed
Integrate a great resource into your lesson plans. The eighth video in an 11-part series explains integration and the relationship between derivatives and integrals. The relationship between a car's velocity and its travel distance helps...
7:54
Lesson Planet

#### Properties of the Definite Integral

For Students 11th - Higher Ed
What are the properties that can be associated with the definite integral? The AP Calculus video lists nine properties of the definite integral and provides explanations of why the properties hold for integrals in language that is simple...
7:42
Lesson Planet

#### Calculus, What Is It Good For?

For Students 11th - Higher Ed
Explore the meaning of differentiation and integration. A video lesson describes in detail what differentiating and integrating an equation means using the momentum and kinetic energy formulas. Progressing from a linear to a quadratic...
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#### More Functions, More Features

For Teachers 10th - 12th Standards
Learners tackle a wide range of intimidating topics in this comprehensive unit that spans piecewise functions, absolute value of functions, and inverse functions (among other topics). Investigative group work alternates with more...
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#### Polar Functions

For Teachers 9th - 12th
Explore the concept of polar functions and learn how to determine the length around a function using polar coordinates. High schoolers use their Ti-calculators to determine the length of a function that looks like a five-leaved rose....
1 In 1 Collection
Lesson Planet

#### Derivative and Integral Rules

For Teachers 10th - Higher Ed
Your calculus learners will enjoy this handy reference sheet containing the most used derivative and integral formulas.
Lesson Planet

#### Properties of Definite Integrals

For Students 11th - Higher Ed
Close your eyes and visualize a definite integral. An interactive provides a visualization of a function and definite integral on a closed interval. Pupils move one point of the interval in either the positive or negative direction....
2:54
Lesson Planet

#### Solving Integrals Using Substitution

For Students 11th - Higher Ed
The concept of solving integrals by substitution with trigonometric functions is modeled by a video designed for AP Calculus classes. The narrator works an example problem using substitution and shows that careful substitution makes...
12:38
Lesson Planet

#### What Does Area Have to Do with Slope? | Essence of Calculus, Chapter 9

For Students 11th - Higher Ed
It seems like area and slope wouldn't have any connection. Here is a video that explains how to find the average value of a function over an interval. It connects the area under a curve to the slope of the antiderivative curve.
Lesson Planet

#### Calculus - Early Transcendentals

For Teachers 9th - Higher Ed Standards
This textbook takes the learner from the basic definition of slope through derivatives, integrals, and vector multivariable calculus. Each section is composed primarily of examples, with theoretical introductions and explanations in...
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For Teachers 12th - Higher Ed
Learners explore and create functions. In this calculus lesson, students create functions with the help of the Voyage 200 PLT. This assignment requires technology.
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#### Calculus Test

For Students 11th - Higher Ed
In this calculus activity, learners integrate various functions, find the sum of a series, and solve differential equations. There are 16 questions including multiple choice and free response.
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#### Piecewise Linear Integral

For Teachers 9th - 12th Standards
Young mathematicians solve and graph piecewise functions in this calculus lesson. They integrate linear functions containing restrictions and graph their answers using the TI.
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#### Calculus

For Students 11th - Higher Ed Standards
Everything you wanted to know about calculus—and more! The resource is a complete Calculus textbook with explanations, examples, and practice problems.
9:00
Lesson Planet

#### Natural Log Function-Integration

For Teachers 11th - Higher Ed Standards
One of the strongest tools in integration is u-substitution, particularly when working with natural logs. In this step-by-step video presentation, a pleasant-voiced instructor walks through a number of integration problems. She begins...
Lesson Planet

#### AP Calculus Practice Exam AB Version-Section1-Part 1

For Students 11th - Higher Ed
In this calculus practice exam, students select the best answer to 28 multiple choice questions. Questions cover information from the entire year of calculus. A suggested time limit of 55 minutes is included.
Lesson Planet

#### Evaluate Integrals

For Students 12th - Higher Ed
In this evaluating integrals worksheet, students solve and complete 12 various types of problems. First, they use the trigonometric identity to rewrite each integral. Then, students use integral formulas for sine function. They also...
Lesson Planet

#### Definite Integral

For Teachers 9th - 12th Standards
Learners investigate the intervals represented by a function in this calculus lesson. They decide what interval of the function will be positive, negative or zero. They are then given graphs of functions and asked to analyze it.
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#### Functions

For Students 12th - Higher Ed
In this functions instructional activity, students solve and complete 5 different sets of problems that include determining functions on intervals. First, they define a definite integral and the Fundamental Theorem of Calculus. Then,...
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#### AP Calculus Practice Exam

For Students 11th - 12th
In this practice exam learning exercise, students solve 17 multiple choice problems. Students find derivatives, points of continuity, maximums, minimums, integrals, and area of enclosed regions.
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#### Calculus Grapher

For Teachers 8th - 12th Standards
Learners draw a graph of any function to see its derivative and integral in this simple, yet informative, simulation. Other options include manipulating the scale, the shape of the graph, and the angle.
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#### InteGreat

For Teachers 9th - Higher Ed Standards
Hands-on investigation of Riemann sums becomes possible without intensive arithmetic gymnastics with this interactive lesson plan. Learners manipulate online graphing tools to develop and test theories about right, left, and midpoint...
4:37
Lesson Planet

#### Trigonometric Integrals

For Students 11th - Higher Ed
Present examples of trigonometric integrals with a video that shows an example of determining a trigonometric integral. The AP Calculus example shows that many of the trigonometric integrals require manipulation and using trigonometric...
1:46
Lesson Planet

#### The Second Fundamental Theorem of Calculus

For Students 11th - Higher Ed
How many fundamental theorems are there? The Second Fundamental Theorem of Calculus is also known as the second part of the Fundamental Theorem of Calculus. Here's a video that introduces the theorem as the method to evaluate a definite...
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#### Integration

