Type 7 Questions: Miscellaneous

Posted on March 26, 2019 by Lin McMullin

Any topic in the Course and Exam Description may be the subject of a free-response or multiple-choice question. There are topics that are not asked often enough to be classified as a type of their own. The two topics listed here have been the subject of full free-response questions or major parts of them. Other topics occasionally asked are mentioned in the question list at the end of the post.

Implicitly defined relations and implicit differentiation

These questions may ask students to find the first or second derivative of an implicitly defined relation. Often the derivative is given and students are required to show that it is correct. (This is because without the correct derivative the rest of the question cannot be done.) The follow-up is to answer questions about the function such as finding an extreme value, second derivative test, or find where the tangent is horizontal or vertical.

What students should know how to do

Know how to find the first derivative of an implicit relation using the product rule, quotient rule, chain rule, etc.
Know how to find the second derivative, including substituting for the first derivative.
Know how to evaluate the first and second derivative by substituting both coordinates of a given point. (Note: If all that is needed is the numerical value of the derivative then the substitution is often easier if done before solving for dy/dx or d²y/dx², and as usual the arithmetic need not be done.)
Analyze the derivative to determine where the relation has horizontal and/or vertical tangents.
Write and work with lines tangent to the relation.
Find extreme values. It may also be necessary to show that the point where the derivative is zero is actually on the graph and to justify the answer.

Simpler questions about implicit differentiation my appear on the multiple-choice sections of the exam.

Related Rates

Derivatives are rates and when more than one variable is changing over time the relationships among the rates can be found by differentiating with respect to time. The time variable may not appear in the equations. These questions appear occasionally on the free-response sections; if not there, then a simpler version may appear in the multiple-choice sections. In the free-response sections they may be an entire problem, but more often appear as one or two parts of a longer question.

What students should know how to do

Set up and solve related rate problems.
Be familiar with the standard type of related rate situations, but also be able to adapt to different contexts.
Know how to differentiate with respect to time. That is, find dy/dt even if there is no time variable in the given equations using any of the differentiation techniques.
Interpret the answer in the context of the problem.
Unit analysis.

Shorter questions on this concept also appear in the multiple-choice sections. As always, look over as many questions of this kind from past exams as you can find.

For some previous posts on related rate see October 8, and 10, 2012 and for implicit relations see November 14, 2012.

Free response questions (many of the BC questions are suitable for AB)

Finding derivatives using the chain rule, the quotient rule, etc. from tables of values: 2016 AB 6 and 2015 AB 6
Implicit differentiation 2004 AB and 2016 BC 4
L’Hospital’s Rule 2016 BC 4
Continuity and piecewise defined functions: 2012 AB 4, 2011 AB 6 and 2014 BC 5
Related rate: 2014 AB4/BC4, 2016 AB5/BC5
Arc length (BC Topic) 2014 BC 5
Partial fractions (BC Topic) 2015 BC 5
Improper integrals (BC topic): 2017 BC 5

Multiple-choice questions from non-secure exams:

2012 AB 27 (implicit differentiation), 77 (IVT), 88 (related rate)
2012 BC 4 (Curve length), 7 (Implicit differentiation), 11 (continuity/differentiability), 12 (Implicit differentiation), 77 (dominance), 82 (average value), 85 (related rate) , 92 (compositions)

Schedule of review postings:

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
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 26 – Type 7 questions – miscellaneous (this post)
Friday March 29 Type 8 questions – Parametric and vector questions (BC topic)
Tuesday April 2 Type 9 questions – Polar equations
Friday April 5 Type 10 questions – Sequences and Series

An Exploration in Differential Equations

Posted on January 18, 2019 by Lin McMullin

This is an exploration based on the AP Calculus question 2018 AB 6. I originally posed it for teachers last summer. This will make, I hope, a good review of many of the concepts and techniques students have learned during the year. The exploration, which will take an hour or more, includes these topics:

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

I also hope that in working through this exploration students will learn not so much about this particular function, but how to use the tools of algebra, calculus, and technology to fully investigate any function and to find all its foibles.

The exploration is here in a PDF file. Here are the solutions.

As always, I appreciate your feedback and comments. Please share them with me using the reply box below.

The College Board is pleased to offer a new live online event for new and experienced AP Calculus teachers on March 5th at 7:00 PM Eastern.

I will be the presenter.

The topic will be AP Calculus: How to Review for the Exam: In this two-hour online workshop, we will investigate techniques and hints for helping students to prepare for the AP Calculus exams. Additionally, we’ll discuss the 10 type questions that appear on the AP Calculus exams, and what students need know and to be able to do for each. Finally, we’ll examine resources for exam review.

Registration for this event is $30/members and $35/non-members. You can register for the event by following this link: http://eventreg.collegeboard.org/d/xbqbjz

The Mean Value Theorem

Posted on October 23, 2018 by Lin McMullin

Another application of the derivative is the Mean Value Theorem (MVT). This theorem is very important. One of its most important uses is in proving the Fundamental Theorem of Calculus (FTC), which comes a little later in the year.

See last Fridays post Foreshadowing the MVT for an a series of problems that will get your students ready for the MVT.

Here are some previous post on the MVT:

Fermat’s Penultimate Theorem A lemma for Rolle’s Theorem: Any function extreme value(s) on an open interval must occur where the derivative is zero or undefined.

Rolle’s Theorem A lemma for the MVT: On an interval if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b) and f(a) = f(b), there must exist a number c in the open interval (a, b) where f ‘(c) = 0.

