The question I pose to students in Introductory Calculus might take on the following form:

A really REALLY REALLY long ship is sailing parallel to a shoreline, 2 km off shore. A lighthouse on the shore projects its beam in a circle; the angular velocity at the point of rotation is given. Determine the linear velocity, from the lighthouse keeper’s perspective, of the beam’s leading edge as it passes by the ship from stem to stern as the ship sails on.  Then consider another scenario; the ship’s frisky dog can run 30 km/h.  Determine the beam’s angular velocity that would allow the dog to keep up to its leading edge as it sweeps past the length of the ship under the following conditions:

(a) the ship is 2 km off shore and positioned at 60 degrees East of North relative to the lighthouse.

(b) the ship is 500 metres off shore at 60 degrees East of North relative to the lighthouse.

(c) the ship is 500 metres off shore due north of the lighthouse.

Although this example is obsolete with the advent of GPS, the mathematics in it is not. This is a very effective “end game” to pursue for students in Introductory Differential Calculus as it requires them to find the instantaneous velocity at a specific point. In fact, many such velocities could be determined from the perspective of an observer on the ship, connecting each to the relative position on the accompanying sinusoidal function……..when is that velocity positive, negative, etc……

The math learned in this example could be applied in the same way to many other similar scenarios.  I will be developing meaningful lessons over the course of the next several weeks, the purpose of which is to empower those taking an interest in solving the problem above……..and many many more.  They will be hand-written and saved as photo files; links to those notes will be added below as they become necessary.

POWER RULE

A good amount of work must be accomplished before we can get to the solution of our light house problem.  To get started, I have written eight pages of notes introducing the notion of derivative using first principles.  To view those notes, click on the link below. 

First Principles (Polynomial & Radical Functions) – Samuelson

The notes found in the link above made reference to the factorization of a difference of cubes shown below:

x^3 – 27 = (x – 3)(x^2 + 3x + 9)

I will illustrate a very powerful method of factoring such things; this method is based on a geometric representation of the factoring problem.  The original difference of cubes (or sum of cubes) represents the area of a rectangle; the factors of that become the side dimensions of said rectangle.  I came across a “gold mine” of information recently in which this method of factoring is included.  James Tanton has put forth a vast collection of video lessons on YouTube to help students and teachers alike to better understand mathematics.  The link below takes you to one such lesson describing the factoring method to which I refer.

James Tanton: Synthetic Division: How to understand it by NOT doing it! (VIDEO!)

The following link contains notes showing more examples of Tanton’s division.  These examples ultimately lead to a deductive proof of the power rule for finding derivatives; read through carefully.

Proof of Power Rule – Samuelson

Our lighthouse problem represents cyclic motion; the function describing this will therefore be a sinusoidal function, perhaps represented by  y=sin(x).   Since we are asked to determine instantaneous velocity at a given point, the derivative of the function will be required.  The four pages of notes found in the link below describe the process of finding that derivative.

Fundamental Trig. Limits – Samuelson

PRODUCT, QUOTIENT & CHAIN RULES

The last thing we need in place so that we can solve our lighthouse problem is the “Chain Rule”.  You’ve already seen the Power Rule and its proof; this rule is inherent in all of Differential Calculus and, as such, will be utilized within the Chain Rule as well.  Clicking on the link below will take you to notes describing this rule and two others; the Product Rule and the Quotient Rule.  I will post examples using these rules to find derivatives of various functions at a later date.

Product, Quotient & Chain Rules – Samuelson

We are getting ever closer to a meaningful solution to the lighthouse problem presented at the outset; the underlying logic behind this solution is very simple.  Several rules have been introduced and proven (hopefully to your satisfaction) and the concept of instantaneous velocity should now be firmly entrenched in your thought process.  In the notes below, I have given a few examples of functions; the differentiation rules are illustrated there in finding the correspnding derivatives.  I’ve also taken this opportunity to introduce “acceleration” from a “derivatives” perspective.

Applications of First & Second Derivatives – Samuelson

IMPLICIT DIFFERENTIATION & RELATED RATES

The notes contained in the link directly below introduce you to Implicit Differentiation and Related Rates; these rate problems will typically involve three variables, one of which is “time”.  Our lighthouse problem is, itself, a related rate and can be solved through the explicit process.  Having said that, the implicit procedure lends itself in a more consistent manner to a wide range of related rate problems.  Consequently, I thought it appropriate to describe that process at this time and provide examples of its use in solving several related rate problems.  These examples will provide the remaining tools required in completing your task.

Implicit Differentiation & Related Rates – Samuelson

Additional links for reference purposes are found below.

Implicit Differentiation

Related Rates

The solution to our lighthouse problem is shown on page 6 of the notes contained in the link directly below.  A description of how angular and linear velocity relate to one another occurs in the pages preceding that.  A firm understanding of this will enable you to reason your way to solution to all such “problems” in a much more meaningful way.