For Teachers 12th
Twelfth graders investigate integration.  In this Calculus activity, 12th graders explore indefinite and definite integrals.  Students use technology (TI-89) to verify the solutions to problems requiring integration by parts.
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#### Integration

For Teachers 11th - 12th
Students integrate integrals.  In this integrating integrals lesson, students integrate definite and indefinite integrals.  Students use their Ti-89 to find the integrals of cubic, inverse, and trigonometric function graphs on a given...
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#### Proof That One Equals Zero (Using Calculus)

For Students 11th - 12th
For this proofs worksheet, students evaluate the integral of 1 function. Students use uv-substitution to prove 0=1.
Lesson Planet

#### Calculus Volume 1

For Students 11th - Higher Ed
Differentiate between calculus concepts. The beginning of a helpful calculus resource begins with a short review of functions and moves right into limits to define derivatives. The six-chapter resource continues with applications of...
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#### Exploration 3: Introduction to Definite Integrals

For Students 11th - 12th
In this integral worksheet, students estimate the definite integral and find the derivative of functions. They graph functions and determine the range of the velocity. This three-page worksheet contains approximately 40 problems.
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#### 2017 AP® Calculus AB Free-Response Questions

For Students 10th - 12th
What is the answer to this question? Pupils use six questions to practice for the AP Calculus AB test. The review questions are divided in two sections, calculator and non-calculator. Additional resources allow teachers to determine how...
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#### Lesson 33- Integration by Substitution continued

For Students 11th - 12th
Humongous Calculus problem of the day - "Two hikers begin ..." Use the properties of indefinite integers and indefinte integrals to solve this problem.  Six practice problems are included with space to do the work, as well as a...
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#### Calculus Practice Test: Velocity, Functions

For Students 10th - 11th
In this calculus worksheet, students observe graphs and identify the limits of the functions listed in the graph. They determine the definite integrals and derivatives. Students use the trapezoid rule to estimate distance. This...
3:23
Lesson Planet

#### Volumes: Solids That are Not Solid of Revolution

For Students 11th - Higher Ed
Viewers learn how to calculate the volume when the cross section is not a circle by using the same principle as if it was a circle. The video goes through an example of finding the volume of a figure where the cross sections are squares...
5:59
Lesson Planet

#### Volumes by Cylindrical Shells

For Students 11th - Higher Ed
Shells sometimes work better than washers. The AP Calculus video presents an example of finding the volume of a rotational solid by using the shell method. The method uses the integral of the lateral surface area of a cylinder as opposed...
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#### Calculus Concepts

For Teachers 12th - Higher Ed
Students use a graphing calculator to do calculations and edit their calculations using script. In this graphing calculator instructional activity, students compute the length of intervals given in parametric form. Students verify the...
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#### Variational Calculus 2

For Students 12th - Higher Ed
In this methods of applied mathematics worksheet, students solve 6 different types of problems related to variational calculus. First, they determine the first integrals of the Euler-Lagrange equations corresponding to the optimizations...
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#### Variational Calculus 1

For Students 12th - Higher Ed
In this methods of applied mathematics worksheet, students solve 4 different problems that relate to variational calculus. First, they determine a second order differential equation for the function y(x) for which the function given...
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#### Nine Definite Integrals and Fundamental Theorem Of Calculus Problems

For Students 11th - Higher Ed
In this definite integral worksheet, student find the value of five definite integrals.  Student then explain why a definite integral inequality holds true.  Finally, students are given the graph of a functions and asked to determine...
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#### A.P. Calculus Test Covering Limits, Derivatives, and Integrals

For Students 12th - Higher Ed
In this calculus worksheet, students solve ten multiple-choice questions and eleven free-response questions covering limits, derivatives, and integrals.  Students should have covered trig limits prior to this worksheet.
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#### Evaluate Integrals

For Students 12th - Higher Ed
In this evaluate integrals instructional activity, students solve and complete 12 different problems. First, they use the trigonometric identity given to rewrite the integral as follows. Then, students substitute when necessary and use...
Lesson Planet

#### Computing Indefinite Integrals

For Teachers 11th - Higher Ed
Provide your students with a framework for introducing indefinite integrals. It takes into account a learner’s prior knowledge of slope fields and definite integrals and uses the notion of an antiderivative to give them an intuitive...
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#### Integration By Substitution

For Teachers 9th - 12th Standards
Young scholars calculate the solution using integrals in this calculus instructional activity. They use the TI to create a visual of how to compute integrals. Learners also use substitution by integration and all its properties.
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#### Calculating Arc Lengths of Simple Functions-I

For Students 10th - Higher Ed
In this calculating arc lengths instructional activity, students are given 4 simple functions and they use the Pythagorean Theorem and the basic techniques of calculus to find the arc-length formula in order to find the arc-length.
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#### 2016 AP® Calculus AB Free-Response Questions

For Students 10th - 12th
Using the common errors and test taker responses, it's easy for teachers to see where to pinpoint instruction throughout the AP Calculus course. Pupils, using the released items, develop an understanding of how future exams assess the...
10:30
Lesson Planet

#### Integration by Partial Fractions

For Students 11th - Higher Ed
Demonstrate a complex integration by using partial fractions with an AP Calculus resource that works an example of integrating a rational function. The function's denominator has linear factors with one of them repeated.
5:41
Lesson Planet

#### Integration by Parts

For Students 11th - Higher Ed
Class members view how to derive the formula to integrate by parts with a video that uses the Product Rule from the beginning of AP Calculus to derive the rule to integrate products. The resource shows the formula in two different forms,...
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#### Challenge Sheet 2: Evaluate Functions

