AP Exam Review
Week 3 - Friday (Day 15)
Focus Area(s)
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Graphs of f’/accumulation function FRQs
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Calculator functions on the AP Calculus Exam
Teaching Tips
So far, our students have been presented with two iconic AP Calculus problems: interpreting tabular data and exploring rate in/rate out functions. Today we look at graphical analysis, another important type of FRQ problem. We chose 2014 AB3 and 2017 AB3, but there are numerous examples from past AP exams available for extra practice. After reviewing your procedures for FRQs, distribute copies of 2014 AB3 and allow students 15 minutes to complete their work. Next week, after discussing the solutions, immediately distribute 2017 AB3 to each student. Allow just 10 minutes for work as a confidence-building formative assessment exercise.
Generally, graph interpretation problems are found in the non-calculator section and most often, the given graph is defined in relation to an integral. Emphasis is placed on relationships between the graphed function and its related functions (those generated by derivatives or integrals). The graph often creates familiar geometric regions with the x-axis, but students are responsible for knowing how to calculate those areas! In addition to a little geometry practice, graphical analysis is likely to require the following:
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Recognizing the slope of the graph is a derivative while the area between the graph and the x-axis represents an accumulation
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Using appropriate slope and function values to write equations for tangent lines
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Naming and justifying the location of extrema (usually not the graphed function, however!) and inflection points
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Naming and justifying concavity on an interval
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Using the FTC to evaluate integrals based on geometric regions
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Interpreting the limits of integration as endpoints of an interval
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ENCOURAGE students to list relationships between any functions they may reference in their work. Discussing g(x) or g’(x) is ambiguous unless students relate g(x) and g’(x) to a known function from the problem stem.
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DISCOURAGE student attempts to write an equation for the given graph. Although it might be possible, the functions are usually messy and unnecessary. Disallow students to create new names for functions without explicitly stating the relationship to the given function. Continue to discourage simplifying algebraic expressions; correct work with an incorrect numeric result won’t earn full credit!
Review Activity 2: Calculator Functions Circuit
Materials
Teaching Tips
One-third of the AP Calculus Exam is calculator active, meaning that students are allowed to use a graphing calculator to complete their work. Although some of these questions won’t require a calculator at all, is it critical that students know how to operate them quickly, accurately, and efficiently when they do. Here are a list of things your students should feel comfortable doing on their calculators:
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Finding intersection points of two curves
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Finding zeros of a function
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Finding maxima/minima from a graph
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Finding numerical derivatives and antiderivatives, both on the graphing screen and on the homescreen
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Storing functions as Y1, Y2, Y3, etc. and then referring to them in their expressions
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Storing values (like coordinates of intersections) and then referring to them in their expressions (like in limits of integration)
By practicing these skills, students can save valuable time on the exam and avoid copy errors from rewriting expressions. By storing values, students also avoid the mistake of rounding off too early.
Today we are having students work on a circuit that focuses exclusively on calculator skills. This activity is one of the many amazing resources created by Vicki Carter, an AP Calculus teacher from South Carolina. Students should not need any additional paper or pencils as each question is designed to be done on a calculator.