### We found 534 resources with the concept derivatives

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Lesson Planet

#### AP Calculus AB

Prepare high school mathematicians for the AP Calculus AB exam with a collection of 38 videos that can be used to introduce the topics or as a review. The videos focus on two processes; differentiating a function and integrating a...

Lesson Planet

#### High School Math Handouts

A collection of different resources can augment any math curriculum. Sheets can be printed, posted on the wall, or emailed to students for use in the classroom.

Lesson Planet

#### Calculus AB/BC - Determining Concavity of Functions over Their Domains

New ReviewTime to take a second look at derivatives finding concavity. While watching the video, learners find out the definition of concavity. Individuals see how to determine whether an interval is concave up or concave down using graphs and the...

Lesson Planet

#### Calculus AB/BC - Derivatives of tan(x), cot(x), sec(x), and csc(x)

New ReviewContinue to differentiate the rest of the trigonometric functions. Pupils see the derivatives of the other four trigonometric functions and begin to memorize the rules. Learners see examples that show that the calculus portion of a...

Lesson Planet

#### Calculus AB/BC - The Quotient Rule

New ReviewSome differentiation rules are just better to memorize. The ninth lesson in a unit of 11 presents the quotient rule to pupils for them to memorize. Learners use the quotient rule to determine the derivative of two quotients after seeing...

Lesson Planet

#### Calculus AB/BC - The Product Rule

New ReviewThere's more to derivatives of multiplication than simply the product of the derivatives. A helpful video gives learners the product rule of finding the derivative of the product of two functions. Pupils find the derivatives of functions...

Lesson Planet

#### Calculus AB/BC - Derivatives of cos(x), sin(x), e^x, and ln(x)

New ReviewThe shortcuts are not just for polynomial functions. Pupils learn the derivatives of the two basic trigonometric functions, cosine and sine. The video provides the derivatives for exponential and logarithmic functions. Learners work...

Lesson Planet

#### Calculus AB/BC - Derivative Rules: Constant, Sum, Difference, and Constant Multiple

New ReviewUse derivatives to find equations of lines. Pupils learn more rules to use as shortcuts to find derivatives. Using the rules, they find the equations of the tangent line and the normal line at a given point. Scholars then apply the new...

Lesson Planet

#### Calculus AB/BC - The Power Rule

New ReviewThere has to be a quicker way. Pupils learn the power rule that allows for a shortcut to take the derivative of terms to a power. Using the newly found rule, scholars find the slope of the tangent line to a function at a given x value....

Lesson Planet

#### Calculus AB/BC - Connecting Differentiability and Continuity

New ReviewDespite what you thought, you can differentiate between continuity and differentiability. Using a short lesson, pupils learn how differentiability and continuity relate to each other. It provides three descriptions of when a function is...

Lesson Planet

#### Calculus AB/BC - Estimating Derivatives of a Function at a Point

New ReviewIt seems like it should be about three. Pupils learn to get an estimate of the derivative of a function at a point by using the derivative functionality of their graphing calculators. They take that information to determine the equation...

Lesson Planet

#### Calculus AB/BC - Defining the Derivative of a Function and Using Derivative Notation

New ReviewPupils learn how to find the derivative of a function by applying the definition using limits. Learners understand that the derivative provides the slope of the tangent line and use that information to find the equation of the tangent...

Lesson Planet

#### 2009 AP® Calculus BC Free-Response Questions

Dive into the exam review. Pupils use questions, including one about diving into a pool, to prepare for the AP® Calculus BC exam. Items include contextual and non-contextual questions with contextual examples requiring a calculator....

Lesson Planet

#### 2007 AP® Calculus BC Free-Response Questions

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...

Lesson Planet

#### 2009 AP® Calculus AB Free-Response Questions

What was one of the big takeaways from 2009? By reviewing the released AP® Calculus free-response questions, pupils and teachers realize that performing mathematics is not all there is to doing well on the exam. Several items require...

Lesson Planet

#### 2007 AP® Calculus AB Free-Response Questions

Grow your knowledge of Calculus AB. Items released from the 2007 AP® Calculus AB exam provide information to teachers about the importance of particular topics. Pupils use the questions to gain an understanding of ways the exam assesses...

Lesson Planet

#### 2006 AP® Calculus AB Free-Response Questions

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...

Lesson Planet

#### 2015 AP® Calculus BC Free-Response Questions

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...

Lesson Planet

#### 2013 AP® Calculus BC Free-Response Questions

Apply the skills to problems to ace the AP test. Free-response questions require scholars to take calculus skills and apply them to situations. Pupils use the questions to see how those situations appear on an upcoming exam. Teacher use...

Lesson Planet

#### 2016 AP® Calculus BC Free-Response Questions

The split is 50-50. Released free-response questions from the 2016 AP Calculus BC show that half the questions are from the AB portion of the course and the others are from the BC portion. Teachers and pupils use the items to prepare and...

Lesson Planet

#### 2018 AP® Calculus BC Free-Response Questions

Don't just rely on the textbook. Released items from the AP Calculus BC provide pupils and teachers insight on how topics appear on the exam. The six items are divided into calculator and non-calculator sets with half of them also...

Lesson Planet

#### 2017 AP® Calculus AB Free-Response Questions

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...

Lesson Planet

#### 2016 AP® Calculus AB Free-Response Questions

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...

Lesson Planet

#### 2015 AP® Calculus AB Free-Response Questions

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...