For Students Higher Ed
In this function activity, students evaluate integrals using the fundamental theorem of calculus. They identify the maxima and minima on intervals. This one-page activity contains seven problems.
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#### Discovering a Fundamental Fact about Integration

For Teachers 11th - 12th
Sketch on coordinate axes to find the exact value of an integral shown. Individuals complete the data grid with the values of integrals returned and use inductive reasoning to guess the formula.
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#### Fundamental Theorem of Calculus

For Students Higher Ed
In this fundamental theorem activity, students give antiderivatives of functions, find the area under a curve, and evaluate continuous functions. This two-page activity contains ten problems.
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#### Calculus Volume 3

For Students 11th - Higher Ed
See calculus to the end. The eBook is the last of a three-volume series covering college-level calculus topics. Scholars begin with parametric equations and polar coordinates before moving to vectors, finally ending with multi-variable...
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#### The Second Fundamental Theorem of Calculus

For Teachers 10th - 12th Standards
Students investigate the fundamental theorem of calculus. For this calculus lesson, students derive the fundamental theorem of calculus. They differentiate between the first and second theorem.
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#### The First Fundamental Theorem of Calculus

For Teachers 9th - 12th Standards
Solve problems using the fundamental theorem. In this calculus lesson, learners solve problems using theorems and proving theorems. They derive the fundamental theorem as they study it.
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#### AP Calculus Exam Prep

For Teachers 11th - 12th Standards
Pupils practice sample questions from the AP Calculus Exam. They solve calculus problems involving bounded regions, integration, derivatives, and limits. They then answer free response and multiple choice questions on the above mentioned...
Lesson Planet

#### Elevator Height as Integral of Velocity

For Teachers 9th - 12th Standards
Pupils investigate integrals and their relationship to velocity in this calculus lesson. They use the idea of a moving elevator to explore vertical motion and use the TI to help them create a visual.
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#### Fundamental Theorem of Calculus

For Teachers 9th - 12th Standards
High schoolers, with the assistance of their TI-84 Plus / TI-83 Plus calculators, explore and assess the connections between an accumulation function, one defined by a definite integral, and the integrand. They summarize that the...
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#### Integration By Parts

For Teachers 9th - 12th Standards
Pupils solve problems using integration by parts in this calculus lesson. Learners apply the product rule and integration by parts. They graph the equation and use the TI to observe the integration process.
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#### Integration of Functions

For Students 10th - 12th
In this calculus learning exercise, students integrate given functions following the integration rules. They simplify their answers with trig. values. There are 8 questions with an answer key.
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#### The Area Function

For Teachers 11th - 12th
Students explore the concept of area under a curve.  In this area under a curve lesson, students find integrals of various functions.  Students use their Ti-Nspire to graph functions and find the area under the curve using the...
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#### An Investigation in Calculus Using the TI-92

For Teachers 11th - 12th
Learners listen to a lecture about defining a function and testing the area on the TI-92 graphing calculator. In this calculus instructional activity, students utilize concepts drawn from their lecture notes, along with screen shots from...
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#### The Fundamental Theorem of Calculus

For Teachers 10th - 12th
Knowledge of definite integrals is fundamental to the understanding of calculus. Instructors of an AP® Calculus course consult a comprehensive guide to the Fundamental Theorem of Calculus. The document provides teaching suggestions and...
2:29
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#### STEMbite: The Calculus of Building Blocks

For Students 9th - 12th
Watch this video with your junior engineers or your higher-level mathematicians. In it, Drew builds half of an arch bridge with building blocks. There happens to be a magnetic board behind the blocks, on which he affixes a magnet for...
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#### 2018 AP® Calculus AB Free-Response Questions

For Students 10th - 12th
Pupils interact with six released questions from the 2018 AP Calculus AB exam. Teachers use the additional materials to determine where the class is in terms of readiness for the exam to help provide additional instruction as needed.
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#### 2017 AP® Calculus BC Free-Response Questions

For Students 10th - 12th
Pupils use the six released free-response questions from an AP Calculus BC exam to prepare for their own exam. The questions cover a range of topics from the BC curriculum, with half of the items coming strictly from AB material and the...
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#### Module 4: Linear and Exponential Functions

For Teachers 8th - 10th Standards
Sequences and series are traditionally thought of as topics for the pre-calculus or calculus class, when learners are figuring out how to develop limits. But this unit uses patterns and slopes of linear functions in unique ways to bring...
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#### Worksheet 28 - Spring 1996

For Students 11th - Higher Ed
In this math learning exercise, students examine the graph in polar coordinates of r = f(θ). Then they sketch a graph of the modified functions indicated.
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#### Worksheet 3

For Students 11th - Higher Ed
In this math learning exercise, students calculate the indefinite integrals. They also find the differential functions. Finally social diffusion is used to describe how information is passed through a sample population.
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#### Worksheet 1

For Teachers Higher Ed
In this math worksheet, students give examples of functions that will satisfy given conditions. Students tell the tabulations for a given function. Students use the definition of a derivative to compute the inverse of a function. they...
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#### Hot Tub

For Teachers 11th - 12th
Students investigate integral calculus.  In this calculus lesson, students explore an application of integrations through a leaking hot tub problem.  The activity emphasizes using the integral of a rate of change to give the accumulated...
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#### Integral Expression

For Students 12th - Higher Ed
For this integral expression worksheet, students solve and complete 4 different parts of a problem. First, they write an integral expression in terms of average. Then, students find and indicate the units of measure in terms of temperature.
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#### The Region Between Two Curves

For Teachers 11th - 12th
Students find the area between two curves.  In this area between two curves lesson, students find the area between the sine and cosine functions.  Students find the volume of the region when it is rotated around the x-axis.  Students...
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#### Riemann Rectangle Errors

