# Other Problems (Type 7)

### AP Questions Type 7: Other topics

Any topic in the Course and Exam Description may be the subject of a free-response or multiple-choice question. The topics discussed here are not asked often enough to be classified as a type of their own. The 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 and/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 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 may appear on the multiple-choice sections of the exam.

Example:

Good Question 17

2004 AB 4

2016 BC 4

2012 AB 27 (implicit differentiation), Multiple-choice

2022 AB 5 (a) Implicit differentiation,

BC classes see Implicit differentiation of parametric equations, and A Vector’s Derivative

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 previous posts on related rates see Related Rate Problems I and Related Rate Problems II.

Examples

2014 AB4/BC4,

2016 AB5/BC5

2019 AB 4 Related Rate

2019 AB 6

2022 AB2 (d), AB4/BC4 (d) Good example that requires using product and evaluation of an expression that include dr/dt and dh/dt.

Good Question 9

Family of Functions

A “family of functions” is defined by an equation with a parameter (sort of an extra variable). Changing the parameter gives a different but similar curve. Questions should be answered in general, that is, in terms of the parameter not a specific value of the parameter. These questions appeared on some exams long ago, may be making a comeback.

Examples:

1995 BC 5

1996 AB4/BC4

Good Question 5: 1998 AB2/BC2

2019 BC 5

Other Topics

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
• L’Hospital’s Rule 2016 BC 4, 2019 AB 3 (Don’t be fooled), 2019 AB 4(c)
• Continuity and piecewise defined functions: 2012 AB 4, 2011 AB 6 and 2014 BC 5
• Arc length (BC Topic) 2014 BC 5
• Partial fractions (BC Topic) 2015 BC 5
• Improper integrals (BC topic): 2017 BC 5, 2022 BC5 (c)

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)

These questions may come from any of the Units in the CED.

Revised March 12, 2021, April 1, and May 14, 2022

# Other Problems (Type 7)

### AP  Questions Type 7: Other topics

Any topic in the Course and Exam Description may be the subject of a free-response or multiple-choice question. The topics discussed here are not asked often enough to be classified as a type of their own. The 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.

Example:

Good Question 17

2004 AB 4

2016 BC 4

2012 AB 27 (implicit differentiation), Multiple-choice

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  Related Rate Problems I and Related Rate Problems II.

Examples

2014 AB4/BC4,

2016 AB5/BC5

2019 AB 4 Related Rate

2019 AB 6

Good Question 9

Family of Functions

A “family of functions” are defined by an equation with a parameter (sort of an extra variable). Changing the parameter gives a different but similar curve. Questions should be answered in general, that is, in terms of the parameter not some specific value of the parameter. These questions appeared on some exams long ago, may be making a comeback.

Examples:

1995 BC 5

1996 AB4/BC4

Good Question 5: 1998 AB2/BC2

2019 BC 5

Other Topics

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
• L’Hospital’s Rule 2016 BC 4, 2019 AB 3 (Don’t be fooled), 2019 AB 4(c)
• Continuity and piecewise defined functions: 2012 AB 4, 2011 AB 6 and 2014 BC 5
• 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)

These question may come from any of the Units in the  2019 CED.

Revised March 12, 2021

# Type 7 Questions: Miscellaneous

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:

# Determining the Indeterminate 2

The other day someone asked me a question about the implicit relation ${{x}^{3}}-{{y}^{2}}+{{x}^{2}}=0$. They had been asked to find where the tangent line to this relation is vertical. They began by finding the derivative using implicit differentiation:

$3{{x}^{2}}-2y\frac{dy}{dx}+2x=0$

$\displaystyle \frac{dy}{dx}=\frac{3{{x}^{2}}+2x}{2y}$

The derivative will be undefined when its denominator is zero. Substituting y = 0 this into the original equation gives ${{x}^{3}}-0+{{x}^{2}}=0$. This is true when x = –1 or when x = 0. They reasoned that there will be a vertical tangent when x = –1 (correct) and when x = 0 (not so much). They quite wisely looked at the graph.

