Mathematics - Refresher
Dynamic Systems
Arithmetic involves numbers that have fixed values. Algebra involves both literal and arithmetic numbers. Although the literal numbers in algebraic problems can change value from one calculation to the next, they also have fixed values in a given calculation. When a weight is dropped and allowed to fall freely, its velocity changes continually. The electric current in an alternating current circuit changes continually. Both of these quantities have a different value at successive instants of time. Physical systems that involve quantities that change continually are called dynamic systems. The solution of problems involving dynamic systems often involves mathematical techniques different from those described in arithmetic and algebra. Calculus involves all the same mathematical techniques involved in arithmetic and algebra, such as addition, subtraction, multiplication, division, equations, and functions, but it also involves several other techniques. These techniques are not difficult to understand because they can be developed using familiar physical systems, but they do involve new ideas and terminology.
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Differentials and Derivatives
While the equation for instantaneous velocity, V = ds/dt, may seem like a complicated expression, it is a familiar relationship. Instantaneous velocity is precisely the value given by the speedometer of a moving car. Thus, the speedometer gives the value of the rate of change of distance with respect to time; it gives the derivative of s with respect to t; i.e. it gives the value of ds/dt.
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The ideas of differentials and derivatives are fundamental to the application of mathematics to dynamic systems. They are used not only to express relationships among distance traveled, elapsed time and velocity, but also to express relationships among many different physical quantities. One of the most important parts of understanding these ideas is having a physical interpretation of their meaning. For example, when a relationship is written using a differential or a derivative, the physical meaning in terms of incremental changes or rates of change should be readily understood.
For the formula for derivatives:
http://en.wikipedia.org/wiki/Table_of_derivatives
Examples:
1. A stone is dropped into a quiet lake, and waves move in circles outward from the location of the splash at a constant velocity of 0.5 feet per second. Determine the rate at which the area of the circle is increasing when the radius is 4 feet.
using
A = pi * r**2
take derivatives on both sides with respect to time,
dA/dt = pi * 2r * dr/dt
substitute the constant velocity in dr/dt and other values,
dA/dt = pi * 2 * 4 * 0.5 sq.ft/sec
The rate at which the area of the circle is increaing is 12.6 sq.ft/sec
2. A ladder 26 feet long is leaning against a wall. The ladder starts to move such that the bottom end moves away from the wall at a constant velocity of 2 feet per second. What is the downward velocity of the top end of the ladder when the bottom end is 10 feet from the wall?
Let x be the floor axis and y be the wall axis. Using Pythagorean Theoram:
x**2 + y**2 = c**2 (where c is constant, i.e., 26 ft)
take derivatives on both side with respect to time:
2x dx/dt + 2y dy/dt = 0
substitute y in terms of x using the above Pythagorean formula,
dy/dt = - (x dx/dt) * 1/sqrt(26**2 - x**2)
given values are: dx/dt = 2 ft/sec; at x = 10 ft then,
dy/dt = 10 * 2 * 1/sqrt(26**2 - 10**2) = 10 * 2 * 1/24 = 10/12
The top end of the ladder is moving at - 0.833 ft/sec. ( - indicates the downward movement)
posted by Siva at 3:54 PM
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