For Teachers 9th - 12th Standards
Students calculate the area under a curve. In this calculus lesson, students integrate the function to find the area under the curve. They use the Ti to create a visual of the graph.
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#### 2012 AP® Calculus AB Free-Response Questions

For Students 10th - 12th
What's on the test? Released items provide insight into the AP Calculus test. The six items are split into two groups, two calculator items and four non-calculator items. Resources allow teachers to see what kinds of answers test takers...
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#### 2011 AP® Calculus AB Free-Response Questions

For Students 10th - 12th
So this is the real test? The resource provides the free-response questions from the 2011 AP Calculus AB exam. The six questions are split into two sections, calculator and non-calculator items. Several resources give teachers...
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#### 2014 AP® Calculus BC Free-Response Questions

For Students 10th - 12th
How much of the material is really important? Pupils use the free-response questions from the 2014 AP Calculus BC exam to see how the questions appear. With the supporting material, teachers understand the topics that appear on the test...
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#### 2015 AP® Calculus AB Free-Response Questions

For Students 10th - 12th
How will the topics be assessed? Using the six released items from the Calculus AB test, pupils see how the exam assesses content from the course. With the additional materials covering test taker's responses and scoring keys, teachers...
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#### 2014 AP® Calculus AB Free-Response Questions

For Students 10th - 12th
Give classes a sneak peek at an upcoming test. Scholars work released items from the AP Calculus AB test to prepare for their own exams. The resource consists of six items, two graphing calculator items, and four non-calculator items....
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#### 2015 AP® Calculus BC Free-Response Questions

For Students 10th - 12th
How will the exam ask this question? The free-response questions from the 2015 AP Calculus BC exam demonstrate how questions appear on the test. Using the questions, pupils gain experience answering similar inquiries to what they see on...
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#### Gateway Arc Length

For Teachers 9th - 12th Standards
Learners differentiate and calculate the length of an arc. They define an arc and relate it to the area of a circle and integrate the function to solve the equations. Young scholars use their TI calculators to graph the equation.
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#### 2006 AP® Calculus AB Free-Response Questions

For Students 10th - 12th
Develop more than just a topical approach to the exam. The six free-response questions from 2006 reveal how the AP exam assesses topics. By using the questions, pupils practice topics such as velocity, sketch slope fields, solving...
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#### 2010 AP® Calculus AB Free-Response Questions

For Students 10th - 12th
Is this going to be on the test? The resource contains the AP Calculus AB free-response questions that were on the 2010 test. Pupils use the six questions to prepare for their tests. Teachers use the included response resources to...
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#### 2007 AP® Calculus BC Free-Response Questions

For Students 10th - 12th
Provide a review for the exam so individuals don't tank. AP® provides the free-response questions for teachers to use to help their pupils prepare for the exam. The six BC questions from 2007 include regions bounded by functions, water...
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#### Accumulation Functions

For Teachers 12th
Twelfth graders explore functions defined by a definite integral.  In this Calculus lesson, 12th graders investigate accumulation functions in which a variable is a limit of integration.  The lesson is meant to be an introduction to the...
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#### Sum of Rectangle Areas to Approximate Integrals

For Teachers 11th - 12th
Students explore the concept of approximating integrals.  In this approximating integrals lesson plan, students approximate integrals by finding the area of rectangles.  Students use a graphing calculator to find the left, right, and...
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#### Volume by Cross Sections

For Teachers 11th - 12th
Eleventh and twelfth graders find the volume of figures using cross sections. They use their Ti-Nspire to find the volume of a solid formed by cross sections of a function. Pupils find the integrals of the shapes using cross sections.
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#### Calculus Volume 2

For Students 11th - Higher Ed
Delve into the middle of calculus. The resource is the second eBook of three that covers the complete calculus content. Seven chapters begin with integrals and different methods of integration before moving on to differential equations....
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#### Integration

For Students 12th - Higher Ed
In this calculus learning exercise, students simplify integrals and solve problems that involve related rates and maximum and minimums.  The one page learning exercise contains twenty-three problems.  Solutions are not provided.
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#### Variational Calculus 3

For Students 12th - Higher Ed
In this methods of applied mathematics worksheet, students solve 5 various types of problems that relate to variational calculus equations. First, they determine a second order differential equation for the geodesic. Then, students write...
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#### Functions Defined by Integrals

For Teachers 10th - 12th
Functions defined by integrals need to be carefully integrated into the course. An AP® Calculus curriculum module discusses functions defined by integrals. It explains several reasons why pupils have difficulty with the concept and...
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#### Average Roller Coaster

For Teachers 11th - 12th
Students explore the concept of average value of a function.  In this average value of a function lesson, students use their Ti-Nspire to determine the average value of a quadratic function.  Students take the integrals of velocity...
22:19
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#### Taylor Series | Essence of Calculus, Chapter 10

For Students 11th - Higher Ed
Sometimes close enough is just perfect. The last video in a series of 11 explores the Taylor series and its connection to derivatives. It shows the process for finding Taylor series approximations for y = sin x and for y = e^x.
15:33
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#### Implicit Differentiation, What's Going on Here? | Essence of Calculus, Chapter 6

For Students 11th - Higher Ed
Teach implicit differentiation explicitly. An engaging video explains implicit differentiation to viewers. It describes how small changes in x and y can be thought of as contributing to the derivative dy/dx. This is the sixth video in a...
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#### Exploring Taylor's Integrals

For Teachers 9th - 12th
Students are introduced to the use of Taylor's polynomials to integrate functions that are not easily integrated by hand. They complete a series of calculations that vary from integrating to finding a Taylor polynomial. Students...
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#### Definite Integrals

For Students 11th - 12th
In this calculus worksheet, students calculate the definite integral given different functions. They evaluate the functions for given values. There are 20 questions.
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#### What is the Number "e"?