Mean Value Theorem I Proof

Mean Value Theorem II Graphical Considerations

Darboux’s Theorem The Intermediate Value Theorem for derivatives.

Mean Tables

Revised from a post of October 31, 2017

Foreshadowing the MVT

Posted on October 19, 2018 by Lin McMullin

The Mean Value Theorem (MVT) is proved by writing the equation of a function giving the (directed) length of a segment from the given function to the line between the endpoints as you can see here. Since the function and the line intersect at the endpoints of the interval this function satisfies the hypotheses of Rolle’s theorem and so the MVT follows directly. This means that the derivative of the distance function is zero at the points guaranteed by the MVT. Therefore, these values must also be the location of the local extreme values (maximums and minimums) of the distance function on the open interval. *

Here is an exploration with three similar examples that use this idea to foreshadow the MVT. You, of course, can use your own favorite function. Any differentiable function may be used, in which case a CAS calculator may be helpful. Answers are at the end.

First example:

Consider the function $f(x)=x+2\sin (\pi x)$ defined on the closed interval [–1,3]

Write the equation of the line through the endpoints of the function.
Write an expression for h(x) the vertical distance between f(x) and the line found in part 1.
Find the x-coordinates of the local extreme values of h(x) on the open interval (–1,3).
Find the slope of f(x) at the values found in part 3.
Compare your answer to part 4 with the slope of the line. Is this a coincidence?

Second example: slightly more difficult than the first.

Consider the function $f\left( x \right)=1+x+2\cos \left( x \right)$ defined on the closed interval $\left[ {\tfrac{\pi }{2},\tfrac{{9\pi }}{2}} \right]$

Write the equation of the line through the endpoints of the function.
Write an expression for h(x) the vertical distance between f(x) and the line found in part 1.
Find the x-coordinates of the local extreme values of h(x) on the open interval $\left( {\tfrac{\pi }{2},\tfrac{{9\pi }}{2}} \right)$
Find the slope of f(x) at the values found in part 3.
Compare your answer to part 4 with the slope of the line. Is this a coincidence?

Third example: In case you think I cooked the numbers

Consider the function $\displaystyle f(x)={{x}^{3}}$ defined on the closed interval $\displaystyle [-4.5]$

Write the equation of the line thru the endpoints of the function.
Write an expression for h(x) the vertical distance between f(x) and the line found in part 1.
Find the x-coordinates of the local extreme values of h(x) on the open interval $\displaystyle (-4,5)$
Find the slope of f(x) at the values found in part 3.
Compare your answer to part 4 with the slope of the line. Is this a coincidence?

Answers

First example:

y = x
$\displaystyle h(x)=f(x)-y(x)=\left( {x+2\sin (\pi x)} \right)-\left( x \right)=2\sin (\pi x)$
${h}'\left( x \right)=2\pi \cos \left( {\pi x} \right)=0$ when x = –1/2, ½, 3/2 and 5/2
$\displaystyle {f}'\left( x \right)=1+2\pi \cos \left( {\pi x} \right)$ , the slope = 1 at all four points
They are the same. Not a coincidence.

Second example:

The endpoints are $\left( {\tfrac{\pi }{2},1+\tfrac{\pi }{2}} \right)$ and $\left( {\tfrac{{9\pi }}{2},1+\tfrac{{9\pi }}{2}} \right)$ ; the line is $y=x+1$
$h\left( x \right)=f\left( x \right)-y\left( x \right)=\left( {1+x+2\cos (x)} \right)-\left( {x+1} \right)=2\cos \left( x \right)$
${h}'\left( x \right)-2\sin (x)=0$ when $x=\pi ,2\pi ,3\pi ,\text{ and }4\pi$
${f}'\left( x \right)=1-2\sin \left( x \right)$ , at the points above the slope is 1.
They are the same. Not a coincidence.

Third example:

The endpoints are (-4, -64) and (5, 125), the line is $\displaystyle y=125+21(x-5)=21x+20$
$\displaystyle h(x)={{x}^{3}}-21x-20$
$\displaystyle {h}'(x)=3{{x}^{2}}-21=0$ when $\displaystyle x=\sqrt{7},-\sqrt{7}$
$\displaystyle {f}'\left( {\pm \sqrt{7}} \right)=3{{\left( {\pm \sqrt{7}} \right)}^{2}}=21$
They are the same. Not a coincidence.

* It is possible that the derivative is zero and the point is not an extreme value. This is similar to the situation with a point of inflection when the first derivative is zero but does not change sign.

L’Hospital’s Rule

Posted on October 16, 2018 by Lin McMullin

Another application of the derivative

L’Hospital’s Rule

Locally Linear L’Hospital’s Demonstration of the proof

L’Hospital Rules the Graph

Good Question An AP Exam question that can be used to delve deeper into L’Hospital’s Rule (2008 AB 6)

Guillaume de l’Hospital
1661 – 1704

Revised from a post of November 7, 2017

There will be two extra posts this week! Check tomorrow for some suggestions on “Teaching Concavity” and on Friday for “Foreshadowing the MVT.”

I made a major update to last Friday’s post On Scaling. It includes a suggestion from a reader of this blog with a Desmos graph that will calculate the Kennedy scale scores for you.

Graphing – an Application of the Derivative.

Posted on October 9, 2018 by Lin McMullin

Graphing and the analysis of graphs given (1) the equation, (2) a graph, or (3) a table of values of a function and its derivative(s) makes up the largest group of questions on the AP exams. Most of the other applications of the derivative depend on understanding the relationship between a function and its derivatives.