Rules of calculus - functions of one variable Derivatives: There are many different ways to indicate the operation of differentiation, also known as finding or taking the derivative.
On plotting the points we have: So this time we see that it is a quadratic curve. The general form of a quadratic curve is: Now if we plug in any of the three ordered pairs into that equation we have: So the required function would be: The slide rule is used primarily for multiplication and divisionand also for functions such as root s, logarithm s and trigonometrybut is not normally used for addition or subtraction.
Slide rules come in a diverse range of styles and generally appear in a linear or circular form with a standardized set of markings scales essential to performing mathematical computations.
Slide rules manufactured for specialized fields such as aviation or finance typically feature additional scales that aid in calculations common to that field.
William Oughtred and others developed the slide rule in the 17th century based on the emerging work on logarithms by John Napier. Before the advent of the pocket calculatorit was the most commonly used calculation tool in science and engineering.
The use of slide rules continued to grow through the s and s even as digital computing devices were being gradually introduced; but around the electronic scientific calculator made it largely obsolete and most suppliers left the business.
Basic concepts In its most basic form, the slide rule uses two logarithmic scales to allow rapid multiplication and division of numbers. These common operations can be time-consuming and error-prone when done on paper.
More elaborate slide rules allow other calculations, such as square root s, exponentialslogarithms, and trigonometric function s. Scales may be grouped in decades, which are numbers ranging from 1 to 10 i.
Thus single decade scales C and D range from 1 to 10 across the entire width of the slide rule while double decade scales A and B range from 1 to over the width of the slide rule. In general, mathematical calculations are performed by aligning a mark on the sliding central strip with a mark on one of the fixed strips, and then observing the relative positions of other marks on the strips.
Number s aligned with the marks give the approximate value of the productquotientor other calculated result. The user determines the location of the decimal point in the result, based on mental estimation. Scientific notation is used to track the decimal point in more formal calculations.
Addition and subtraction steps in a calculation are generally done mentally or on paper, not on the slide rule.
Most slide rules consist of three linear strips of the same length, aligned in parallel and interlocked so that the central strip can be moved lengthwise relative to the other two. The outer two strips are fixed so that their relative positions do not change.
Some slide rules "duplex" models have scales on both sides of the rule and slide strip, others on one side of the outer strips and both sides of the slide strip which can usually be pulled out, flipped over and reinserted for conveniencestill others on one side only "simplex" rules.
A sliding cursor with a vertical alignment line is used to find corresponding points on scales that are not adjacent to each other or, in duplex models, are on the other side of the rule. The cursor can also record an intermediate result on any of the scales.
The answer, 6, is read off the bottom scale where 3 is on the top scale. In general, the 1 on the top is moved to a factor on the bottom, and the answer is read off the bottom where the other factor is on the top.
In such cases, the user may slide the upper scale to the left until its right index aligns with the 2, effectively multiplying by 0. Here the user of the slide rule must remember to adjust the decimal point appropriately to correct the final answer.
So the true answer is not 1. Use the double-decade scales A and B. Use the folded scales. In this example, set the left 1 of C opposite the 2 of D.
Move the cursor to 7 on CF, and read the result from DF. Use the CI inverted scale. Position the 7 on the CI scale above the 2 on the D scale, and then read the result off of the D scale, below the 1 on the CI scale.
Since 1 occurs in two places on the CI scale, one of them will always be on-scale. Use both the CI inverted scale and the C scale.
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