For Teachers 11th - 12th
Pupils explore natural logarithms.  In this calculus lesson plan, students investigate exponential growth and integral calculus which leads to the natural logarithm function. Pupils solve problems involving exponential growth.
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#### Multivariable Calculus: Implicit Differentiation

For Students Higher Ed
In this implicit differentiation worksheet, students compute the determinant of the Jacobian matrix and solve equations by implicit differentiation. This two-page worksheet contains definitions, examples, and explanations. It contains...
7:38
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#### Volumes by Washers

For Students 11th - Higher Ed
Here's an AP Calculus resource that works an example problem of finding the volume where the cross sections of a solid are washer shaped. In the example the axis of rotation is vertical, creating horizontal washers.
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#### I'm Lovin' It: Finding Area Between Curves

For Teachers 11th - Higher Ed Standards
Students explore the concept of finding the area between two curves. In this finding the area between two curves instructional activity, students model the logos of McDonalds, Nike, and Motorola on grid paper. Students find functions to...
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#### Shortest Path

For Teachers 9th - 12th
High schoolers explore an application of definite integrals. They graph three functions that go through the same three points. Learners use the symbolic capacity of their calculators to find the shortest path through the three points and...
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#### Filling The Urn

For Teachers 9th - 12th Standards
Students calculate the area under a curve. In this calculus instructional activity,students calculate the area by integrating the equation. They calculate the volume of cylindrical shapes created between graphs.
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#### Integral Test

For Teachers 12th
Twelfth graders investigate convergence of an infinite series.  In this Calculus lesson, 12th graders apply the integral test to determine convergence of an infinite series.
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#### Shortest Path

For Teachers 12th
Twelfth graders explore an application of the definite integrals.  In this calculus lesson, 12th graders graph three functions on the same domain and each goes through the same three points.  Students use the symbolic capacity of their...
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#### Area under curve

For Teachers 12th
Twelfth graders investigate an application of definite integrals.  In this Calculus lesson, 12th graders use the symbolic capacity of the TI-89 calculator and the concept of the definite integral to explore the area bounded by a function...
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#### Lab 2: Euler's Method and Riemann's Sums

For Students Higher Ed
In this Euler's Method and Riemann's Sum worksheet, students identify the best situation in which to use Euler's Method and Riemann's Sum. They identify the initial value of a problem and use the accumulation formula to determine the...
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#### Little Red Corvette

For Teachers 11th - 12th
Students investigate the acceleration of a car.  In this acceleration of a car lesson plan, students make a scatter plot of corvette acceleration data.  Students find a model to fit the data by taking the integral of the curve created by...
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#### Basic Calculus II: Fundamental Theorem of Calculus

For Students Higher Ed
In this fundamental theorem worksheet, students find the antiderivative of a function. This two page worksheet contains five multi-step problems.
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#### 2003 AP® Calculus AB Free-Response Questions

For Students 10th - 12th
Take a free look. Released free-response questions from the 2003 AP® Calculus AB exam show how questions cover different topics. Pupils and teachers review the topics of bounded regions, particle movement, rates, relationships between...
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#### 2001 AP® Calculus AB Free-Response Questions

For Students 10th - 12th
Take a closer look at the exam. Released items from the 2001 Calculus AB exam allow teachers and pupils to view how the exam presents questions. Scholars use the questions to prepare for the test and compare their responses to the...
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#### 2009 AP® Calculus BC Free-Response Questions Form B

For Students 10th - 12th
Let them eat cake. The free-response questions from an AP® Calculus BC exam cover several topics that include finding the area under a curve within the context of creating a birthday cake. Calculus concepts include finding rates of...
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#### 2002 AP® Calculus BC Free-Response Questions Form B

For Students 10th - 12th
Though the items may be older, they still provide information. Released free-response items from the 2002 AP® Calculus BC exam provide teachers and pupils with information about how topics appear on the exam. Questions come from the AB...
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#### 1999 AP® Calculus BC Free-Response Questions

For Students 10th - 12th
Get a 20/20 view of the exam. Free-response questions from the 1999 Calculus BC exam provide an insight to the exams. Pupils review the questions to see how topics show up on the test. Topics range from particle motion along a function,...
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#### 2010 AP® Calculus BC Free-Response Questions Form B

For Students 10th - 12th
Keep moving along a curve. Two items in the set of released free-response questions from the 2010 AP® Calculus BC exam involve movement along a graph. One involves particle motion along a polar curve while the other uses a squirrel...
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#### Engineering in a Calculus Classroom

For Teachers 11th - 12th
Learners solve several differential equations. In this calculus lesson, students test their modulus calculation by testing different weights or beam lengths. They share their findings in class.
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#### Solving Differential Equations via the Fundamental Theorem of Calculus

For Teachers 11th - 12th
Students use their TI-89 calculator to compute derivatives and anti-derivatives. In this differential equation lesson plan, students follow detailed directions to complete one table of derivative/anti-derivative values. They compute the...
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#### Integration Crossing

For Students 11th - 12th
In this Calculus worksheet, students use the TI-89 calculator to compute derivatives and anti-derivatives in order to solve a crossword puzzle. The two page worksheet contains twenty-one problems. Answers are not provided.
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#### Trapezoidal and Midpoint Approximations: Area of a Skirt

For Students 11th - Higher Ed
When are trapezoids better than rectangles? Using trapezoids pupils approximate the area of fabric defined by a function. Just like with rectangles, learners realize the more trapezoids the more accurate the approximation. Scholars use...
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#### Approximating Integrals with Riemann Sums

For Teachers 9th - 12th
Students practice calculating and analyzing Riemann sums and illustrate when Riemann sums will over/under-approximate a definite integral. They view how the convergence of Riemann sums as the number of subintervals get larger.
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#### Categorize Twelve Integrals

