Archive for the ‘Differential Equations’ Category

Here’s problem involving a falling object and the speed at which its shadow travels along the ground. As usual, in related rates, once a relationship between the variables involved has been established, the calculus required to reach its conclusion is very straight forward.

In order to make efficient use of time, these problems provide students the opportunity to practice simple differentiation procedures. In addition, the graphs provided below open the door to a discussion on the Mean Value Theorem of differential calculus, serving to either introduce or reinforce that concept.


Falling Ball


Click on the link provided here to interact with the falling ball and its shadow.



The ball’s displacement from its release point was provided in the image above. As a review (since integral calculus has already been introduced), that displacement formula is once again derived through basic differential equations; this is shown directly below.

Acceleration, Velocity and Displacement



I’ve included solutions for  t=1  and  t=2  below. In keeping with my belief that students can learn effectively through comparison and contrast, three varied methods are shown.



Thanks for reading.

The main subject of this entry was originally planned as an optimization problem involving differential calculus only; its been slightly modified. This more interesting approach provides the derivative up front, presenting students with three separate tasks to pursue from that point. As a consequence, students are reintroduced to differential equations and curve sketching.

A talking point emerges as well: Is there a difference between derivatives and differential equations?


Inscribed Triangle of Maximum Area

Inscribed Triangle animation


Click on the link provided here to explore area of the inscribed triangle.


Thanks for looking.

In the spirit of consistency, several additional examples supplementing my introduction to related rates are included here. As seen in the first example below, past concepts are revisited and, in turn, connected to newly introduced ones. Mixing things up from that point on provides students with a variety of perspectives on the same theme that serves to further cement understanding.


Separable Differential Equation

Circle area Derivative


Three Approaches to Related Rates

Circle area Differential Equation


Application to Conical Container

DiffEqua Realted Rates2



Rate of Change of Arc Length



The approach to solving each of these related rate problems is the same; identify what is known, what is desired and then connect the two.


Thanks for reading.

Surface areas of curved 3-dimensional solids tend to be much more difficult for students to conceptualize than those whose sides do not stray from a “level” plane. These will eventually be addressed but we will first discover how to calculate lengths of curves.

The circle will once again be called upon to initiate this exploration; the image below illustrates, in part, the method of exhaustion that Archimedes utilized to arrive at his estimate for π.

Archimedes Circumference2

Click on the link here to interact with what Archimedes revealed.


……..and now this. Was Archimedes wrong???



If the fellow above had joined pairs of points at each successive corner with a line segment (hypotenuse) and based his calculation for circumference on the sum of those, he would have found that Archimedes was correct all along.


Arc Length Formula

As  Δs approaches zero in this exploration, its length becomes a more accurate estimate for the arc length near the “point” of tangency (there are always two points in very close proximity). The end result through the limiting process shown directly below is the formula for arc length.

ArcLength Circle

Its always beneficial for students to work through several examples to cement their understanding of new concepts and related procedures; my preference is to provide examples that are already familiar to them. Calculating arc length (in this case) can then serve as a verification and acceptance of the new concept is achieved with confidence. The offering directly below and the link that follows connects the arc length formula to the Pythagorean theorem.

Arc Length

The formula for arc length is based on the Pythagorean theorem; it is therefore not surprising that they produce the same lengths on linear functions.


The real power of the formula for arc length lies in its applications to curves. Since students have known the circle’s circumference for several years, it is appropriate to now derive 2πr using our new tool. This is shown below and once again brings trigonometric substitution into play.

Circumference of the Circle

Arc Length (Circumference of Circle)

Once the circle’s circumference has been established using the arc length formula, the integration process can be further solidified by using arc length to once again calculate the circle’s area.


The formula for arc length will once again be employed in deriving the formulas for surface area of the sphere and cone.


Thanks for reading.


Having side-by-side comparisons can be valuable experience as similarities (and differences)  are more readily apparent; students can gain a deeper understanding of the nuances from one to the next. For example, its worth noting that the rate of change of a circle’s area with respect to its radius is equal to that circle’s circumference; a similar relationship exists between the volume and surface area of a sphere.

Investing some thought into these and other subtleties can go a long way towards increasing one’s intuitive feel for, and enjoyment of this discipline.


Another comparison worth checking out is quadratic vs exponential growth.

Thanks for reading.

In order to achieve some degree of continuity, I continually strive to weave together concepts, not only within my own “area of influence”, but across other disciplines as well. Physics is naturally folded into the fabric of calculus for obvious reasons; others disciplines, not so much.

I wanted to raise awareness in students of how calculus appears in applications relating to Biology and Chemistry; logistic growth is the obvious choice for the former and is relatively straight forward once students have a feel for differential equations.

The notes directly below make clear (I hope) the distinction between two types of growth from the context of differential equations.  The exponential growth model below will be expanded upon to eventually derive the well-known formula for logistic growth.

Separable Diff Equations2


The application to Chemistry that was alluded to earlier will require First Order Differential Equations, another “diversion” that can be pursued when a change of pace is needed. We will hopefully be afforded the time to develop an adequate understanding of this before semester’s end.

Thanks for reading.

A basic understanding of differential equations has already been established through our introduction to integration. In addition, rates of change have also been linked to our brief study of differential calculus. Students now require a period of time to work through basic problem solving scenarios relating to both differentiation and integration to develop an acceptable degree of fluency.
Every now and then, a diversion from the “daily grind” can be well-received. Separable differential equations and an introduction to slope fields will be one such “diversion” and will be shared with students when deemed appropriate. Since students are very familiar with quadratic functions, will we begin there.
The equation dy/dx=x will be presented and analyzed from a “rate of change” perspective at various positions on the Cartesian plane. Once this slope field has been sketched, the shape of the parent function becomes readily apparent; the need for initial conditions arises to uniquely define each member from the family.
The image directly below sees through the completion of the scenario described above. This procedure is then repeated for other basic separable differential equations, all of which produce slope fields that are recognizable to students; these appear below our parabola example.


Differential Equation: dy/dx=kx, where k=2

Constant of Proportionality Given

Quadratic 2x(2)



Differential Equation: dy/dx=kx^2, where k=3

Constant of Proportionality Given

Cubic 3x^2 (2)


Constant of Proportionality Not Given

Cubic 3x^2(3)



Differential Equation: dy/dx=-x/y

Circle -x over y (2)

For the circle above, I’ve included two acceptable treatments.

The indefinite integrals require that initial conditions be substituted in after the fact to solve for the constant of integration.  The second version has the initial conditions included as bounds of integration, resulting in definite integrals; same result.


Solving these differential equations tie together quite nicely the two sides of calculus to which students have been introduced. These examples also set up other such equations and problems that will be presented in the not too distant future, such as  Quadratic vs Exponential Growth.

Thanks for reading.