Angular & Linear Velocity – Samuelson

Our lighthouse problem has now been solved.  The procedures and thought patterns established through this process can be applied in the same manner to set up and solve many such problems; I may add some of those to this page at a later date.  For now, I will turn my attention to Integral Calculus; my goal in this will be to derive the arc length of a circle of given radius.

Please visit  James Tanton, The Republic of Math and Cut The Knot for all things related to Mathematics.

As a footnote to our exploration above, the link below may be of interest.

Superluminal Motion

The ultimate goal of this section will be to derive the following:

(a) Formulae for volume of the cylinder, cone, sphere, and various parabaloids using two methods; the slab (disk) method and the shell method.

(b) Formulae for surface area of the cone and sphere using similar methods to those mentioned above.

(c) Formulae for area of the circle and ellipse.

(d) Formula for arc length of a circle given its central angle. For this, we will reference the Mean Value Theorem and our knowledge of differentiaion.

Along the way we will be introduced to several methods of integration, all of which will be called upon when needed in meeting each of our goals stated above.

I will be developing notes once again over the course of the next several weeks addressing this topic. These notes will be hand-written, scanned and saved as photo files. I will place links at appropriate intervals throughout this page that will lead you to those notes. I will also include other links that will serve to reinforce the contents of my notes; these links will be added to the sidebar of my Math 31 blog for easy referencing if you choose to search them out at a later date.

POWER RULE OF INTEGRATION

Integration and differentiation are inverse processes of one another. Differential Calculus was not the first of the two to be developed but is typically easier to learn; that is why it is taught prior to its Integral counterpart. Differential Calculus is often referenced when integrating functions because of this. Having said that, I have attempted to explain Integral Calculus on its own with minimal reference to differentiation. I did “slip” and found myself making just such a reference on page 12 in an attempt to make a justification. Hopefully I will be fogiven for this small indiscretion. The link directly below contains 12 pages of hand-written notes illustrating how I introduce the concept of integration. More notes will follow over the next several weeks as we progress through more complex functions.

Introduction to Integral Calculus – Samuelson

There are many very good references availble to further assist in the understanding of this topic. The information contained in the links below will resemble much of what is contained in my notes and ellaborate further on those points.

The Fundamental Theorem of Calculus

Fundamental Theorem (2)

What “dx” Actually Means

Riemann Sum

Anti-Differentiation & the Power Rule

Antiderivatives/Indefinite Integrals

Before we can derive the formula for volume (or area) of anything, a thorough understanding of the process of integration is required. There are several different methods of integration available to us; the method chosen depends on the type of function being integrated. We will explore these various types of functions and the appropriate method of integration to be used in each case; upon completion of this requirement, we will be equipped to reach our goal.

INTEGRATION BY PARTS

Let us reflect back on the process of differentiation for a few moments. This entire process began with our wish to find an expression representing instantaneous velocity at any given point on a function; we found that this value was represented by the slope of the tangent to the function at that point. The expression for this was discovered through first principles (delta method), and is now referred to as the derivative of the function; we used first principles to find derivatives of several different types of functions in order to more fully conceptualize the notion. As our functions became more complex, this process became much more arduous and time consuming so we were introduced to some helpful rules to assist us in our work; they are the power rule, the product/quotient rule, and the chain rule.

Integration is essentially the “inverse” process of differentiaion and has its own set of “rules” that can assist us in determining the area, among other things, under a curve in a given interval. We have already been introduced to the power rule for integration earlier on this page. The power rule works very well for relatively simple functions; as these functions become more complex the neccessity for other rules emerges, just as it did with differential calculus. The next “rule” that we will familiarize ourselves with is referred to as Integration by Parts, a process tied directly to the product rule of differentiation; the notes contained in the link below will illustrate this. In these notes, a comparison between integration using the power rule and integration by parts will be made early on as they relate to simple polynomial funtions. The purpose behind this is to help us become accustomed to this new “rule”; as we work through the notes, the functions being integrated will evolve into more complex ones, hopefully leaving us with a deeper appreciation for this new process.

Integration by Parts – Samuelson

For additional information on Integration by Parts, click on the links below. For easy reference, these links can also be found on my Math 31 Blog in the sidebar under Integral Calculus.

Integration by Parts

Integrals Tutorial

INTEGRATION BY PARTIAL FRACTIONS

We are now ready to move on to the next stage of our journey, that being Integration by Partial Fractions. This process often results in a situation requiring the natural logarithm and hence, some knowledge of its inverse, the Euler constant. That being the case, a brief overview of those concepts would be appropriate at this time. In the links directly below, James Tanton first illustrates Euler’s constant (e), relating it to the compound interest formula. This is followed by the derivation of Euler’s Formula by means other than the Taylor Series; although this is not a requirement at the high school level, it is nonetheless very enlightening and should be viewed by all. Beneath Tanton’s offerings are two additional links, one providing an overview of the natural logarithm; the final link directly below provides a description of Integration by Partial Fractions. A small investment of time spent in each of these will serve us well as we proceed on through this process.