For Students 11th - Higher Ed
In this integral worksheet, students categorize the given twelve integrals to determine which of the four basic integral functions provided each follows.
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#### Exploration 4: Definite Integrals by Trapezoidal Rule

For Students 10th - 12th
In this integral worksheet, learners geometrically estimate distance in given problems. They use the trapezoidal rule to determine definite integrals. Students explore distance and velocity. This three-page worksheet contains...
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#### The Integration of Exponential Functions

For Students 11th - 12th
For this calculus worksheet, students solve problems using integration. They solve exponential functions using differentiation formulas. There are 12 problems with an answer key.
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#### Integration By Parts

For Teachers 11th - 12th
This lesson plan provides an introduction to integration by parts. It helps learners first recognize derivatives produced by the product rule and then continues with step-by-step instructions on computing these integrals. It also shows...
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#### The Method of Integration by Parts

For Students 11th - 12th
In this calculus worksheet, students perform integration by parts. They solve differential equations as they use integration by part to solve unlike terms. There are 23 problems with an answer key.
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#### The Method of Integration by Partial Fractions

For Students 9th - 10th
In this calculus worksheet, students solve problems using integration by partial fractions. They add fractions to get a common denominator, then take the derivative. There are 20 questions with an answer key.
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#### The Limit Definition of a Definite Integral

For Students 11th - 12th
In this calculus worksheet, students calculate the derivative and integral of different functions. They use interval to mark the beginning and end of their calculations. There are 15 questions with an answer key.
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#### Worksheet 1

For Students 11th - Higher Ed
In this math worksheet, students give an example of a function which is continuous but not differentiable. They also write a definition for differentiable.
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#### What is an Antiderivative? Slope Fields

For Teachers 10th - 12th
Students take the derivative of a function. In this calculus lesson, students define slope fields and what they are used for. They relate slope fields to the real world.
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#### Worksheet 28 - Fall 1995

For Students 11th - Higher Ed
For this math worksheet, learners define a definite integral. Then they draw a picture or sequence of pictures with progressively finer partitions..
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#### Sturm Liouville Theory

For Students 12th - Higher Ed
In this Sturm Liouville Theory worksheet, students solve one multi-step problem that transforms the original differential equation into the Sturm-Liouville form. First, they apply the ideas expressed for each equation given in...
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#### Worksheet 13

For Students 11th - Higher Ed
In this math worksheet, students find the solution for the Euler equation. Then they simplify the two differential for the functions of g and yp.
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#### Worksheet 25

For Students 11th - Higher Ed
In this math learning exercise, students state the Fundamental Theorem of Calculus. Then they evaluate the functions and test them against the theorem.
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#### Calculus Exam

For Students 11th
In For this calculus worksheet, student solve 6 multi-step problems involving volume, intervals, acceleration, and graphing. A graphing calculator will be necessary to complete the worksheet.
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#### Twenty-Eight Multiple Choice AP Calculus BC Exam Preparation Questions

For Students 11th - Higher Ed
In this AP Calculus review worksheet, students solve twenty-eight multiple choice questions in preparation for the AP Calculus BC exam.
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#### Integration By Parts

For Students 11th - Higher Ed
in this math worksheet, students read the example for the product rule. Then they apply the rule individually or in groups to the following integrals.
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#### Twenty Various Multiple Choice and Free-response Indefinite Integrals

For Students 11th - Higher Ed
In this integrals worksheet, students integrate various indefinite integrals in six multiple choice and fourteen  free response problems.  These problems include: trigonometric functions, u substitutions, and Ln.
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#### Integrals

For Students Higher Ed
In this college level calculus worksheet, students evaluate integrals and determine whether it is best to use Green’s Theorem, the Fundamental Theorem of Line Integrals, direct integration, or a combination.  The two page worksheet has...
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#### Integrals Involving Trig. Functions

For Students 12th - Higher Ed
In this calculus worksheet, student evaluate twenty-one integrals.  This worksheet is designed as a test with multiple-choice as well as free response problems.
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#### Quartic Regions

For Teachers 11th - 12th
Students explore the concept of quartic regions.  In this quartic regions lesson, students find the area under a curve using their Ti-89 calculator.  Students find definite integrals of two curves.  Students determine the points of...
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#### Riemann Sums

For Teachers 11th - 12th
Learners explore Riemann Sums.  In this Calculus lesson, students investigate Riemann Sums as an approximation of the value of the definite integral.
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#### AP Calculus Practice Exam

For Students 11th - 12th
In this Calculus worksheet, learners are provided with practice problems for their exam.  Topics covered include derivatives, area bounded by a curve, local maximum, instantaneous rate of change, and the volume of a solid of revolution. ...
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#### Simpson's Rule

For Students Higher Ed
In this Simpson's Rule worksheet, students use the Fundamental Theorem of Calculus to determine the antiderivatives and evaluate integrals. They find the midpoint of given polygons and sketch the graph of the function. This nine-page...
8:48
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#### Volumes by Slicing

For Students 11th - Higher Ed
Find the volume of a rotational solid by integrating the area of a slice. The AP Calculus resource works through an example of finding the volume of a solid created by rotating a function about the x-axis. The presentation breaks the...
8:18
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#### Areas Between Curves

For Students 11th - Higher Ed
There are times that it is necessary to add areas to find the area between the curves. The video for AP Calculus presents how to find the area between curves by subtracting the area under the lower curve from the area under the upper...
16:11
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#### Finding The Area Under a Curve Using Rectangles

For Students 11th - Higher Ed
Here's an AP Calculus video that provides a discussion on determining an approximate for the area under a curve by calculating the sum of areas of rectangles that is smaller than the area and by finding the sum of areas of rectangles...
5:17
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#### Separation of Variables and Initial Value Problem

