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Transcript of Derivative Divas
Divas What is rate-of-change? What is mathematics? What is calculus? Ambrocya Burge
Andrea Steevens Math is greek from máthema. Mathmatics: what one learns; what one gets to know. Math is anything from addition to geometry to trigonometry. What is a limit? How can calculus ideas be explained graphically? Math is learned at all ages and applies to one's entire life. What is a derivative? What are some applications of limits? What causes a limit to not exist? Math can teach you how to dance! How does calculus help us determine it? Example Is it an abstraction? Limits in non-mathematical terminology the value that a function "approaches" as the input or index approaches some value Those who do not appreciate math are those who do not understand what math is all about. That is why the nature of math desperately needs to be explained. Simply put, math is about solving problems.
Calculus is Latin for stone, and the ancient
Romans used stones for counting and arithmetic. Calculus was developed by two different men in the seventeenth century. Gottfried Wilhelm Leibniz (1646-1716), a self-taught German mathematician, and Isaac Newton (1642-1727), an English scientist, both developed calculus in the 1680s. Calculus is the study of how things change.
It provides a framework for modeling systems in which there is change,
and a way to deduce the predictions of such models.
Calculus can find a way to
increase money and also to find acceleration. How are they applied? A derivative is a measure of how a function changes as its input changes. More practically, a derivative is the slope of the tangent line drawn at any point of a function. Most real-world applications of derivatives involve optimization. Example Derivatives are also used in the world of physics. The derivative of position is velocity. The derivative of velocity is acceleration. The derivative of acceleration is jerk (the rate of change of acceleration). We need to enclose a field with a fence. We have 500 feet of fencing material and a building is on one side of the field. Determine the dimensions of the fence that will enclose the largest area. A=xy 500=x+2y x=500-2y A=(500-2y)y A=500y-2y² A'=500-4y y=125 x=250 Rate-of-change is the rate at which f(x) is changing with respect to x at some point. The derivative is essentially a measure of this. dy is the
rate at which y changes with respect to x. dx ----- A 10 m ladder is against a wall. The bottom of the ladder is moving away from the wall at a rate of 4 m/s. What is the rate-of-change of the top of the ladder downward when the bottom of the ladder is 8 m from the wall? What is the relationship
between calculus and physics? 10m x=8m y 4 m/s x² + y² = 100 dx = 4 m/s dy = ? d (x² + y²) = d (100) --- --- dt dt A limit doesn't exist if the function is not continuous at that point.
2x(dx) + 2y(dy) = 0
The way to find out if a limit of a certain function exists or not is to approach the limit from the left and the right side.
---- ---- dt dt For example: Take the limit of the function f(x) as x approachs 0. If you approach 0 from the left and it equals infinity and when you approach 0 from the right and it equals infinity then the limit of f(x) as x approachs 0 doesn't exist.
In this case it doesn't exist because it is infinite discontinous. 2*8*4 + 2*6*dy = 0 --- dt dy ---- dt = -5.33 m/s Limits are used almost everyday. They are used in medicine to set up the boundaries of a dosage of a particular medicine, in constructing buildings and even in baking goods.
Mathematical methods are used in Physics. Mathematics is not only the “language”of Physics (i.e. the tool for expressing, handling and developing logically physical concepts and theories), but also, it often determines to a large extent the content and meaning of physical concepts and theories themselves.
Physical concepts, arguments and modes of thinking are used in Mathematics. That is, Physics is, not only a domain of application of Mathematics, providing it with problems “ready-to-be-solved” mathematically by already existing mathematical tools. It also provides, ideas, methods and concepts that are crucial for the creation and development of new mathematical concepts, methods, theories, or even whole mathematical domains. Physics is the fundamental study of Nature, in other words, in physics we want to find out how stuff works. Mathematics, on the other hand, is harder to pin down since it exists only in the human mind. Mathematics epitomizes the word "abstract." Voltage is also a limit... Yes, it is a concept used to define a value on a function. Derivatives F(x) F'(x) Limits When we are computing limits the question that we are really asking is what y value is our graph approaching as we move in towards on our graph. Finding the derivative of a function given graphically and finding local maxima and minima.
For the function of f :
(b) Where does change its sign
(c) Where does have local minima and maxima
Describe the relationship between the following features of the function of :
(a) the local maxima and minima of
(b) the points at which the graph of changes concavity
(c) the sign changes of
(d) the local maxima and minima of 0=500-4y
Is it possible for a function to be continuous everywhere but differentiable nowhere? Yes. The Weierstrass Function is a famous example. Like a fractal, the graph is self-similar; when you zoom in, the graph always looks the same. This continues infinitely. The Blancmange function is also self-similar and nowhere differentiable. ---- ---- dt dt