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Further functions,Suppose . = sin 2.. Then gives a new function of ., and we can differentiate it again, to obtain ..

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Integration,Suppose that we wish to compute the area of the region bounded by the lines . = 0, . = ., . = ., and . = ., where ., . and . are constants. This region is shown shaded in Figure 6.1, and it is clear that we can evaluate its area as .(. − .) simply by using the rule for the area of a rectangle.

Dappled

Further integration,A rational function where .(.) and .(.) are both polynomials can often be easier to work if it is split up into fractions with the factors of .(.) as denominators. For example

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Linear equations and matrices,We start by looking at some examples of simultaneous equations that will indicate the various possibilities.

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Derogate

Complex numbers,Consider the four number sets, ℕ, ℤ, ℚ and ℝ. Each one has algebraic shortcomings, illustrated by the following table:

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Differential equations,Students coming across the topic for the first time often have difficulty in appreciating just what a differential equation is. We shall postpone discussion of this problem until we have looked at some simple examples, which also serve to give motivation for the solution of differential equations.

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Socio-Economic Approach to Managemente how we should show the function at . = 2. We cannot give it two values, since the value of . (2) is unique, and specified by the definition to be 4. In graphical terms, we could depict this by putting a small open circle at (2, 5) and a filled circle at (2, 4) as in Figure 3.1.

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Sachin Chaturvedi,Krishna Ravi Srinivasome from the standard functions, calculus gives us a straightforward technique for solving these problems. We use the first and second derivatives to help decide whether a function is increasing or decreasing and to determine the shape of the curve.