Lesson Planet

#### 2014 AP® Calculus AB Free-Response Questions

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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#### 2013 AP® Calculus AB Free-Response Questions

What is the important stuff? The free-response questions from the 2013 Calculus AB exam highlight the topics in calculus assessed that year. Items provide pupils the opportunity to experience questions similar to what they will see on...

Lesson Planet

#### 2012 AP® Calculus AB Free-Response Questions

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...

Lesson Planet

#### 2011 AP® Calculus AB Free-Response Questions

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...

Lesson Planet

#### 2014 AP® Calculus BC Free-Response Questions

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...

Lesson Planet

#### 2017 AP® Calculus BC Free-Response Questions

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...

Lesson Planet

#### 2018 AP® Calculus AB Free-Response Questions

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.

Lesson Planet

#### 2010 AP® Calculus AB Free-Response Questions

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...

Lesson Planet

#### Derivative and Integral Rules

Your calculus learners will enjoy this handy reference sheet containing the most used derivative and integral formulas.

Lesson Planet

#### Introduction to Rate-of-Change Problems

Sal solves an example for finding the rate of change of the height of water in a cone at a specific point when it is being filled at a given rate. In this video, Sal reviews the volume of a cone and the chain rule and then, uses these to...

Lesson Planet

#### Maxima Minima Slope Intuition

Before solving any problems, Sal gives an overview of what happens to the slope, first derivative, and second derivative at local maxima and minima to give the viewer a more intuitive feel for these types of problems.

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#### Optimization Example 4

Sal does another optimization example, this time, minimizing the total cost of an open rectangular box. In this problem, the volume can be defined in terms of a single variable and given a cost model, he builds a cost equation. He finds...

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#### Graphing with Calculus

In a problem where you are not given the original function, but rather only three known points on the graph and a few additional pieces of information about when the first and second derivatives are positive or negative, Sal shows how...

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#### Optimization with Calculus 3

Using what one learned about finding the minimum and maximum of functions, the optimization problem to find two numbers whose product is -16 and whose sum of squares is a minimum is solved. Sal starts by writing the equations and...

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#### Proof: d/dx(x^n)

Using the binomial theorem and definition of a limit, Sal shows a proof that the derivative of xn equals nxn-1.

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#### Proof: d/dx(sqrt(x))

Using the definition of a limit, Sal proves the derivative of �x or x1/2 is equal to _ x-1/2.

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#### Proof: d/dx(ln x)=1/x

Using the definition of a limit, various properties of logarithms, and a definition of e, Sal shows the proof of derivative of ln x = 1/x. Note: The video titled "Proof of Derivatives of Ln(x) and e^x,Ó has a clearer explanation of this...

Lesson Planet

#### Proof: d/dx(e^x) = e6x

Using the derivative of ln x, the chain rule, and the definition of a limit, Sal shows a proof that derivative of ex = ex. Note: The video titled "Proof of Derivatives of Ln(x) and e^x,Ó has a clearer explanation of this proof.

Lesson Planet

#### Calculus: Derivatives 3

This video covers the differential notation dy/dx and generalizes the rule for finding the derivative of any polynomial. It also extends the notion of the derivatives covered in the Khan Academy videos, "Calculus Derivatives 2Ó and...

Lesson Planet

#### Calculus: Derivatives 2

Sal continues where he left off with the last video, "Derivatives 1,Ó by looking at the equation y = x^2 and examining the slope of the secant line at a specific point, and again defining the limit as x approaches zero to get the slope...

Lesson Planet

#### Calculus: Derivatives 2.5 (new HD version)

By defining the formal definition of a derivative, f�(x), Sal is able to find the general form of the derivate function for the example f(x) = x2. He continues to stress the importance of an intuitive understanding of derivative functions.

Lesson Planet

#### Calculus: Derivatives 2 (new HD version)

Sal continues where he left off with the last video, "Derivatives 1,Ó by looking at the equation y = x2 and examining the slope of the secant line at a specific point. He continues with defining the limit as x approaches zero to get the...

Lesson Planet

#### Calculus: Derivatives 1

Sal defines the term derivative by taking the listener on a well-organized tour of slope. First, he reviews the concept of slope of a line from algebra, then extends this idea to look at the slope of the curve by first examining a secant...

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#### Calculus: Derivatives 1 (new HD version)

Lesson Planet

#### Calculus: Derivative of x^(x^x)

Sal starts with an example of finding dy/dx of y = x2 and builds to showing the solution to the more complicated implicit differentiation problem of finding the derivative of y in terms of x of y = x ^ x ^ x .

Lesson Planet

#### Proofs of Derivatives of Ln(x) and e6x

In this video, Sal takes on the challenge of proving both the derivative of ln x = 1/x and of ex = ex, showing that no circular logic is used in the proof. It contains a clearer version of both proofs shown in the videos titled, _Proof...

Lesson Planet

#### Proof: d/dx(e^x) = e6x

Using the derivative of ln x, the chain rule, and the definition of a limit, Sal shows a proof that derivative of ex = ex. Note: The video titled ï¿½Proof of Derivatives of Ln(x) and e^x,ï¿½ has a clearer explanation of this proof.

Lesson Planet

#### Product Rule

Sal defines the product rule and then shows two examples of how it is used. He then shows an example of finding the derivative by using both the chain rule and product rule together.

Lesson Planet

#### Exploring Derivatives Using Lists

Learners explore lists and determine the derivative for that list. They use regressions to discover equations.

Lesson Planet

#### Differentiation Quiz - Form D

In this differentiation quiz worksheet, students solve 10 short answer questions. Students take the derivatives of ten functions using differentiation rules and uv substitution.

