Transformations Limit Continuity Derivative Tutorial 12.12.2014 in math Transformations Limit Continuity Derivative

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Transformations in biology (molecules) and computers (computer graphics)

Computer graphics is often reduced to projections, which are often end up being stretches and other geometrical transformations

Odd function graph is invariant to 180 degrees rotation around (0, 0) or around OZ.

Even function graph is invariant to reflection in OY or 180 degrees rotation around OY.

Congruence for translation, rotation, reflection

Similarity for enlargement

Stretches cause the image not to be congruent and not to be similar to the original shape

Moore's law "Moore's law" is the observation that, over the history of computing hardware, the number of transistors in a dense integrated circuit doubles approximately every two years. The observation is named after Gordon E. Moore, co-founder of the Intel Corporation, who described the trend in his 1965 paper. His prediction has proven to be accurate, in part because the law now is used in the semiconductor industry to guide long-term planning and to set targets for research and development. The capabilities of many digital electronic devices are strongly linked to Moore's law: quality-adjusted microprocessor prices, memory capacity, sensors and even the number and size of pixels in digital cameras. All of these are improving at roughly exponential rates as well. This exponential improvement has dramatically enhanced the effect of digital electronics in nearly every segment of the world economy. Moore's law describes a driving force of technological and social change, productivity, and economic growth in the late twentieth and early twenty-first centuries. The period is often quoted as 18 months because of Intel executive David House, who predicted that chip performance would double every 18 months (being a combination of the effect of more transistors and their being faster). Although this trend has continued for more than half a century, "Moore's law" should be considered an observation or conjecture and not a physical or natural law. Sources in 2005 expected it to continue until at least 2015 or 2020. The 2010 update to the International Technology Roadmap for Semiconductors predicted that growth will slow at the end of 2013, however, when transistor counts and densities are to double only every three years.

Complexity in biology and computing

Limits exercises

Continuous function In mathematics, a continuous function is, roughly speaking, a function for which small changes in the input result in small changes in the output. Otherwise, a function is said to be a discontinuous function. A continuous function with a continuous inverse function is called a homeomorphism.

Continuous function (continued) Continuity of functions is one of the core concepts of topology, which is treated in full generality below. The introductory portion of this article focuses on the special case where the inputs and outputs of functions are real numbers. In addition, this article discusses the definition for the more general case of functions between two metric spaces. In order theory, especially in domain theory, one considers a notion of continuity known as Scott continuity. Other forms of continuity do exist but they are not discussed in this article.

Continuous function (continued) As an example, consider the function h(t), which describes the height of a growing flower at time t. This function is continuous. By contrast, if M(t) denotes the amount of money in a bank account at time t, then the function jumps whenever money is deposited or withdrawn, so the function M(t) is discontinuous.

Difference quotient

Homework is due 17 December 2014. It is on the web site.