Topic 4.2 Rates of Change & Differential Equations

Syllabus

Applications of Differentiation

4.2.1 use implicit differentiation to determine the gradient of curves whose equations are given in implicit form

4.2.2 examine related rates as instances of the chain rule: \frac{dy}{dx}=\frac{dy}{du}\times\frac{du}{dx}

4.2.3 apply the incremental formula \delta y\approx\frac{dy}{dx}\delta x to differential equations

4.2.4 solve simple first order differential equations of the form \frac{dy}{dx}=f(x); differential equations of the form \frac{dy}{dx}=g(y); and, in general, differential equations of the form \frac{dy}{dx}=f(x)g(y), using separation of variables

4.2.5 examine slope (direction or gradient) fields of a first order differential equation

4.2.6 formulate differential equations, including the logistic equation that will arise in, for example, chemistry, biology and economics, in situations where rates are involved

Modelling Motion

4.2.7 consider and solve problems involving motion in a straight line with both constant and non-constant acceleration, including simple harmonic motion and the use of expressions \frac{dv}{dt}, v\frac{dv}{dx}\text{ and }\frac{d}{dx}\left(\frac12 v^2\right) for acceleration

Lessons

Implicit Differentiation

View the Prezi on Implicit Differentiation.

Differentiating Parametric Equations

View the Prezi on Differentiating Parametric Equations.

(A nice, short lesson, this one.)

Tangent Equations

View the Prezi on Equations of Tangent Lines.

(Another short lesson)

See this page for interactives that step through the process.

Solving Differential Equations

Small Change

The following Prezi is about the incremental formula \delta y\approx\frac{dy}{dx}\delta x. This is part of the year 12 Methods course, and you should already be familiar with it. The Specialist course covers the application of this to differential equations (see the following Prezi).

View the Prezi on Small Change and Differential Equations.

Note: this Prezi has no audio.

Related Rates

Also see the sequence of lessons and exercises at https://www.khanacademy.org/math/differential-calculus/derivative-applications/rates-of-change/v/rates-of-change-between-radius-and-area-of-circle.

See also Mister Woo’s series of three videos on Rates of Change.

Slope Fields

See the sequence of videos and exercises at https://www.khanacademy.org/math/differential-equations/first-order-differential-equations/differential-equations-intro/v/creating-a-slope-field.

See the lesson sequence starting at http://www.casio.edu.shriro.com.au/lesson20a.php for details of how to generate slope fields on the Classpad. Note that this lesson assumes use of Classpad Manager software that students will not have, but you should be able to follow along using your classpad instead.

Formulating Differential Equations

See the sequence of videos at https://www.khanacademy.org/math/differential-equations/first-order-differential-equations/logistic-differential-equation/v/modeling-population-with-differential-equations.

Rectilinear Motion

Some of the basics are covered nicely (with practice questions) at http://17calculus.com/integrals/linear-motion/. There is also a wikibooks page that covers the basics: https://en.wikibooks.org/wiki/Physics_with_Calculus/Mechanics/Motion_in_One_Dimension. There’s a youtube playlist http://www.youtube.com/watch?v=g5PpEM7B4uA&list=PLfviI50k54GA-O2zzVSHTWzogNLAaNoHI – again, just the basic, straightforward bits of the topic. (This link and the previous one are referenced on the 17calculus page.)

Khan Academy has a video that covers the basics: https://www.khanacademy.org/math/integral-calculus/indefinite-definite-integrals/indefinite-integrals/v/antiderivative-acceleration and a sequence of videos and exercises that covers mean value of a function in a way that relates it nicely to area under a curve: https://www.khanacademy.org/math/integral-calculus/solid-revolution-topic/function-average-value/v/average-function-value-closed-interval. KA also has a nice “distance travelled” example at https://www.khanacademy.org/math/differential-calculus/derivative-applications/motion-along-line-derivatives/v/total-distance-traveled-by-a-particle.

Simple Harmonic Motion

Some of the basics are covered in a video from iitutor: http://youtu.be/urpo-MbpBWQ and another from Dr C Davies: https://www.youtube.com/watch?v=FQZgwygOeO8

A very nice animation can be found at http://www.montereymotiongraphics.com/samples/mmg_ph05.swf (but it’s Flash, so will not work on many devices).

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