Lesson Planet

#### Calculus 3.6 - The Chain Rule

Greg Kelly puts together another great slide presentation to demonstrate ways to combine derivative rules to evaluate more complicated functions. This pattern is called the chain rule. He shows step by step ways to solve these...

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#### Calculus AB/BC - The Fundamental Theorem of Calculus and Accumulation Functions

New ReviewConnect integrals with differentiation. The beginning of the video introduces the definition of an accumulation function. Scholars learn about the relationship between integrals and derivatives. Using what is commonly known as Part Two...

Lesson Planet

#### Calculus AB/BC Unit 4 Review: Analytical Applications of Differentiation

New ReviewIt's nice to know that there are actual real-world uses of derivatives. Scholars review their knowledge of the applications of derivatives from the unit. They work on a set of practice problems to test their understanding.

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#### Calculus AB/BC - L'Hopital's Rule

New ReviewDirect substitution fails. Now what? The seventh of eight videos in Unit 4 - Contextual Applications looks at L'Hopital's rule. Scholars learn to take the derivative of both the numerator and denominator of a function when substitution...

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#### Calculus AB/BC - Approximating a Function's Value Using Local Linearity

New ReviewNo need to be exact every time. Pupils watch an engaging video on approximating function values. They see how to write the equation of a tangent line, use it to calculate approximations, and then determine whether it is an overestimate...

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#### Calculus AB/BC - Solving Related Rates Problems

New ReviewA solution to a lack of resources on related rates awaits. Viewers of an educational video learn how to solve related rates problems. They build on their prior experience of related rates to write equations for contextual situations,...

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#### Calculus AB/BC - Introduction to Related Rates

New ReviewHere's a video for those who can relate to being confused about related rates. The fourth of eight lessons in Unit 4 - Contextual Applications looks at related rates. Individuals learn to write equations that show a relationship between...

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#### Calculus AB/BC - Rates of Change in Applied Contexts Other Than Motion

New ReviewStay a while—and learn about rates of change. Scholars learn about rates of change not related to motion by watching an informative video. They see how to interpret derivatives in different contexts, paying particular attention to...

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#### Calculus AB/BC - Straight-Line Motion: Connecting Position, Velocity, and Acceleration

New ReviewMove the video on motion to the front of the line. The second of eight lessons in Unit 4 - Contextual Applications looks at straight-line motion. Scholars learn how position, velocity, and acceleration are related by derivatives before...

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#### Calculus AB/BC - Interpreting the Derivative in Context

New ReviewPut math in a real-life context to increase connections to the concept. Pupils view a short video that first reviews the connection between derivatives and rates of change. They learn how to interpret the derivative in contextual...

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#### Calculus AB/BC - Calculating Higher-Order Derivatives

New ReviewDerivatives are so fun that sometimes one wants to take the derivative multiple times. The sixth of seven installments in Unit 3 - Differentiation focuses on finding higher-order derivatives. After learning about the notation, scholars...

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#### Calculus AB/BC - Selecting Procedures for Derivatives

New ReviewDon't get lost trying to memorize all those derivative rules. Scholars review the different methods at their disposal to determine derivatives from Units 2 and 3. They fill out a graphic organizer and then complete a set of practice...

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#### Calculus AB/BC Mid-Unit 6 Review: Integration and Accumulation of Change

New ReviewAccumulate the content so far. Learners review the content from the first five lessons dealing with accumulation functions. The eight problems go over finding maximums, minimums, concavity, points of inflection, and intervals of increase...

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#### Calculus AB/BC - Interpreting the Behavior of Accumulation Functions Involving Area

New ReviewAccumulation functions should behave like other functions. Using content from Unit 5, pupils analyze the first and second derivative to determine the behavior of an accumulation function. Learners find out how to calculate the relative...

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#### Calculus AB/BC Unit 5 Review: Analytical Applications of Differentiation

New ReviewTime to put it all together in one sheet—well five. Pupils review all the topics presented in the resources for Analytical Applications of Differentiation. The five pages summarize the concepts with explanations and examples.

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#### Calculus AB/BC - Exploring Behaviors of Implicit Relations

New ReviewPut everything together implicitly. Pupils use their knowledge of implicit differentiation to determine whether a curve is increasing or decreasing or its concavity. Given a differential equation, scholars determine whether a specific...

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#### Calculus AB/BC - Solving Optimization Problems

New ReviewMove beyond setting up equations. Pupils continue with optimization problems and use their knowledge of derivatives to find solutions. They watch as the video uses equations from previous examples and find solutions to answer specific...

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#### Calculus AB/BC - Introduction to Optimization Problems

New ReviewLet's get the equation set. Pupils see the importance of setting up an one-variable equation in the first of two lessons on optimization problems. The presenter works four different types of problems before individuals use the included...

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#### Calculus AB/BC - Sketching Graphs of Functions and Their Derivatives

New ReviewFind deeper meaning in graphs. Pupils use the knowledge gained from the previous sections in the unit to sketch graphs of a function's derivative. Learners also see how to sketch a graph of a function given the graph of its derivatives....

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#### Calculus AB/BC Mid-Unit 5 Review: Analytical Applications of Differentiation

New ReviewIt's time to put analytical applications to work. Learners use the eight questions of the mid-Unit 5 review to refresh their memories of the key points. Pupils apply what they have learned to use the Mean Value Theorem, find extrema...

