Type 3 Questions: Graph and Function Analysis

The long name is “Here’s the graph of the derivative, tell me things about the function.”

Students are given either the equation of the derivative of a function or a graph identified as the derivative of a function but no equation is given. It is not expected that students will write the equation (although this may be possible); rather, students are expected to determine important features of the function directly from the graph of the derivative. They may be asked for the location of extreme values, intervals where the function is increasing or decreasing, concavity, etc. They may be asked for function values at points.

The graph may be given in context and student will be asked about that context. The graph may be identified as the velocity of a moving object and questions will be asked about the motion. See Type 2 questions – Linear motion problems

Less often the function’s graph may be given and students will be asked about its derivatives.

What students should be able to do:

• Read information about the function from the graph of the derivative. This may be approached by derivative techniques or by antiderivative techniques.
• Find and justify where the function is increasing or decreasing.
• Find and justify extreme values (1st and 2nd derivative tests, Closed interval test aka.  Candidates’ test).
• Find and justify points of inflection.
• Find slopes (second derivatives, acceleration) from the graph.
• Write an equation of a tangent line.
• Evaluate Riemann sums from geometry of the graph only.
• FTC: Evaluate integral from the area of regions on the graph.
• FTC: The function, g(x), maybe defined by an integral where the given graph is the graph of  the integrand, f(t), so students should know that if,  $\displaystyle g\left( x \right)=g\left( a \right)+\int_{a}^{t}{f\left( t \right)dt}$ then  ${g}'\left( x \right)=f\left( x \right)$  and  ${{g}'}'\left( x \right)={f}'\left( x \right)$. In this case students should write ${g}'(t)=f\left( t \right)$ on their answer paper, so it is clear to the reader that they understand this.

Not only must students be able to identify these things, but they are usually asked to justify their answer and reasoning. See Writing on the AP Exams for more on justifying and explaining answers.

The ideas and concepts that can be tested with this type question are numerous. The type appears on the multiple-choice exams as well as the free-response. Between multiple-choice and free-response this topic may account for 15% or more of the points available on recent tests. It is very important that students are familiar with all the ins and outs of this situation.

As with other questions, the topics tested come from the entire year’s work, not just a single unit. In my opinion many textbooks do not do a good job with integrating these topics, so be sure to use as many actual AP Exam questions as possible. Study past exams; look them over and see the different things that can be asked.

For some previous posts on this subject see October 1517192426 (my most read post), 2012 and  January 2528, 2013

Free-response questions:

• Function given as a graph, questions about its integral (so by FTC the graph is the derivative): 2014 AB 3/BC 3
• Function given as an equation: 2016 AB 6
• Function given as a graph 2016 AB 3/BC 3
• Table and graph of function given, questions about related functions: 2017 AB 6,
• Derivative given as a graph: 2016 AB 5 and 2017 AB 3

Multiple-choice questions from non-secure exam. Notice the number of questions all from the same year; this is in addition to one free-response question (~25 points on AB and ~23 points on BC out of 108 points total)

• 2012 AB: 2, 5, 15, 17, 21, 22, 24, 26, 76, 78, 80, 83, 82, 84, 85, 87
• 2012 BC 3, 11, 12, 15, 12, 18, 21, 76, 78, 80, 81, 84, 88, 89

A good activity on this topic is here. The first pages are the teacher’s copy and solution. Then there are copies for Groups A, B, and C. Divide your class into 3 or 6 or 9 groups and give one copy to each. After they complete their activity have the students compare their results with the other groups.

Schedule of review posts

• Tuesday February 27 – AP Exam Review
• Friday, March 2 – Resources for reviewing
• Tuesday March 6 – Type 1 questions – Rate and accumulation questions
• Friday March 9 – Type 2 questions – Linear motion problems
• Tuesday March 13 – Type 3 questions – Graph analysis problems (this post)
• Friday March 16 – Type 4 questions – Area and volume problems
• Tuesday Match 20 Type 5 questions – Table and Riemann sum questions
• Friday March 23 Type 6 questions – Differential equation questions
• Tuesday March 27 – Type 7 questions – miscellaneous
• Friday March 30 Type 8 questions – Parametric and vector questions (BC topic)
• Tuesday April 3 Type 9 questions – Polar equations
• Friday April 6 Type 10 questions – Sequences and Series