# Good Questions

My choices for the Good Question series are somewhat eclectic. Some are chosen because they are good, some because they are bad, some because I learned something from them, some because they can be extended, and some because they can illustrate some point of mathematics.

Difficult Problems and Why We Like Them (6-10-2013)

These posts contain a discussion of the questions and suggestions for using and adapting them. After writing about several questions, I hit upon the name “Good Question,” so while some have other names they are all interesting. The annotations give details and topics related to the question and the course description. They are in no particular order.

BEFORE CALCULUS

My Favorite Function (7-31-13) A pre-calculus question on finding roots and why calculators can’t do this one.

A Problem with 4 Solutions and 2 Morals. (6-6-2014). On simplifying an expression with radicals. Pre-calculus, calculator use.

LIMITS AND CONTINUITY

Good Question 5: 1998 AB 2 (7-29-2015) End behavior, limits at infinity, max-min, range, family of function. The correct use of infinity and DNE. Some students found a totally not expected way to do the last part without using calculus. A question I used in almost every presentation to teachers.

Dominance (8-8-2012) and Far Out (10-31-2012) A really fun limit to investigate after student know how to find extreme values, end behavior, and points of inflection

DERIVATIVES

At Just the Right Time (9-12-2013) A simple textbook question that I assigned before the class was ready for it with good results. slope, tangent line.

A Standard Problem? (5-14-2013) Find the closes point on a parabola. And then go from there. An investigation.  Max-min. Also see The Marble and the Vase below for more on the same problem.

Mean Tables (9-16-2014) A discussion of 2003 AB 90 in which students which short table of values could be those for a function describes in the stem. Mean Value Theorem, Graph analysis.

The Marble and the Vase (10-23-2014) A further discussion of “A Standard Problem? ” – see above. Max-min, with graphs.

Related Rate Problems II (10-10-2012) Two problems you won’t find in textbooks

Extremes without Calculus (10-13-2014) A student’s question.

Good Question 1 2008 AB 6  (1-21-2015) Tangent lines, critical points, point of inflection and limit at infinity, relationship between f, f’ and f’’.

Good Question 3: 1995 BC 5 (7-8-2105) A questions designed for calculator solution for the first year graphing calculators were required on the exams. It a shame there were not more like this. Second derivative, concavity.

Good Question 7: 2009 AB 3 (10-5-2015) The “Mighty Cable Company” question, Accumulation (of money), how to determine meaning of a definite integral, max-min.

Determining the Indeterminate (12-6-2015) Implicit differentiation and analysis of a curve where the derivative is 0 and where it is an indeterminate form and what it all means.

Good Question 9 (2-14-2106) A related rate question that got me thinking. Max/min

Good Question 10: The Cone Problem (11-8-2016) Cutting a sector from a circle and making a cone of maximum volume. Domain of the model.

INTEGRALS

Good Question 11 – Riemann Reversed (11-29-2016) Given a Riemann sum find the associated function and its domain so you can find the integral. This is the reverse of the usual problem when one finds the Riemann sum first and is becoming a common question on the AP Calculus Exams. There are several examples and a discussion of the concern that the answer is never unique, which makes it a poor question.

Good Question 4: 2008 AB 10 (7-15-2015) A multiple-choice question on comparing several Riemann sums: left-, right-, midpoint, and trapezoid sums. Lots of ways to extend and adapt for your class.

Most Triangles are Obtuse!  (1-18-2013) Using the average value of a function concept a 10th grade BC student proved this fact. Integration.

Challenge and Solution (3-29-2013 and 4-8-2013) Find when a vase is half-full. Two different methods with two very different correct solution. Why? Volume, washer method, shell method.

Variations on a Theme by ETS (6-14-2013) Adapting simple multiple-choice question (2008 AB 9) on accumulation and definite integration

Variations on a Theme – 2 (6-28-2013) Riemann sums and ideas on how to adapt this problem to get more out pf int. Also, discussed under Good Question 4: 2008 AB 10 below.

Lin McMullin’s Theorem (7-10-2013) A sighting of the Golden Ratio in the points of inflection of any quartic polynomial and More Gold (7-17-2013) another sighting of the Gold Ratio in cubic polynomials by David Tschappat

Half-full and Half-empty  (1-16-2015) A quick thought experiment leading to the concept of the average value of a function.

Good Question 6: 2000 – AB 4 (8-25-2015) Another of my favorite questions for working with teachers and teaching accumulation. Finding the function from its derivative done two ways.

Good Question 8 or not? (1-5-2016) A textbook question that is not good, but from which you can learn a lot. Unit analysis, accumulation, CAS work, approximation (and not a good one), reading a Riemann sum and its units, periodic functions.

Good Question 12 – Parts with a Constant (12-13-2016) You’re always telling students, “Don’t forget the + C, the constant of integration.” So now we do integration by parts and don’t worry about the + C. Why?

Good Question 13 (12-12-2017) An antiderivative 4 ways: u-substitution, integration by parts, a different u-substitution and adding zero in a convenient form. All are correct, but the answers are not the same, or are they?

Good Question 15: 2018 BC 2(a) (5-23-2018) A accumulation question with rather strange units. Making sense of the units. A BC question suitable for AB.

DIFFERENTIAL EQUATIONS

Good Question 2: 2002 BC 2 (2-17-2015) Differential equations: Suitable for AB:  slope field, particular solutions, max-min, second derivative test, a clever solution, and for BC only:  Euler’s Method. Continued in A Family of Functions next below.

A Family of Functions (2-21-2015) Jumping off from Good Question 2: 2002 BC 2 immediately above, a look at all the solutions. End behavior, maximum points, graphing.

Euler’s Method for Making Money (2-25-2015) Relating exponential growth and compound interest to Euler’s Method.

Good Question 16: 2018 BC 2(b) (5-29-2018) A density problem. Using units to find the solution. A BC question suitable for AB.

Summer Fun  (6-12-18) links to an exploration on a question similar to 2018 AB 6. Finding the general solution of the differential equation by separating the variables, checking the solution by substitution, using a graphing utility to explore the solutions for all values of the constant of integration, C, finding the solutions’ horizontal and vertical asymptotes, finding several particular solutions, finding the domains of the particular solutions, finding the extreme value of all solutions in terms of C, finding the second derivative (implicit differentiation), considering concavity, investigating a special case or two. The exploration is here and the solutions are here.

SEQUENCES AND SERIES

Good Question 14 (2-23-2018) The Integral Test for convergence or divergence of an infinite series. Why it works. (Similar to 2016 BC 92)

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