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#### Calculus AB/BC - Using the First Derivative Test to Determine Relative Local Extrema

New ReviewCritically apply intervals of increasing and decreasing. Pupils watch a video to see how finding a critical point and finding intervals of increasing and decreasing work together to identify local extrema. Scholars use the first...

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#### Calculus AB/BC Unit 3 Review: Differentiation

New ReviewIt's time to take stock of the class's knowledge of derivatives. Individuals review how to take the derivatives of composite, implicit, and inverse functions. They complete some problems involving the Chain Rule, implicit...

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#### Calculus AB/BC - Differentiating Inverse Trigonometric Functions

New ReviewDifferentiating inverse trigonometric functions is quite different from applying the skill inverse functions. Scholars learn how to differentiate inverse trigonometric functions in the fourth of seven video installments in Unit 3 -...

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#### Calculus AB/BC - Differentiating Inverse Functions

New ReviewReverse any loss of knowledge on inverse functions. Individuals watch a video to learn about differentiating inverse functions. They memorize the rule, then see how to apply it in finding derivatives and determining equations of tangent...

Lesson Planet

#### Calculus AB/BC - The Chain Rule

New ReviewUnshackle oneself from the burden of not understanding derivatives. The first of seven lessons in Unit 3 - Differentiation looks at the Chain Rule. By watching a video, scholars learn when to apply the rule and how to use it to determine...

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#### Calculus AB/BC - Using the Candidates Test to Determine Absolute (Global) Extrema

New ReviewAbsolutely! Find the global maximum and minimums. Pupils watch a video showing how to find the absolute maximums and minimums of a function on an interval by testing critical points. The learners see how to determine whether a critical...

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#### Calculus AB/BC - Determining Intervals on Which a Function is Increasing or Decreasing

New ReviewGoing up? Wait, it might be going down! Learners watch a video to see how to use the derivative and critical points to find where a function is either increasing or decreasing. Individuals use the rate of change to solve real-world...

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#### Calculus AB/BC - Using the Mean Value Theorem

New ReviewAt some point the rate of change must be the average. Pupils first see the definition of the Mean Value Theorem and hear an explanation of it in simple terms. The helpful video then compares the Mean Value Theorem with the Intermediate...

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#### Calculus AB/BC - Connecting a Function, Its First Derivative, and Its Second Derivative

New ReviewUncover what the first and second derivatives reveal about the function. Pupils move beyond sketching graphs to determine what derivatives tell about the function itself. The video reminds learners about particle motion and how to...

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#### Calculus AB/BC - Using the Second Derivative Test to Determine Extrema

New ReviewTake a second look at extrema of functions. Pupils learn how to use the second derivative to determine whether a critical point is a maximum or a minimum. Learners use concavity to identify critical points as either a max or a min and...

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#### Calculus AB/BC - Extreme Value Theorem, Global Versus Local Extrema, and Critical Points

New ReviewIt is critical to find all the extreme points. Pupils analyze graphs in the presentation to determine whether there are points that are either maximums and minimums. Using derivatives, learners find out how to find critical points from...

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#### Calculus AB/BC - Unit 2 Review: Differentiation

New ReviewTie up everything in one differentiable bow. Pupils work through 20 problems to review the concepts from the unit. The problems range from finding the average rate of change to finding the derivatives of various functions that require...

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#### Integrals - Calculus

New ReviewIntroduce the opposite of differentiation. Pupils learn the basic definition of integration and how to integrate a polynomial. The video shows learners how the integral is the opposite of the derivative. Individuals see how to confirm...

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#### Fundamental Theorem of Calculus Part 2

New ReviewUse the Fundamental Theorem, not just look at it. Learners watch as the video walks viewers through finding the definite integral of a trigonometric function. The presenter provides helpful suggestions as she evaluates the example...

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#### Part 1 of the Dreaded Fundamental Theorem of Calculus!

New ReviewUse the Fundamental Theorem of Calculus to find a derivative. With an example, pupils observe how being to derive a formula to help determine the derivative of any given integral. The video derives the formula using the first part of the...

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#### Dimensions That Maximize the Area of the Rectangle

Opt to learn about optimization. Young mathematicians view an educational video that posits the question: For a given perimeter, what are the dimensions that maximum the area of a rectangle? With the use of optimization skills, they...

Lesson Planet

#### Maxima and Minima On a Closed Interval

Close the chapter on extrema on a closed interval. Watching an informative video provides scholars with an opportunity to learn about extrema. Specifically, they see how to find the global maxima and minima on a closed interval by first...

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#### First Derivative Test to Find the Max and Min

It only takes a minimum amount of effort to understand optimization. Pupils watch an illuminating video to learn about applications of the first derivative test. They see how to find critical points and then check to see whether each...

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#### Inflection Points and Concavity

Confused about concavity? Here's the perfect video. Young mathematicians watch a video that explains the process of using the second derivative to determine inflection points and the concavity of a function. An example reinforces the...

Lesson Planet

#### Where Is the Function Increasing and Decreasing?!

Increase understanding of increasing and decreasing behavior. Viewers of an instructional video learn how to identify intervals where a function is increasing and decreasing using critical points and the first derivative test. They watch...

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#### Estimate the Root with Linear Approximation

Root for those learning about roots. An installment of an extensive playlist covering derivatives focuses on using linear approximations to estimate roots. The example problem shows how to find the fourth root of 15 without a calculator.

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#### 2011 AP® Calculus BC Free-Response Questions Form B

Take a second look. Form B of the AP® Calculus BC free-response questions allow pupils and teachers to see another set of items. The six questions expand the set of items from the 2011 free-response questions. Form B items include a...

