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

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

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

Determining the Indeterminate 2

Posted on December 6, 2015 by Lin McMullin

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?

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.
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

Posted on March 13, 2013 by Lin McMullin

AP Type Questions 7

Implicitly defined relations and implicit differentiation

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

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

Teaching Calculus

The pleasure lies not in discovering the truth, but in searching for it.

Tag Archives: implicit relations

Determining the Indeterminate 2

AP Questions Type 7: Other topics

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AP Questions Type 7: Other topics

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Implicitly defined relations and implicit differentiation

Related Rates

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