Monthly Archives: February 2016

Many Roads Give Same Derivative

A recent post in the AP Calculus Community expressed some confusion about different ways to compute \displaystyle \frac{dy}{dx} at (0,4) for the function x=2ln(y-3).  I share below the two approaches suggested in the original post, proffer two more, and a slightly more in-depth activity I’ve used in my calculus classes for years.  I conclude with an alternative to derivatives of inverses.

Two Approaches Initially Proposed

1 – Accept the function as posed and differentiate implicitly.

\displaystyle \frac{d}{dx} \left( x = 2 ln(y-3) \right)

\displaystyle 1 = 2*\frac{1}{y-3} * \frac{dy}{dx}

\displaystyle \frac{dy}{dx} = \frac{y-3}{2}

Which gives \displaystyle \frac{dy}{dx} = \frac{1}{2} at (x,y)=(0,4).

2 – Solve for y and differentiate explicitly.

\displaystyle x = 2ln(y-3) \longrightarrow y = 3 + e^{x/2}

\displaystyle \frac{dy}{dx} = e^{x/2} * \frac{1}{2}

Evaluating this at (x,y)=(0,4) gives \displaystyle \frac{dy}{dx} = \frac{1}{2} .

Two Alternative Approaches

3 – Substitute early.

The question never asked for an algebraic expression of \frac{dy}{dx}, only the numerical value of this slope.  Because students tend to make more silly mistakes manipulating algebraic expressions than numeric ones, the additional algebra steps are unnecessary, and potentially error-prone.  Admittedly, the manipulations are pretty straightforward here, in more algebraically complicated cases, early substitutions could significantly simplify work. Using approach #1 and substituting directly into the second line gives

\displaystyle 1 = 2 * \frac{1}{y-3} * \frac{dy}{dx} .

At (x,y)=(0,4), this is

\displaystyle 1 = 2 * \frac{1}{4-3}*\frac{dy}{dx}

\displaystyle \frac{dy}{dx} = \frac{1}{2}

The numeric manipulations on the right side are obviously easier than the earlier algebra.

4 – Solve for \frac{dx}{dy} and reciprocate.

There’s nothing sacred about solving for \frac{dy}{dx} directly.  Why not compute the derivative of the inverse and reciprocate at the end? Differentiating first with respect to y eventually leads to the same solution.

\displaystyle \frac{d}{dy} \left( x = 2 ln(y-3) \right)

\displaystyle \frac{dx}{dy} = 2 * \frac{1}{y-3}

At (x,y)=(0,4), this is

\displaystyle \frac{dx}{dy} = \frac{2}{4-3} = 2 , so

\displaystyle \frac{dy}{dx} = \frac{1}{2}.

Equivalence = A fundamental mathematical concept

I sometimes wonder if teachers should place much more emphasis on equivalence.  We spend so much time manipulating expressions in mathematics classes at all levels, changing mathematical objects (shapes, expressions, equations, etc.) into a different, but equivalent objects.  Many times, these manipulations are completed under the guise of “simplification.”  (Here is a brilliant Dan Teague post cautioning against taking this idea too far.)

But it is critical for students to recognize that proper application of manipulations creates equivalent expressions, even if when the resulting expressions don’t look the same.   The reason we manipulate mathematical objects is to discover features about the object in one form that may not be immediately obvious in another.

For the function x = 2 ln(y-3), the slope at (0,4) must be the same, no matter how that slope is calculated.  If you get a different looking answer while using correct manipulations, the final answers must be equivalent.

Another Example

A similar question appeared on the AP Calculus email list-server almost a decade ago right at the moment I was introducing implicit differentiation.  A teacher had tried to find \displaystyle \frac{dy}{dx} for

\displaystyle x^2 = \frac{x+y}{x-y}

using implicit differentiation on the quotient, manipulating to a product before using implicit differentiation, and finally solving for y in terms of x to use an explicit derivative.

1 – Implicit on a quotient

Take the derivative as given:$

\displaystyle \frac{d}{dx} \left( x^2 = \frac{x+y}{x-y} \right)

\displaystyle 2x = \frac{(x-y) \left( 1 + \frac{dy}{dx} \right) - (x+y) \left( 1 - \frac{dy}{dx} \right) }{(x-y)^2}

\displaystyle 2x * (x-y)^2 = (x-y) + (x-y)*\frac{dy}{dx} - (x+y) + (x+y)*\frac{dy}{dx}

\displaystyle 2x * (x-y)^2 = -2y + 2x * \frac{dy}{dx}

\displaystyle \frac{dy}{dx} = \frac{-2x * (x-y)^2 + 2y}{2x}

2 – Implicit on a product

Multiplying the original equation by its denominator gives

x^2 * (x - y) = x + y .

Differentiating with respect to x gives

\displaystyle 2x * (x - y) + x^2 * \left( 1 - \frac{dy}{dx} \right) = 1 + \frac{dy}{dx}

\displaystyle 2x * (x-y) + x^2 - 1 = x^2 * \frac{dy}{dx} + \frac{dy}{dx}

\displaystyle \frac{dy}{dx} = \frac{2x * (x-y) + x^2 - 1}{x^2 + 1}

3 – Explicit

Solving the equation at the start of method 2 for y gives

\displaystyle y = \frac{x^3 - x}{x^2 + 1} .

Differentiating with respect to x gives

\displaystyle \frac{dy}{dx} = \frac {\left( x^2+1 \right) \left( 3x^2 - 1\right) - \left( x^3 - x \right) (2x+0)}{\left( x^2 + 1 \right) ^2}

Equivalence

Those 3 forms of the derivative look VERY DIFFERENT.  Assuming no errors in the algebra, they MUST be equivalent because they are nothing more than the same derivative of different forms of the same function, and a function’s rate of change doesn’t vary just because you alter the look of its algebraic representation.