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#### 2010 AP® Calculus BC Free-Response Questions Form B

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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#### 2009 AP® Calculus BC Free-Response Questions Form B

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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#### 2008 AP® Calculus BC Free-Response Questions Form B

This is integral to the test. Pupils realize that every question in the released free-response items from the 2008 AP® Calculus exam Form B requires integration. By reviewing the additional resources, scholars learn how important it is...

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#### 2007 AP® Calculus BC Free-Response Questions Form B

There is just a single real-world problem. Released free-response items from the 2007 AP® Calculus BC Form B contains only one real-world question. The question involves rate of change of wind chill scenario. The mathematical problems...

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#### 2005 AP® Calculus BC Free-Response Questions Form B

Let's put it in context. Pupils use the AP® Calculus BC free-response questions to practice for the exam. Five of the six released items deal with mathematical problems, with one using a real-world situation. Learners use differentiation...

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#### 2004 AP® Calculus BC Free-Response Questions Form B

Take another look at the exam. A set of questions is a different form for the 2004 AP® Calculus BC free-response question section. The questions are evenly split into calculator and non-calculator sections. Pupils use their knowledge of...

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#### 2012 AP® Calculus BC Free-Response Questions

Get better prepared. Individuals use the six released AP® Calculus BC free-response questions to practice for the actual exam. The questions contain two calculator items and four non-calculator items. Topics include particle movement,...

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#### 2011 AP® Calculus BC Free-Response Questions

Does the exam look pupils expected? Released free-response questions from the AP® Calculus BC exam allow teachers and pupils to see and practice with actual exam questions. Three of the questions come from the AB portion of the course...

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#### 2010 AP® Calculus BC Free-Response Questions

What do a zoo, snow, and a moving particle have in common? The six free-response items from the 2010 AP® Calculus BC exam allow teachers and pupils see how items may show up in context. Three of the questions are from only the BC portion...

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#### 2011 AP® Calculus AB Free-Response Questions Form B

Half are real-world. The 2011 AP® Calculus AB free-response questions let pupils and teachers see how content appears on the exam. Half of the questions contain real-world context, and four items do not allow calculators. Real-world...

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#### 2010 AP® Calculus AB Free-Response Questions Form B

Most say that practice makes perfect. Scholars use the released free-response questions to practice for the AP® Calculus AB exam. Each question comes with several parts and either do or do not allow calculators. Items include topics such...

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#### 2008 AP® Calculus AB Free-Response Questions Form B

How would those items score? The resource provides pupils an opportunity to answer actual free-response questions and determine their scores. Items from the AP® Calculus AB exam from the 2008 administration demonstrate how topics appear...

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#### 2007 AP® Calculus AB Free-Response Questions Form B

What do the questions look like? Six free-response questions provide pupils with knowledge of how calculus topics appear on the AP® Calculus exam. The questions involve calculator and non-calculator problems by providing mathematical and...

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#### 2005 AP® Calculus AB Free-Response Questions Form B

Grab a peak inside the test. The 2005 free-response questions from the AP® Calculus exam allow learners to see how topics appear on the tests. Topics cover differential equations, bounded areas, particle motion, and slopes of tangent lines.

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#### 2004 AP® Calculus AB Free-Response Questions Form B

The test is not all about skills. The six free-response questions contain two real-world application problems. Those items deal with populations and velocities. The other four questions ask pupils to show their knowledge of calculus...

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#### 2009 AP® Calculus AB Free-Response Questions Form B

Get a free peek into a calculus exam. Released free-response items from the AP® Calculus AB exam provide teachers and pupils examples of how topics may appear. The six questions cover topics such as growth rates, volumes of rotations,...

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#### 2006 AP® Calculus AB Free-Response Questions Form B

An exam divides the six free-response questions into calculator use and no calculator use items. The non-calculator items are conceptual in nature and test whether pupils understand the calculus concepts. Scholars explain their answers...

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#### 2003 AP® Calculus AB Free-Response Questions Form B

Get a better grasp of the intended content. The six free-response questions provide insight into the purpose of the AP® Calculus content. Two questions have real-world context while the remaining are void of context. The items range from...

Lesson Planet

#### Calculus Volume 3

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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#### Calculus Volume 1

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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#### Fundamental Theorem of Calculus

Does it make more sense to teach the Second Fundamental Theorem of Calculus before the First Fundamental Theorem of Calculus? An AP® Calculus curriculum module explores the interesting question. A set of five worksheets first looks at...

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#### Beginning the Year with Local Linearity

Local linearity isn't the first thing that comes to mind to start off an AP® Calculus course. A scholarly article discusses one possible beginning to the AP® Calculus course: investigating and introducing derivatives through activities...

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#### Approximation

Sometimes close enough is appropriate. A curriculum document for AP® Calculus examines the importance of providing approximation questions throughout the course. It looks at approximating derivative values, approximating definite...

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#### Reasoning from Tabular Data

Don't table the resource—use it now. An AP® Calculus curriculum module encourages the use of tabular data throughout the course. It provides some example topics, such as rate of change, net change, and average value of a function, where...

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#### 2008 AP® Calculus BC Free-Response Questions

The AP® exam is not just about the calculations. By reviewing the free-response questions from 2008, teachers and pupils see that several items require justifications and explanations to gain full credit. The chief reader even points out...