Euler’s Constant “e”

Deriving Euler’s Formula

Natural Logarithm

Integration by Partial Fractions

The link below contains 23 pages of hand-written notes describing the Integration of Rational Functions. The focus of these notes is centered on the decomposition of rational functions into the sum of partial fractions; all but one of the examples chosen require the introduction of the natural logarithm to the integration process. For this reason, the first several pages in this set of notes focus on that function and its inverse, y = e^x. The example shown on pages 20 and 21 does not require the natural logarithm; it is integrated by Trigonometric substitution instead, a process that will be explained at a later date. For now we will work towards the mastery of Integration by Partial Fractions.

Integration by Partial Fractions – Samuelson

INTEGRATION BY TRIGONOMETRIC SUBSTITUTION

We are now within reach of fulfilling our goals stated at the outset; these goals are to derive the following:

(a) Formulae for volume of the cylinder, cone, sphere, and various parabaloids using two methods; the slab (disk) method and the shell method.

(b) Formulae for surface area of the cone and sphere using similar methods to those mentioned above.

(c) Formulae for area of the circle and ellipse.

(d) Formula for arc length of a circle given its central angle. For this, we will reference the Mean Value Theorem and our knowledge of differentiaion.

Before proceeding any further a “new” method of integration must be introduced, that being Integration by Trigonometric Substitution; this was eluded to in the previous set of notes. The example illustrated on pages 20 & 21 in my notes on Integration by Partial Fractions revealed that the antiderivative of the expression “[3/(x^2+1)]” was “[3arctan(x)]“. This method will be explained in my next set of notes; we will also be made aware of why the “Natural Logarithm Method” WILL NOT work in finding the antiderivative of this expression. My notes on Integration by Trigonometric Substitution can be found in the link directly below.

Integration by Trigonometric Substitution – Samuelson

Visit the links below for additional information on this method of integration. The first one takes you to a lecture given by a guest instructor at MIT.

Trigonometric Substitution & Polar Coordinates

Integration by Trigonometric Substitution (1)

Integration by Trigonometric Substitution (2)

Integrating Algebraic Functions

CIRCLE AREA & ARC LENGTH

The notes contained in the link directly below show the derivation of circle area using some of the integration techniques illustrated earlier; the formula for arc length is also derived, with reference to the Mean Value Theorem. This formula is applied to several functions to determine arc length over a given interval and is ultimately used to prove the formula for circumference of a circle. This circumference formula is then used to once again prove the formula for area of circles, this time using the “shell” method of integration.

Circle Area & Arc Length – Samuelson

The links below reinforce the ideas presented in my notes and provide much more information on those topics.

Arc Length

Arc Length – Riemann Sum

Parametric Equation

Mean Value Theorem – Interactive

Arc Length, Area, and the Arcsine Function

SURFACE AREAS & VOLUMES OF REVOLUTION

When the graph of a 2-dimensional function is revolved about a vertical or horizontal axis, the result is referred to as a 3-dimensional solid.  By applying some of the concepts we have already learned, the surface areas and volumes of many such solids of revolution can be determined with relative ease.  The notes contained in the link directly below call on our knowledge of arc length to assist in determining the area of surfaces of revolution; direct reference is made here to the “onion proof” that was introduced in the previous set of notes.  The premise behind this approach is to find the sum of the areas of infinitely many, infinitely narrow “bands” that are “wrapped” around the surface of revolution.  If one of these “bands” was cut and then stretched out, it would form a rectangle whose length would be equivalent to the circumference (arc length) of the surface of revolution at that point; the width of this “band” would ultimately be represented by “dx” or “dy”, depending on the axis of revolution. 

Several examples involving the calculation of surface areas are illustrated in my notes; following those are two methods of determining volumes of solids of revolution, the first being the “Shell Method”.  This method once again reflects back on the “onion proof” that was used previouly to derive the formula for area of a circle.  The idea behind the “onion proof” is that “layers” (bands) are added uniformly around the circumference until the desired radius is attained.  The areas of these infinitely narrow “layers” are then added to determine the area of the circle; this circle essentially forms the base of the cylindrical shells that emerge as a result of revolving a function about an axis.  The “height” of these cylindrical shells is determined by the function itself and is always perpendicular to its circular base; the volume of each cylindrical shell is the product of its “height” and the area of its circular base.  As layers are added to the circular base, the “height” of these cylindrical shells also changes, governed by the function itself; the volumes of these cylindrical shells, having infinitely thin lateral surfaces, are ultimately added to determine the volume of the solid itself.  After working through several examples using the Shell Method,  the volumes of many of the same solids are calculated using the Slab (Disk) Method.  With this method, cylindrical “slabs” of infinitely small “height” are stacked together, their individual volumes added to determine the volume of the solid of revolution.

Surface Areas & Volumes of Revolution – Samuelson

Additional links have been included below that will reinforce (and go beyond) the notions presented above.

Area of a Surface of Revolution

Volumes of Revolution

Torus

The Ellipsoid

Arc Length, Surface Area and Polar Coordinates

Polar Coordinates & Parametric Equations