For Students 11th - Higher Ed
Ap Calculus classes learn how to solve a differential equation by getting like variables together with a video that presents an example of how to solve a differential equation by using the method of separation of variables. The example...
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#### Exploring Functions with Calculus and Dynamic Software

For Teachers 11th - Higher Ed
Calculus can be project-based and inquiry-centered by using dynamic software.
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#### Approxiamte Solutions to Differential Equations-Slope Fields (graphical) and Euler's Method (numeric)

For Teachers 12th
Twelfth graders explore differential equations.  In this calculus instructional activity, 12th graders explore Euler’s Methods of solving differential equations.  Students use the symbolic capacity of the TI-89 to compare Euler’s Method...
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#### MVT for Integrals

For Teachers 10th - 12th
Students identify the meaning of the Mean Value Theorem. In this calculus lesson, students solve problems using the definite integral. They calculate the average value of a function.
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#### Graphing Calculators

For Teachers 12th
Twelfth graders graph functions using paper and pencil. They observe the proper use of graphing technology. Students complete more difficult problems using the graphing calculator. They compare their hand graphs to the calculator...
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#### The Fundamental Theorem of Calculus and Properties of the Definite Integral

For Teachers 11th - Higher Ed
Students discuss the importance of fundamental theorems in math. In this calculus lesson plan, students define the fundamental theorem of calculus and discuss why it is so important they understand it. They work problems to model how...
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#### Functions Defined by Integrals

For Students 9th - 11th
In this calculus worksheet, students solve functions by taking the derivative of the function. There are 3 questions with and answer key.
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#### Fundamental Theorem of Calculus

For Students 11th - 12th
In this calculus worksheet, students observe the graph the function and identify the interval as increasing or decreasing. They identify the critical points and perform integration. There are 4 questions.
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#### Inverse Trigonometric Functions (Review)

For Teachers 11th - Higher Ed
Students discuss the importance of inverse functions and derivatives. In this calculus lesson, students analyze inverse trig functions by solving for missing angles. They graph the functions and discuss integration by parts.
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#### AP Calculus Practice Exam

For Students 12th - Higher Ed
For this calculus worksheet, students calculate the derivative, find the invertible given the derivative and review basic concepts for their AP calculus exam. There are 17 questions.
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#### The Definite integral and Area Under a Curve

For Teachers 11th - 12th
Students define integrals and explore how they relate to the areas under a curve. In this calculus lesson, students solve word problems relating curves and integrals. They sing a song to help them remember the formula for the area under...
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#### Integration Involving Inverse Trig Functions

For Teachers 12th - Higher Ed
Students review the steps for integration and solve equations using integration by parts. In this calculus lesson, students apply their knowledge of integration to inverse trig function. Students review their trig and inverse trig...
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#### Integrals involving Trig Substitution

For Teachers 11th - 12th
Students define integrals and list the trig values. In this calculus lesson, students integrate functions using trig substitution. They solve integrals containing sums and differences of squares.
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#### Integration By Parts

For Teachers 12th
Twelfth graders define integration by parts and solve integrals. In this calculus lesson, 12th graders integrate functions using the product rule. They work example problems with the teacher before doing it on their own.
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#### Integration Quiz

For Students 11th - 12th
In this calculus worksheet, students solve functions using integration. They find the absolute maxima and minima. There are 2 questions.
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#### Average Value of a Function

For Teachers 11th - 12th
Young scholars investigate the average value of a function given different interval values. For this calculus lesson, students explore word problems and apply their knowledge of average value to find the answer. They apply previous...
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#### I Have Who Has Cards - Integration Formulas

For Students 11th - 12th
In this calculus instructional activity, students rewrite a word problem using algebraic symbols and calculate the derivative of a function to find the correct solution. There are 20 matching questions with an answer key.
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#### Antiderivatives: Finding Position and Velocity

For Students 11th - Higher Ed
This AP Calculus resource works through an example that shows how to find the velocity function and the position function given acceleration. It concludes by covering how to use them to solve problems.
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#### Cubic Regions

For Teachers 11th - 12th
Students apply what they have learned about derivatives and integrals. In this algebra lesson, students explore this concept further through real life applications and the TI calculator. This assignment is geared towards advanced...
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#### Riemann Sums

For Teachers 10th - 12th Standards
Students calculate the area under a curve. In this calculus lesson, students use Riemann Sums to find the area under a curve. They apply the boundaries of intervals as they solve for each area.
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#### The Region Between Two Curves

For Teachers 9th - 12th Standards
Learners find the area of a region between two curves. They integrate functions such as sin(x) and cos(x) on a given interval and find the area bounded by the two functions. Pupils use their TI-89 to aid them in finding the bounded...
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#### Listen and Line Up

For Teachers 9th - 12th Standards
High schoolers practice the derivative using an interactive activity. In this calculus lesson they explore the derivative by playing a game. Each student gets a turn at being the caller.
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#### Solids of Revolution - Disks

For Teachers 9th - 12th Standards
Young mathematicians solve the volume using the disk method in this calculus lesson. They revolve a graph around the x-axis and calculate the volume. They use the TI to see the revolving around the x-axis and create a virtual cross...
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#### Substitution Method

For Students 11th - 12th
In this calculus worksheet, students solve the function using the derivatives. They use properties of derivative and trigonometry to help them find the trig values. There are 28 problems.
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#### Applying the Fundamental Theorem of Calculus

For Students Higher Ed
In this fundamental theorem worksheet, students identify the anti-derivative and write equations as an explicit function. This two-page worksheet contains two multi-step problems.
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#### The Inverse Trigonometric Functions