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#### 2006 AP® Calculus BC Free-Response Questions

Move along with the review. Teachers and scholars use the six released free-response questions to review for their upcoming AP® exam. Questions from the 2006 exam contain problems dealing with cars, objects moving along a curve, and...

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#### 2005 AP® Calculus BC Free-Response Questions

Differentiate between the exams. Using the Q&A document, teachers compare the free-response questions that BC test-takers see to those that AB test-takers see. The pupils see which questions come from AB content and also those that...

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#### 2004 AP® Calculus BC Free-Response Questions

Given a set of questions, pupils realize the way the Calculus BC free-response exam asks questions. The items include both contextual and non-contextual questions. Both types are split into the calculator and non-calculator sections for...

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#### 2003 AP® Calculus BC Free-Response Questions

There are shaded areas, particles, polar equations, oh my! Teachers use the materials to see how their pupils compare to the AP® Calculus BC exam expectations. Scholars use the released items to practice on several BC topics and become...

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#### 2001 AP® Calculus BC Free-Response Questions

Take an abbreviated look at the exam. Class members review released items to prepare for the BC exam. The released questions are the six free-response questions from the 2001 exam. Three of the items have context, while the other three...

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#### 2000 AP® Calculus BC Free-Response Questions

How are concepts divided? Pupils use the AP® Calculus free-response questions to see how the exam addresses concepts. The concepts are divided into those that use calculators and others that do not. Calculator items require finding...

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#### 1999 AP® Calculus BC Free-Response Questions

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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#### 1998 AP® Calculus BC Free-Response Questions

Provide a test run. Pupils use the questions to practice taking the free-response section of the AP® Calculus BC exam. Teachers see the different ways questions appear on the exam to prepare individuals for whatever comes their way on...

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#### 2008 AP® Calculus AB Free-Response Questions

Get the most out of released items. Teachers use the included materials to learn about test-takers' performance on released free-response questions. Pupils respond to the questions and compare their responses to sample scored responses...

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#### 2005 AP® Calculus AB Free-Response Questions

Develop an awareness of the exam. Pupils and teachers use the six release free-response questions to see how the exam covers topics. The questions are divided among calculator and non-calculator items. Scholars see the importance of...

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#### 2003 AP® Calculus AB Free-Response Questions

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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#### 2002 AP® Calculus AB Free-Response Questions

Peek behind the curtain. Using a helpful resource, teachers and pupils see the way concepts appear on the AP® Calculus exam. Scholars use the questions to prepare for their own exams, gaining quality practice before test day. Questions...

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#### 2000 AP® Calculus AB Free-Response Questions

Gain an upper hand in preparation for the exam. Teachers and pupils use six released questions to gain an understanding of the AP® Calculus exam. By answering the questions, scholars see how different concept of the course show up on the...

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#### 1999 AP® Calculus AB Free-Response Questions

Gain an understanding of the test before taking your own. Scholars develop an understanding of how calculus concepts appear on the AP® exam. The six released questions come from across the curriculum and include particle movement and...

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#### 2004 AP® Calculus AB Free-Response Questions

Develop a deeper understanding of calculus. The released item and support materials allows teachers and pupils to dig deeper into calculus topics. Scholars use the commentary and descriptions of the items to gain a better understanding...

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#### 2001 AP® Calculus AB Free-Response Questions

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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#### 1998 AP® Calculus AB Free-Response Questions

Take a peek into the exam. Released items from the Calculus AB exam provide pupils insight into how questions appear on the exam. The six questions range from finding the area under a curve and the volume of the revolution to working...

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#### Vectors: A Curriculum Module for AP® Calculus BC

It is all in a matter of time. The model curriculum module uses parametric equations and vectors to solve problems dealing with particle motion along a curve. At the beginning of the lesson, the unit reviews parametric equations and...

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#### Broken Spreadsheet II

Work in reverse with the product becoming the given. Using a spreadsheet image of the graph of a trigonometric function, young scholars investigate methods of creating spreadsheet data that results in the given graph. The catch? The data...

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#### Going Up

Going on up—and up and up! An open-ended task asks learners to model the movement of an amusement ride with parametric equations. They then analyze their equations to determine how the shadow of the ride's car moves as it rises at a...

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#### Cultural Growth

Scholars read and interpret a graph relating bacterial growth in a culture over time. They apply knowledge of derivatives, estimation, and graphing to the skill practice questions.

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#### Introduction to Calculus

Don't let your class's heart rates rise as you introduce them to differentiation ... an inquiry-based lesson helps them keep it in check! The second lesson in a three-part series asks learners to analyze the rate of change of different...

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

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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#### Man vs. Beast: Approximating Derivatives using Human and Animal Movement

What does dropping a ball look like as a graph? An engaging activity asks learners to record a video of dropping a ball and uploading the video to software for analysis. They compare the position of the ball to time and calculate the...

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#### Using Motion Sensors to Explore Graph Sketching

Get moving to a better understanding of graphs of derivatives. Using motion sensors, scholars vary their velocities to create graphs of the first derivative of a function. The activity challenges groups to first create a script of the...

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#### Derivative Grapher

Follow the slope in real time. Using graphs created with Desmos, pupils compare the graph of a function to the graph of its derivative. Pairs then work together to create derivative graphs of their own.

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#### Make It Fit

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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#### Derivate

Derive tangent lines through derivatives. Scholars use an app to graph functions and investigate derivatives and tangent lines. The interactive automatically graphs a tangent line at a specified point and can also produce tables of values.

