College Mathematics Archives - Page 8 of 30

The 0.0001 Time Rate Approach, an Extended Application

April 25, 2013 Armand Gelig Macapagong Calculus and Analysis, College Mathematics

In the previous post, I introduced to you the 0.0001 Time Rate approach This post is an extended application of the said technique on curvatures and radius of curvatures.

Curvature is defined (by The Facts on File Dictionary of Mathematics) as the rate of change of the slope of the tangent to a curve. For each point on a smooth curve there is a circle that has the same tangent and the same curvature at that point. The radius of this circle called the “radius of curvature”, is the reciprocal of the curvature, and its center is known as the center of curvature. If the graph of a function y=f(x) is a continuous curve, the slope of the tangent at any point is given by the derivative dy/dx and the curvature is given by

$C = \displaystyle\frac{\displaystyle\frac{d^2y}{dx^2}}{(1+(\displaystyle\frac{dy}{dx})^2)^{3/2}}$

and the radius of curvature is given by

$RC = \frac{1}{C} = \displaystyle \frac{(1+(\displaystyle \frac{dy}{dx})^2)^{3/2}}{\displaystyle \frac{d^2y}{dx^2}}$

where: $\frac{dy}{dx}$ is the derivative of $y$ with respect to $x$ and $\frac{d^2x}{dx^2}$ is the second derivative of y with respect to x or in simpler terms, it is the derivative of $\frac{dy}{dx}$ . » Read more

Leave a comment derivative of a function, derivatives, radius of curvature, time rate approach

13 Calculus Tutorial Sites, 22 000+ Solved Problems

March 11, 2013 GB Calculus and Analysis, High School Calculus, Resources and Freebies

If you want to learn Calculus, the websites below will most likely help to you. Most of these websites contain conceptual and intuitive explanations of Calculus concepts and most of them are interactive.

All in all, they contain more than 22,000 calculus tutorials and computations. » Read more

2 comments calculus help, calculus tutorials, calculus video tutorials, interactive calculus help, interactive calculus tutorials, paul's online math notes, visual calculus

Fractals: A Different Type of Geometry

March 3, 2013 GB College Geometry, College Mathematics

The Dimensions in Between

The mathematics that we have learned since elementary school is a bit beautiful. A line is perfectly straight, the path of the ball thrown upward is a parabola and the shape of the earth we see in books is a sphere. The truth is, the lines, parabolas, and spheres are “flawless models” of the real world. A straight line that can be drawn using a ruler or a meter stick is not perfectly straight (try using a magnifier). The shape of the path of the ball thrown upward is not all the time a symmetric parabola. Lastly, Earth is not a perfect sphere, it bulges in the equator, it has mountains, valleys, and trenches.

Clearly, the perfect mathematics that we know is an ideal notion and it does not have a very close resemblance of reality. It also appears that the mathematics, particularly the Geometry that we have learned in school is not enough to describe the real world. The dimensions that we have talked in the last post, cannot simply apply to these types of “irregularities.” » Read more

Leave a comment fractal geometry, fractals, hilbert's curve, space filling curve, the koch's snowflake, what are fractals

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