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If you use this opening section, you will want a quiz to test students. Students usually struggle with it, but I always thought that it let students know that this course isn't a pushover. Even if you don't use the opening section, there are 3 problems in this quiz that test students on average and instantaeous rate of change using formulas and tables as well as a trapezoidal rule problem to estimate the definite integral. |
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The concept of derivatives doesn’t exist for students yet, but they are asked to find the slope of the secant line between 2 points on a curve as well as the slope and equation of the tangent line at a point. Also average and instantaneous rate of change at a given time. Answers given to this one
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Same as above.
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Look at a graph and determine the limit as x approaches a number as well as positive or negative infinity.
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Find limits as x approaches a number or infinity of algebraic function. Students are required to use problem notation as well as splitting limits into left and right-hand limits.
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04a. Limits AP Type
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Student's first exposure to an AP-type free response problem. |
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Power, product, and quotient rule, as well as finding derivatives by definition. Finding equation of tangent lines too.
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Students are aware that calculators can find derivatives of functions at specific values. This quiz, using tables, tests students on their knowledge of the power, product, quotient, and chain rules where calculators do them little good.
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Emphasizes the basic rules as well as the chain rule and implicit differentiation.
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Same as above.
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It is difficult to find full AP free response problems that only involve limits and derivatives. Here are 5 of them and you might want to use these as a Take-Home type exam to give the kids a taste of the real thing. |
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Emphasizes the chain rule as used with trig functions. Implicit differentiation at a point or in general tested as well. You get the answers to this one.
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Same as above
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Given a piecewise function, determine whether it is continuous, differentiable, both or neither at a point. Also find values of constants to make a piecewise function differentiable.
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A 30-question multiple choice exam testing all concepts up through differentiability.
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A seven question related rates exam using problems similar to the ones in the manual.
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Students dread related rates questions but they are important. This calculator active problem is similar to one that appeared in a recent AP FR exam.
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Given a position function, find velocity, acceleration and whether the particle is speeding up or slowing down. Also vertical motion problems.
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For the topic of function analysis, students need to be able to examine the derivative of a function and determine whether the function is increasing or decreasing, concave up or down, and to sketch the shape of the function. These problems are given, sometimes all within one week to nail down this difficult concept for students.
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See above.
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See above
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See above
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A very standard problem - given a graph of the derivative of a function, determine information about the function. It is wise to give this problem right after the last 4 quizzes to reinforce the concept.
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Another standard type problem with a trig expression is given and questions about absolute max and mins are given along with inflection point. Some algebraic manipulation is necessary. |
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Given a function, find the absolute maximum or minimum value of the function on an interval.
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Most of my tests in this section have been in the form of a short quiz. This is not an accident. I like to give these quizzes and then have the students grade them in class giving immeidate reinforcement. This practice test puts a lot of concepts together: Rolles, Mean Value, given the derivative, finding invervals of increasing/decreasing, concavity, inflection points, relative ins and maxes as well as absolute max and min. And this practice exam has the solutions as well as a bonus! |
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An exam exactly in the same format as 19a. above. |
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A six question optimization exam using problems similar to the ones in the manual.
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Optimization is my favorite AP topic. Kids so relate to it (maximizing their grade while minimizing their efforts!). Unfortunately we can't spend that much time on it. Here are 6 problems with subparts that are interesting, realistic, and make the kids think. They have to do 3 of the 6 problems but you can make them do as many as you wish. A good weekend or holiday assignment. |
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I like to give students problems to do over the winter holiday to keep their skills sharp. These are 40 question multiple choice that emphasize max/mins and concavity.
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Just a small exam emphasizing the power rule and trig for integration as well as a differential equation.
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You announce a midterm but do kids really know how to study? I like to give them a sense of how much we have done this year with this 30 question multiple choice practice exam that gives them an idea what they know and don't know. This could be used for an exam as well.
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Time to put all their acts together with a big exam. This is a 45 question multiple choice exam testing all concepts through basic integration.
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Basic indefinite integration emphasizing u-substitution.
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Riemann sums as well as the trapezoidal rule. Calculators are needed. This can be given as a take-home exam as each problem as three possibilities so that students can get different versions of the exams.
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Given a piecewise curve, students are asked to find definite integrals between various values utilizing geometric formulas. Also, examine the accumulation function based on this graph. You get answers to this one.
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Same as above.
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Finding definite integrals of algebraic functions, u-substitution and changing the limits, the average value formula as well as the 2nd Fundamental Theorem. You get answers to this one.
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Same as above.
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Now that students understand accumulation and the 2nd Fundamental Theorem, they need to see this important type of question incorporated into a typical free response question. This is the same question as #13a above but a subpart involding a definite integral has been added.
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By the time this topic is taught, time is at a premium. I have a 2-page exam for the concepts with 3 variations for each page. This can be given as a take-home with cheating difficult to accomplish or an in-class exam.
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Derivatives and integrals using the natural log (ln) function as well as the exponential function.You get solutions to this one.
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Same as above
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Find the derivative of the inverse of a function at a point (with and without calculator) as well as the derivative of an inverse trig function and an integral resulting in an inverse trig function.
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Creating a slope field, solving separable DEQ’s as well as solving an exponential growth problem.
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