This is a collaborative and challenging activity to practice evaluating indefinite integrals by using Integration by Parts. It aims to develop and consolidate student’s skills in integration by this special method. Partners work through 8 sections (or less if preferred). Each section contains one integral. Activity Directions: A partner start solving an integral while the other partner is waiting for the answer to integrate it. In the next section, partners take turns and it goes the same way. An alternative way ( I think it is better way) to use this activity: Partner A solves an integral of a section while Partner B solves an integral of the other (the next in line) section. Then they swap papers and each partner integrates the function that his partner has obtained previously. The problems are well thought so the obtained functions after the first integrating can be also integrated by parts! A HINT is provided in each section that gives the sum of the partners’ answers. Answer keys are included. I hope this activity will be stimulating and beneficial for your calculus students.

This is a complete rigorous lesson on Advanced Techniques of Integration for students enrolled in AP Calculus BC or Calculus 2. ( The resource contains 47 problems - 17 with full typed solutions and 30 with answer keys) Lesson Objective: Find indefinite integrals and evaluating definite integrals using integration by parts. It includes A. Detailed Guided Notes (5 pages) including derivation of Integration by Parts formula and 7 worked out examples with indefinite and definite integrals as 4 examples perform Integration by Parts once 3 examples including repeated Integration by Parts( twice and for third time). The students are guided through the process of picking u, du, v, and dv. B. 20 Classifies Practice Problems ( 10 come with full typed solutions and other 10 are similar to them for students to try their own. Students are given the answers. ) C. 12 Assignment Problems with answer keys ( 6 indefinite and 6 definite integrals ). D. 2 versions Multiple Choice Exit Tickets /Quizzes/ Assessments ( with answer keys)

This product includes 24 integration problems to be solved by u-substitution. Your students can work alone, in pairs, or small groups to complete the problems placed on 12 cards ( there are 2 problems on each card – one indefinite and one definite integral to be evaluated as both integrals have the same integrand). Students recording sheets are provided and created as worksheets so the product can be used for cooperative learning, extra practice, homework assignment or even as an assessment. Typed solutions (appropriate for teachers!) are provided.

In this fun Valentine’s themed activity, students will practice evaluating definite integrals using u- substitution. Activity Directions: Students are instructed to evaluate 12 definite integrals using u- substitution. The corrected values are recorded on twelve of 13 hearts on a given picture. Every time students arrive at a correct answer, they will “win the heart” having this answer written on it. Students are also asked to find the only one heart who can NOT be won. The product can possibly be used as a partner or a group activity (group of 2 and 3). Student recording sheets and answer keys are provided. I hope you and your students enjoy! ♡ Happy Valentine’s Day! ♡

In this fun Valentine’s themed activity, students will practice evaluating definite integrals using the properties of integrals and the table of basic integrals. Activity Directions: Students are instructed to evaluate 12 definite integrals using the table of basic integrals. The corrected values are recorded on twelve of 13 hearts on a given picture. Every time students arrive at a correct answer, they will “win the heart” having this answer written on it. Students are also asked to find the only one heart who can NOT be won. The product can possibly be used as a partner or a group activity (groups of 2 or 3). Student recording sheets and answer keys are provided. I hope you and your students enjoy! ♡ Happy Valentine’s Day! ♡

This is a collaborative activity to practice evaluating indefinite integrals by using their BASIC properties and the table of common integrals. Partners work through 20 sections each containing one integral. Activity Directions: A partner start solving an integral while the other partner is waiting for the answer to check it by differentiating. In the next section, partners take turns and it goes the same way. An alternative way to use this activity: Partner A solves an integral of a section while Partner B solves an integral of the other (the next in line) section. Then they swap papers and differentiate the obtained functions to check each other’s work. If extra practice is needed Partner A and Partner B can change places with each other and continue solving. Applying both integrating and differentiating help students gain a better understanding of integration. I hope it will be beneficial for your calculus students. Answer keys and full solutions (typed and handwritten clearly) are included.

Students will practice condensing logarithms with this multiple-choice game. On each page there are two problems each in a table. In the table there are given a picture of a parrot, a logarithmic expression that students have to condense and four answer choices. Name of parrot species corresponds to each answer choice. If students solve the problem correctly they will find out what parrot species is this one on the picture. The problem are 12 and vary in difficulty. The distractors are well thought. Answer key is provided.

This is a collaborative and challenging activity to practice finding derivatives by implicit differentiation. All of the equations are selected to be solved for y to get the form y= f(x). Functions included are polynomial, rational, including radicals, exponential, logarithmic, trigonometric and inverse trigonometric. Activity Directions: Partners work through 8 sections (or less if preferred). Each section contains one equation in two variables x and y. Partner A solve it for y and then differentiate the function y=f(x) directly to find the derivative of y(x). Partner B deals with the same equation using implicit differentiation to find the derivative of y. When partners are ready, they check that their derivatives are the same. In the next section, Partner A will use implicit differentiation while Partner B applies differentiation directly. Once the work is completed, partners compare their derivatives again. Partners are supposed to get the same answers in each section. This activity can be used once more like that - next time partners can practice the same examples exchanging their roles – Partner A takes the position of Partner B and Partner B takes the position of Partner A. In such a way each partner will solve each of the problems using two solution techniques.

In this 24 questions- activity, students apply L’Hospital’s Rule to evaluate limits. There are included the following indeterminate forms - 0/0, infinity / infinity, infinite minus infinity and the product of zero and infinity. All functions are included from polynomial to square root, from exponential to log, from trig to inverse trig functions. There are two similar versions of this practice each consisting of two sections. Each section contains three groups of two limits. The problems IN EACH SECTION have THE SAME ANSWER! In section1, two problems have the form 0/0; the next two have the form infinity/infinity and the last two problems have the form infinity minus infinity. In section 2, two problems have the form 0/0; the next two have the form infinity/infinity and the last two problems have the form the product of zero and infinity. ( Not all of the questions require L’Hospital’s Rule (i.e. another valid method could be used) however students are instructed to use only L’Hospital’s Rule to find the limits.) This activity can be used for class work, independent or grouped (groups of 2 or 4). It can be used as an assessment and homework as well. Answer keys are included.