Substituting the y-as-a-function-of-x equation from method 3 into the first two derivative forms converts all three into functions of x.  Lots of by-hand algebra or a quick check on a CAS establishes the suspected equivalence.  Here’s my TI-Nspire CAS check.

implicit

Here’s the form of this investigation I gave my students.

Final Example

I’m not a big fan of memorizing anything without a VERY GOOD reason.  My teachers telling me to do so never held much weight for me.  I memorized as little as possible and used that information as long as I could until a scenario arose to convince me to memorize more.  One thing I managed to avoid almost completely were the annoying derivative formulas for inverse trig functions.

For example, find the derivative of y = arcsin(x) at x = \frac{1}{2}.

Since arc-trig functions annoy me, I always rewrite them.  Taking sine of both sides and then differentiating with respect to x gives.

sin(y) = x

\displaystyle cos(y) * \frac{dy}{dx} = 1

I could rewrite this equation to give \frac{dy}{dx} = \frac{1}{cos(y)}, a perfectly reasonable form of the derivative, albeit as a less-common  expression in terms of y.  But I don’t even do that unnecessary algebra.  From the original function, x=\frac{1}{2} \longrightarrow y=\frac{\pi}{6}, and I substitute that immediately after the differentiation step to give a much cleaner numeric route to my answer.

\displaystyle cos \left( \frac{\pi}{6} \right) * \frac{dy}{dx} = 1

\displaystyle \frac{\sqrt{3}}{2} * \frac{dy}{dx} = 1

\displaystyle \frac{dy}{dx} = \frac{2}{\sqrt{3}}

And this is the same result as plugging x = \frac{1}{2} into the memorized version form of the derivative of arcsine.  If you like memorizing, go ahead, but my mind remains more nimble and less cluttered.

One final equivalent approach would have been differentiating sin(y) = x with respect to y and reciprocating at the end.

CONCLUSION

There are MANY ways to compute derivatives.  For any problem or scenario, use the one that makes sense or is computationally easiest for YOU.  If your resulting algebra is correct, you know you have a correct answer, even if it looks different.  Be strong!

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Straightening Standard Deviations

This post describes a bivariate data problem I introduced last month in my AP Statistics class, but it easily could have appeared in any Algebra 2 or PreCalculus course, particularly for those classes adapting to the statistics strands of the CCSSM and new SAT standards.  While I used the lab to introduce standard deviations of random samples, the approach also could be used if your bivariate statistics unit is occurs later in your sequencing.

My class started a unit on sampling when we returned in January.  They needed to understand how larger sample sizes tended to shrink standard deviations, but I didn’t want to just give them the formula

\displaystyle \sigma_{\overline{x}}=\frac{\sigma}{\sqrt{n}} .

I know many teachers introduce this relationship by selecting samples with perfect square sizes and see the population standard deviations shrink by integer factors (quadruple the sample size = halve the standard deviation, multiply the sample size by 9 = standard deviation divides by 3, etc.), but I didn’t want to exert that much control.  My students had explored data straightening techniques in the fall and were used to sampling and simulations, so I wanted to see how successfully they could leverage that background to “discover” the sample standard deviation relationship.

My AP Statistics students use TI Nspire CAS software on their laptops, so I wrote their lab using that technology.  The lab could easily be adapted to whatever statistics technology you use in your class.  You can download a pdf of my lab here.

LAB RESULTS AND REFLECTIONS

The activity drew samples from a normal distribution for which students were able to define their own means and standard deviations.  Students could choose any values, but those who chose integers tended to make the later connections more easily.

Their first step was to draw 2500 different random samples of sizes n=1, 4, 10, 25, 50, 100.  From each 2500 point data set, students computed sample means and standard deviations.  In retrospect, I should have let students select all or most of their own sample sizes, but I’m still quite satisfied with the results.  If you do experiment with different sample sizes, definitely run the larger potential sizes on your technology to check computation times.

One student chose \mu = 7 and \sigma = 13.  Her sample means and standard deviations are

sd3

It was pretty obvious to her that no matter what the sample size, \overline{x} \approx \mu, but the standard deviations were shrinking as the sample sizes grew.  Determining that relationship was the heart of the activity.  Obviously, the sample size (SS) seemed to drive the sample standard deviation (SD), so my student graphed her (SS, SD) data to get

sd1

We had explored bivariate data-straightening techniques at the end of the fall semester, so she tried semi-log and log-log transformations to check for the possibilities that these data might be represented by an exponential or power function, respectively.  Her semi-log transformation was still curved, but the log-log was very straight.  That transformation and its accompanying linear regression are below.

sd2

Her residuals were small, balanced, and roughly random, so she knew she had a reasonable fit.  From there, she used her CAS to transform (re-curve) the linear regression back to an equation for the original data.

sd4

It made sense that this resulting formula not only depended on the sample size, but also originally on the population standard deviation my student had earlier chosen to be \sigma = 13.  Within reasonable round-off deviations, the numerator appeared to be the population standard deviation and the exponent of the denominator was very close to \frac{1}{2}, indicating a square root.  That gave her the expected sample standard deviation formula, \displaystyle \sigma_{\overline{x}} = \frac{\sigma}{\sqrt{n}}.

I know this formula is provided on the AP Statistics Exam, but the simulation, curve straightening, linear regression, and statistical confirmation of the formula were a great review and exercise.  I hope you find it useful, too.