For Students 12th
In this Calculus instructional activity, 12th graders determine the derivative of inverse trigonometric functions and determine integrals.  The one page instructional activity contains seven problems.  Answers and solutions are provided.
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#### Worksheet Two

For Teachers Higher Ed
In this math worksheet, students solve differential equations. Students use similar equations, and analyze ways they differ from one another. Students draw directional fields and sketch solutions. The final three questions are...
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#### Getting Ready to Find Antiderivatives

For Teachers 10th - 12th
Young scholars find antiderivatives.  In this antiderivative instructional activity, students use their calculator to compare given derivatives.  They graph derivatives and explore a family of functions.
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#### NEXT GENERATION SPACECRAFT-Orion

For Teachers 11th - 12th
Learners investigate the volume of a solid.  For this calculus lesson, students use integration to find the volume of a solid generated by a region, the Orion crew module.
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#### Timed Caluculus Exam

For Students 10th
For this Calculus worksheet, 10th graders have 45 minuets to solve 6 multi-step problems involving velocity, integrals, and graphing.
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#### Riemann Sums and the Fundemental Theorem of Calculus

For Teachers 11th - 12th
Students explore the Fundamental Theorem of Calculus. In the Calculus lesson, students investigate indefinite and definite integrals and the relationship between the two, which leads to the discovery of the Fundamental Theorem of Calculus.
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#### Worksheet 17

For Students 11th - Higher Ed
For this math worksheet, students suppose that a function f and its derivative f1 are continuous and that there are constants K, a, and M such that |f(t)|< Keat for t ≥ M.
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#### Let's Integrate All Things Easy

For Teachers 12th
Twelfth graders investigate integration by parts.  In this Calculus lesson, 12th graders explore the use of a mnemonic device designed to tell students which expression to use for integration by parts.
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#### Integrations Applications

For Students 11th - 12th
In this algebra worksheet, students solve functions using integration and the derivative to find the area and volume of a curve. There are 11 questions.
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#### Area Under the Curve with TI InterActive! Document

For Teachers 11th - 12th
Students explore the area under the curve using the TI InterActive! document.  In this calculus lesson, students input various functions and create tables. They answer questions based on their results.
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#### Matrices

For Students 9th - 10th
For this calculus worksheet, students practice solving equations of area and integrals. There are 14 multi-step problems.
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#### Numerical Analysis: Dimensional Vectors

For Students Higher Ed
In this dimensional vector worksheet, students prove dimensional vectors. They write systems of equations, verify solutions, and sketch graphs of the functions. This two-page worksheet contains approximately ten problems.
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#### Even More Calculus

For Students 12th
In this calculus instructional activity, 12th graders differentiate and integrate basic trigonometric functions, calculate rates of change, and integrate by substitution and by parts.  The twenty-two page instructional activity contains...
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#### The Calculus Whiz Who Loved Candy

For Teachers 9th - 12th
Students develop an equation for finding the volume of a commonly known piece of candy (M&M, Hershey's Kiss, Tootsie Roll Pop, Life Saver, etc.) by using calculus.
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#### Integration

For Students Higher Ed
In this college level calculus worksheet, students evaluate the given integrals.  Problems include trigonometric and logarithmic functions.   The two page worksheet contains twenty-six problems.  Answers are not included.
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#### Area Between the Curves-The Game

For Teachers 12th
Twelfth graders explore the area between two curves.  In the Calculus lesson, 12th graders play a game in which paired students pick a function, and then predict the area between the two curves, without knowing the other person’s...
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#### Differential Equations

For Students 11th - Higher Ed
In this differential equations worksheet, students solve and complete 3 sets of various problems. First, they solve each of the differential equations. Then, students use the given information to evaluate the constant of integration....
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#### An Introduction to Solving Differential Equations

For Teachers 12th
Twelfth graders investigate differential equations.  In this calculus lesson, 12th graders are presented with a step-by-step illustrated review of the process used in solving differential equations and an application problem.  Students...
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#### Barehands Integration

For Students 11th - 12th
In this calculus worksheet, students use integration to calculate the solution of functions. They use the correct notation and formula to calculate where the derivative is constant. There are 10 questions.
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#### AP Calculus Practice Exam

For Students 11th - 12th
In this Calculus worksheet, students are provided with questions that are reflective of the content of their exam.  Topics covered include derivatives, volume of a solid of rotation, local maximum and minimum, and integration.  The one...
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#### A.P. Calculus BC Section II, Part A and Part B

For Students 12th - Higher Ed
In this A.P. Calculus BC worksheet. students solve six practice problems.  A graphing calculator is required on the first three questions and not allowed on the next three.  Directions are provided.
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#### AP Calculus Exam

For Students 10th - 11th
In this calculus learning exercise, learners review important rules of derivatives and integrations. They graph their functions using symmetry and coordinate pairs. There are 20 questions.
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#### A.P. Calculus Practice Exam AB Version- Section I- Part B

For Students 11th - Higher Ed
In this A. P. Calculus AB worksheet, students solve seventeen multiple-choice problems with a graphing calculator.  This worksheet is designed as a practice test for the A.P. Exam and should be timed.
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#### Make It Fit

For Teachers 11th - 12th
Of course, calculus can help you build the largest box possible. Scholars play a digital game to gain familiarity with situations where optimization would be appropriate. After gaining a solid conceptual understanding of optimization,...
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#### Worksheet 37

For Students 11th - Higher Ed
In this math worksheet, students sketch the regions that give rise to the integrals. Then rewrite them as double integrals with the integration order reversed.
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#### Multivariable Calculus: Introduction to Path Integrals

For Students Higher Ed
In this path integral worksheet, students compute line integrals under given conditions, define vector fields, and explore gradient fields, as well as a potential function. This two-page worksheet contains five problems.