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#### Logarithmic Differentiation: Graphing the Derivative of a Logarithm

Log the values of the derivative of a logarithm. The interactive plots the derivative of the natural logarithm. Learners first determine the derivative of natural logarithm and the general logarithm. Using the formulas for the...

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#### Equation of the Normal Line at a Point

Is it normal to find a perpendicular line to the tangent line? The short video provides a problem to find the equation of the normal line to a function. Beginning with an explanation of the normal line, the 27th installment in a 31-part...

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#### Differentiability and Horizontal and Vertical Tangent Lines

Use slope to find the location of horizontal and vertical tangent lines. Using the derivative of a radical function, the presenter finds the points where the derivative is zero or undefined on a given interval. The 26th in a series of 31...

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#### Value that Makes Two Tangent Lines Parallel

Perform parallel calculations to find equal slopes. Using the formula for the equation of the tangent line to a point, the resource sets out to find two points with a difference of one that have parallel tangent lines.The 25th video in a...

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#### Product Rule for Derivatives

Find the derivative of a complex function involving two factors. The video presents a function involving the product of two complex binomials. Using the product rule, the presenter highlights the two factors and their representations...

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#### Equation of the Tangent Line

Given a function and a point on the function, the video shows how to determine the equation of the tangent line at that point. Using the function equation, the presenter determines the y-coordinate for the point. The 24th video in a...

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#### How to Use Logarithmic Differentiation to Find the Derivative

Use the natural logarithm to simplify differentiation. The 22nd resource in a series of 31 provides an example of a problem that would be best differentiated by using logarithms. The presenter takes the natural logarithm of both sides....

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#### Derivatives of Logarithms

It is only natural to take the derivative of a logarithmic function with a complex argument. The 21st installment in a series of 31 provides an example of taking the derivative of a natural logarithmic function. First simplifying the...

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#### Finding the Derivative of a Function with e^x

Go beyond the basics with exponential functions. Using a function that contains an exponential function, the video shows the steps involved in finding its derivative. The 20th installment in a series of 31 uses an example with the...

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#### Derivatives of inverse hyperbolic functions

Apply knowledge of derivatives and inverse hyperbolic functions. Using the formula for the derivative of an inverse hyperbolic function, the resource finds the derivative of a more complex function. The complex function is comprised of a...

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#### Derivatives of Inverse Trig Functions - Arccot

Get two functions for the cost of one. The short video is the 16th in a series of 31 video on calculus topics. The presenter works through finding the derivative of a function involving two arccot functions. One of the functions requires...

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#### Derivatives of Inverse Trig Functions - Arcsin

Put another link in the chain. With the aid of the chain rule, the presentation shows how to take the derivative of an arcsin function with a binomial argument. The 15th video in a 31-part series goes on to work through the...

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#### Derivatives of Trig Functions

What are the most important things about trigonometric derivatives? The 13th video in a series of 31 highlights three important aspects of trigonometric derivatives, the first being the derivatives of sine and cosine. The second is the...

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#### Chain Rule for Derivatives, with Trig Functions

Break the chain in finding derivatives of trigonometric functions. The short video provides a demonstration of finding the derivative of a trigonometric function where the argument is a function. The 12th video in a series of 31 focuses...

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#### Chain Rule for Derivatives, with Product Rule

Unchain the product rule. The 10th video in a series of 31 provides an example of the chain rule in conjunction with the product rule. Using a binomial and a trinomial raised to powers as factors, the presenter works through the process...

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#### Chain Rule Combined with Power Rule

How does the chain rule work with the power rule? Using a radical function, the video shows how the chain rule works along with the power rule. The resource continues to simplify the derivative and places it in a familiar form. The...

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#### Reciprocal Rule

Alter the reciprocal rule to work with more functions. The eighth segment in a series of 31 introduces the reciprocal rule for finding a derivative. Using the quotient rule, the video derives the reciprocal rule and creates a rule that...

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#### Product Rule - 3+ functions

Will the product rule work for more than two functions? The sixth video in a series of 31 provides a function that is the product of three other functions. Making connections to the product rule with two functions, the presenter shows...

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#### Definition of the Derivative to Find the Derivative

Serve up the definition of the derivative two ways. Video three in a series of 31 begins with making the distinction between the difference quotient and the definition of the derivative. The presentation shows two ways the definition is...

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#### Antiderivative: Piecing it Together

Build a function backwards. Given a graph of the derivative of a function, pupils piece together a graph of the original function, the antiderivative. Learners use their graphs and the graphs of the derivatives to answer questions about...

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#### Absolute Extrema and Optimization: Building the Biggest Box

Optimally, you want the largest box. Given a square piece of box material, pupils determine the size of congruent squares to cut out of the corners to create a box with the greatest volume. Learners determine the equation of the volume...

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#### Absolute Versus Local Extrema

Get the class to take an extreme look at functions. The interactive presents a function on a closed interval with a movable tangent line. Using the given function, pupils determine the extrema, critical points, and points of inflection.

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#### Derivatives with Mean Value Theorem and Rolle's Theorem: Maxima and Minima

Rolle with the mean values and derivatives. Scholars complete the statements of Rolle's Theorem and the Mean Value Theorem. Using the interactive to illustrate scenarios, pupils respond to questions concerning the theorems in terms of...

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#### Higher Order Derivatives—Acceleration and Jerk

Accelerate your class through finding the second derivative. Using a bank of equations, pupils determine the equations for distance, velocity, and acceleration and their associated function notations. With the equations, learners answer...

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#### Chain Rule

Keep it on the chains. Pupils work through the steps in the chain rule using the interactive as a guide. Learners identify the two composed functions and their associated derivatives from a set of choices. They continue on to build the...