These are 12 challenging practice problems on limits at infinity. Functions included are only exponential, natural logarithms and inverse tangents. The product can be used as independent practice, extra practice, enrichment and homework assignment. The practice sheets have enough room for students to show work. Answer keys are included.

In this 24 questions – collaborative partner activity, students are to find the limits of various functions (polynomial, rational, radical, exponential, logarithmic) as x approaches infinity. To determine the limits, students will need to use a combination of techniques. The limits in this activity can all be found without L’Hopital’s rule. Each partner has his own set of 12 problems. Corresponding partners’ problems are thought carefully so that the answer of each of Partner’s A problems is the same as the answer of its corresponding problem of Partner’s B. almost all of them to be similar examples - limits of functions which are of the same type Students are instructed to show all work and check whether their results match. Answer keys are included.

In this 16 questions - partner activity, students apply L’Hospital’s Rule to evaluate limits. There are included the following indeterminate forms - 0/0, infinity / infinity, infinity minus infinity, zero ⨯ infinity, zero^zero, infinity^zero and 1^infinity. The functions included are polynomial, exponential, logarithmic, trig and inverse trig functions. Each partner has his own set of 8 problems. Partners will use L’Hospital’s Rule to evaluate their first two limits directly. They will need to manipulate to make zero/zero or infinity/infinity and then to apply L’Hospital’s Rule to evaluate the next six limits. ► THE FORM of the limit and THE ANSWER of each of Partner’s A problems are THE SAME as the form of the limit and the answer of each Partner’s B corresponding problem. (Problems A1 and B1 are corresponding, so as problems A2 and B2 and so on). Students are instructed to show all work and check whether their results match. Full solutions are provided.

This engaging and fun activity has the student find the equation to a tangent line at a given value of the variable x. The functions included are polynomial, rational, involving radicals, exponential, logarithmic and inverse trigonometric. Students work independently or in groups of two or three to solve 13 problems. They use a table to find out the syllables corresponding to each of their answers. Then students record the syllables in another table and read the resulting sentence. It is a motivational quotation by the Professor Steven Strogatz about calculus. Student recording sheet and answer keys are included.

This is an engaging and collaborative group activity on finding the absolute extrema of a function (extrema on an interval). The functions included are polynomial, rational, involving radicals, exponential, logarithmic and trigonometric. There are 10 sections - 10 functions. Students work in groups of two, three or/and four. In each section, each member of a group is asked to find the absolute extrema of the given function on a specified closed interval . Thus one and the same function will be examined for absolute extrema at most in four different particular intervals. The collaborative part of this activity is in the initial stage of solving where partners find the first derivative of the given function and determine all the critical values. Students compare their results, find and fix any errors to continue solving in the right direction. They evaluate the function at the critical points found and the end points again having the opportunity to compare their calculations. As a consequence their final (different) answers must be all correct. Student recording sheets with steps that lead to solution and answer keys are included. The product can be also used as an independent/extra practice with 4 different forms and assignment or homework.

This resource contains a total of 16 problems. Students will practice taking the first derivatives of y(x) for each of given implicit functions by implicit differentiation. The packet has 2 worksheets each containing 8 various examples including polynomial, rational, exponential, logarithmic, trigonometric and inverse trigonometric functions. The worksheets can be used as a class practice, for an extra practice or enrichment, an assessment or homework assignment. It can be also used as a partner activity – like that: ⟡ Partner A will solve WS # 1 while Partner B solves WS # 2 , then they swap papers and Partner A will solve WS # 2 while Partner B solves WS # 1 . Once they have completed the work, they compare their results. If there are different answers to one and the same problem, students have to identify and correct any errors. Typed solutions to all of the problems are provided.

This resource contains a total of 24 problems. Students will practice taking the derivatives of some complicated functions by logarithmic differentiation. The packet has 2 worksheets: ⟐ The first worksheet has the students finding the first derivatives of 12 complicated functions using natural logarithms. ⟐ The second worksheet also contains 12 problems of finding the derivatives by logarithmic differentiation as some of the examples are more complex. The worksheets can be used as class practice, for an extra practice or enrichment, an assessment or homework assignment. It can be also used as a partner or group activity – like that: ⟡ Partner A will solve WS #1 (page 1) while Partner B solves WS # 1 (page 2), then they swap papers and Partner A will solve WS #1 (page 2) while Partner B solves WS #1 (page 1). Once they have completed the work, they compare their results. If there are different answers to one and the same problem, students have to identify and correct any errors. Partners continue working through WS # 2 in the same manner with pages 2 and 3 or pages 1 and 2… ⟡ Worksheet # 2 can be used as a group activity ( for groups of 3 as its pages are three and each page contains 4 problems). Full solutions (handwritten clearly) are provided.

This resource contains total of 30 problems. Students will practice higher order differentiation of common and composite functions (with and without the chain rule). The packet has 4 worksheets: ⟐ The first worksheet has the students finding the second derivative of 10 various common and composite functions. ⟐ The second worksheet is finding the third derivative of 6 functions (without using the chain rule). ⟐ The third worksheet is finding the fourth derivative of 6 common functions (without using the chain rule). ⟐ The third worksheet is finding the first four derivatives of 8 composite functions using the chain rule. The worksheets can be used in class for group work or an independent practice, for enrichment, an assessment or homework assignment. Detailed typed answer keys are provided.