${{x}^{3}}-{{y}^{2}}+{{x}^{2}}=0$

The graph appears to run from the first quadrant, through the origin into the third quadrant, up to the second quadrant with a vertical tangent at x = –1, and then through the origin again and down into the fourth quadrant. It looks like a string looped over itself.

What’s going on at the origin? Where is the vertical tangent at the origin?

The short answer is that vertical tangents occur when the denominator of the derivative is zero and the numerator is not zero.  When x = 0 and y = 0 the derivative is an indeterminate form 0/0.

In this kind of situation an indeterminate form does not mean that the expression is infinite, rather it means that some other way must be used to find its value. L’Hôpital’s Rule comes to mind, but the expression you get results in another 0/0 form and is no help (try it!).

My thought was to solve for y and see if that helps:

$y=\pm \sqrt{{{x}^{3}}+{{x}^{2}}}$

The graph consists of two parts symmetric to the x-axis, in the same way a circle consists of two symmetric parts above and below the x-axis. The figure below shows the top half.

$y=+\sqrt{{{x}^{3}}+{{x}^{2}}}$

So, the graph does not run from the first quadrant to the third; rather, at the origin it “bounces” up into the second quadrant. The lower half is congruent and is the reflection of this graph in the x-axis.

So, what happens at the origin and why?

The derivative of the top half is $\displaystyle \frac{dy}{dx}=\frac{3{{x}^{2}}+2x}{2\sqrt{{{x}^{3}}+{{x}^{2}}}}$. Notice that this is the same as the implicit derivative above. Now a little simplifying; okay a lot of simplifying – who says simplifying isn’t that big a deal?

$\displaystyle \frac{dy}{dx}=\frac{x\left( 3x+2 \right)}{2\sqrt{{{x}^{2}}\left( x+1 \right)}}=\frac{x\left( 3x+2 \right)}{2\left| x \right|\sqrt{x+1}}=\left\{ \begin{matrix} \frac{3x+2}{2\sqrt{x+1}} & x>0 \\ -\frac{3x+2}{2\sqrt{x+1}} & x<0 \\ \end{matrix} \right.$

Now we see what’s happening. As x approaches zero from the right, the derivative approaches +1, and as x approaches zero from the left, the derivative approaches –1.  This agrees with the graph. Since the derivative approaches different values from each side, the derivative does not exist at the origin – this is not the same as being infinite.  (For the lower half, the signs of the derivative are reversed, due to the opposite sign of the denominator.)

The tangent lines at the origin are x = 1 on the right, and x = –1 on the left, hardly vertical.

What have we learned?

1. Indeterminate forms do not necessarily indicate an infinite value. An indeterminate form must be investigated further to see what you can learn about a function, relation, or graph.
2. Sometimes simplifying, or at least changing the form of an expression, is helpful and therefore necessary.

Extension: Using a graphing utility that allows sliders (Winplot, GeoGebra, Desmos, etc)  enter $A{{x}^{3}}-B{{y}^{2}}+C{{x}^{2}}=0$ and explore the effects of the parameters on the graph.

_________________________

The Man Who Tried to Redeem the World with Logic

WALTER PITTS (1923-1969): Walter Pitts’ life passed from homeless runaway, to MIT neuroscience pioneer, to withdrawn alcoholic. (Estate of Francis Bello / Science Source)

I ran across this article that you might find interesting. It is about Walter Pitts one of the twentieth century’s most important mathematicians we, or at least I, have never heard of.  It is from the February 5, 2015 of the science magazine Nautilus

# Implicit Relations & Related Rates

AP Type Questions 7

### 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, the 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 the 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.)
• 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.

### 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, using any of the differentiation techniques.
• Interpret the answer in the context of the problem.
• Unit analysis.

Shorter questions on both these concepts 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 relation s see November 14, 2012