## Derivative of e^x: Additional Perspectives

Posted: February 26, 2016 in Calculus: An Introduction
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In accordance with normal procedures, previously addressed topics are once again reflected on in this entry. Directly below, the derivative of e^x is dealt with from the perspectives of slope fields and differential equations. Beneath that, the derivative of e^x is considered as a Taylor Series.

Slope Field

Taylor Series

It is easy for high school students to determine that dy/dx = y for the Taylor Series representation of e^x above. This sets the stage very nicely for an exploration into power series which, in turn, lead directly to some other very cool “discoveries”. Follow the link provided for a more conventional extraction of the derivative of e^x.

Reference

Courant, Richard., John, Fritz (1999).  Introduction to Calculus and Analytics: Classics of Mathematics. New York, NY:  Springer-Verlag Berlin Heidelberg.

## Separable Differential Equations and Slope Fields

Posted: February 15, 2016 in Differential Calculus, Differential Equations, Slope Fields
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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

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

Constant of Proportionality Given

Constant of Proportionality Not Given

Differential Equation: dy/dx=-x/y

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.