27 Apr 2006 - 23:38

Dr. Ian's overview of calculus

I was just looking up the meaning of 'delta' at Math Forum, and I found Dr. Ian's overview of calculus.

I haven't read it, but I have a distant memory Dr. Ian is one of the Math Forum people whose explanations I've liked before.

Also, I need an overview of calculus. Or I will be a couple of years from now.

-- CatherineJohnson - 27 Apr 2006

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How much does this overview mean to anyone who hasn't done calculus already?

-- TracyW - 28 Apr 2006

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I agree with Tracy.

Besides, it was a bad answer. You have to know what degree program "Pat" is going into, and more importantly, what math Pat already had. If Pat only had one year of algebra in high school, then it's not going to be an easy path. I suspect Pat is trying to get a feel for how much work it's going to require. The answer assumed that Pat knew quite a bit.

As for an overview of calculus, I don't like it. Dr. Ian's answer will just scare and confuse most.

I like to keep it simple. In algebra, you learn about variables, functions, and graphing - that they have shape. In trig you learn about fancier functions that have useful shapes, like waves on the ocean. Shapes represent real life and it's useful to define these shapes with equations and study them. In calculus, you learn that there is much more information you can figure out from the equation (shape) of these functions, like slope and area. Calculus has two parts. Differentiation gives you the slope of a function at any location and integration gives you the area under the function. It's a good idea to get really good at algebra before you dive into calculus.

If you have a function that defines the cross-section shape of a hill, a student can find out where the steepest part (largest slope) of the hill is using differentiation. If you have the equation of the curved shape to one edge of a garden and you wish to calculate the area to buy fertilizer, you would integrate the equation to find the area under the curve.

In either case, if you don't have the equation for a shape, you can figure it out. A common method is to measure points along a curved path in terms of X and Y using some convenient reference line or coordinate system. You can then use curve-fitting techniques to fit a curve through the points to create this equation.

As an aside, if you fit a curve through all of the points in a data set, then it is called interpolation. For statistics, with hundreds or thousands of data points (usually scattered), one uses some form of best or least-squares fit of the data - the curve does not go exactly through each point. Interpolation is used to describe real objects. Reverse engineering in geometry is the process of collecting very many data points from real objects so that a computer/mathematical model (equations) can be constructed. Once you have these equations you can use calculus to figure out all sorts of things about the object, like surface area, volume, centroids, and curvatures. You can even use the model to create photo-realistic 3D images. You use calculus and the equations of the model to determine the normal to the surface at any point. This allows you to figure out how light rays will reflect off of the surface. It all starts with the equation(s) of the object. Algebra and trig gives you the basic skills and calculus lets you have have fun - do some real work.

Back to the original problem about delta. Delta is generically used to describe a change, usually small. This is not to be confused with epsilon , which is used just to describe a small amount. Delta is important in calculus because slope (differentiation) can be looked at as a delta; a change in Y divided by a change in X.

-- SteveH - 28 Apr 2006

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Dr Ian's answer causes fear and confusion because that is the typical experience I would imagine of someone jumping into calculus. Congratulations! You have entered Calculus! A class where we jump from strange new theory(limits) to strange new theory(infintesially small changes in variable x) only to end up practicing differentiation and integration for months and months.

Calculus, the class, deals with the analysis of continuous functions. Algebra allows one to determine the value of a function at any point while calculus allows one to find the slope of a function at any point or the area contained underneath an function between any two points.

Of course this new technique requires the mastery of a new set of tools, which are mainly differentiation and integration. Mastery learning rears its ugly head again because the class revolves around simplifying equations into a form that one can apply the known differentiation/integration rules. Like Dr. Ian says "you learn 47,000 different tricks for computing derivatives".

To succeed in calculus you need to master functions: plotting, solving and especially factoring. Lots of rote routine work. The learning of the math really occurs in an application class such as physics:mechanics when instantaneous velocity or acceleration are discussed.

-- SeanPrice - 28 Apr 2006

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I'll have to read it as a naive calculus aspirant & see how it affects me....

-- CatherineJohnson - 29 Apr 2006