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#### Constant, Identity, and Power Rules: Power Rule

Learn to gain power in finding derivatives. The resource introduces the power rule. Using the sliders in the interactive, pupils create polynomial functions and find their derivatives. Given the derivative of a polynomial, learners...

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#### Derivatives of Trigonometric Functions: Derivative of sin(x)

Graphically find the derivative of sin(x). Using the interactive, pupils graph the slope of the tangent line to the sine function. Class members use the resulting graph to determine the derivative of the sine function. They verify their...

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#### Tangent Line Approximation: Estimating Square Roots

Estimating a square root is as easy as evaluating a linear equation. Using the derivative of the square root function, pupils calculate an estimation of square roots. Class members determine the equation of the tangent line at the value...

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#### Derivative of x^x

Use this trick to find the derivative of x raised to the x power. The video shows how to find a complicated derivative by rewriting the function using properties of logarithms. After rewriting the function, the derivative is simply...

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#### Implicit Differentiation Completely Explained!

Explicitly explain how to use implicit differentiation. Using a trigonometric function defined using y and x, the resource works through an example of implicit differentiation to explain the process. The 19th video in a 31-part series...

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#### Derivatives of Hyperbolic Functions

Trig provides assistance in finding the derivatives of hyperbolic functions. The presentation provides an example of finding the derivative involving hyperbolic functions. Using the chain rule, the video finds the derivative of a...

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#### Derivative of cot^4(x/2)

Apply the chain rule, the power rule, and the derivative of a trigonometric function all in one. Using a function raised to a power, the 14th video in a series of 31 shows how several rules work together to find a derivative. The...

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#### Chain Rule for Derivatives, with Quotient Rule

Work on the chain gang. The 11th installment in a series of 31 demonstrates how the chain rule and the quotient rule work together to find a derivative. Using a trigonometric function of a quotient involving exponentials as an example,...

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#### Quotient Rule

It takes a few more steps than just dividing. Given a function that is the quotient of two other functions, the resource finds the derivative. The seventh video in a series of 31 introduces the quotient rule to find derivatives. The...

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#### Power Rule for Derivatives

Find shortcuts to finding derivatives. The resource introduces the power rule for finding the derivatives of polynomial functions. The fourth video in a series of 31 starts by applying the rule to monomials and then shows how to combine...

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#### Derivatives

Instantly find the rate of change at a point. The video presents an example illustrating how to find the rate of change at a point on a function. Using graphs, the presenter illustrates the method of approximating the slope of the...

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#### Slope of a Tangent Line: Slope of the Tangent and Secant Lines

Learn to find the slope through a single point. The interactive provides a visualization of how to find the slope of a tangent line. With the aid of the visualization, pupils see the definition of the derivative in action. Class members...

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#### Average and Instantaneous Rates of Change

How can you determine the rate of change on a curve? Pupils use the interactive to discover what happens with the average rate of change as the point move closer to the other. Using the definition of the derivative, learners find that it...

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#### Higher Order Derivatives | Footnote, Essence of Calculus

Take the derivative. Then repeat. Scholars learn about higher order derivatives in the 10th video of an 11-part series. They see how the second derivative connects to physics concepts such as acceleration.

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

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...

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#### Limits | Essence of Calculus, Chapter 7

There's no limit on the importance of the limit in calculus. Scholars learn the formal definition of derivative and the epsilon-delta definition of limit by watching a video. They also listen to an explanation of l'Hospital's Rule.

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#### Derivatives of Exponentials | Essence of Calculus, Chapter 5

See why exponential functions are special. Viewers learn how the derivatives of exponential functions are proportional to their original functions. They see that e^x is a special case where the constant of proportionality is one.

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#### Derivative Formulas Through Geometry | Essence of Calculus, Chapter 3

Geometry to the rescue. By watching a video, scholars see how geometric concepts help explain derivative formulas: areas of rectangles and volumes of rectangular prisms for the Power Rule and similarity and trig ratios for the Sine Rule.

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#### Implicit Differentiation, What's Going on Here? | Essence of Calculus, Chapter 6

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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#### Visualizing the Chain Rule and Product Rule | Essence of Calculus, Chapter 4

Take a look at the product rule and chain rule through the eyes of geometry. The fourth video in an 11-part series continues to explore derivative rules using geometry. An area model and a set of number lines help viewers understand the...

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#### The Paradox of the Derivative | Essence of Calculus, Chapter 2

It turns out that "instantaneous rate of change" isn't an accurate representation of the derivative. Viewers of the second video in a series of 11 learn about the concept of the derivative, from the limit of a difference quotient to the...

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#### Solar Toasters

Help a company maximize their profits! A detailed project-based learning activity examines two production scenarios. Your young scholars write a linear demand function. They then apply the function to develop a revenue, cost, and profit...

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#### Calculus Grapher

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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#### Implicit Differentiation

Model for class members how to find both a first and second derivative of a trigonometric function with a video tutorial that works through a problem that uses the Quotient Rule to evaluate the derivative. The instructor shows the steps...

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#### Chain Rule

Class members learn how to evaluate derivatives of functions using the Chain Rule through a video that shows the steps of finding a derivative with the Chain Rule. The two examples include both a trigonometric and polynomial function.

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#### The Product, Power, and Quotient Rules

Teach the rules of finding derivatives using the video lesson. The instructor explains in detail the Product, Power, and Quotient Rules. He describes each rule and then illustrates several examples.