A Tour of the Calculus by David Berlinski

By David Berlinski

In its biggest point, the calculus services as a celestial measuring tape, in a position to order the countless expanse of the universe. Time and house are given names, issues, and boundaries; probably intractable difficulties of movement, progress, and shape are lowered to answerable questions. Calculus was once humanity's first try and symbolize the realm and maybe its maximum meditation at the topic of continuity. Charts and graphs all through.

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The inclination, which we denote by (the Greek letter theta), will be measured in degrees, denoted by the symbol Moreover, any line parallel to the x-axis, including the x-axis itself, will be regarded as having the inclination 0° (zero degrees). Since vertical angles are equal (see Figure 15), it doesn't matter e 0. y x L Figure 15. 74), we have tacitly assumed that the line L slopes up to the right and that the point PI lies to the left of the point P2• As an exercise, consider the modifications required in the other cases.

Of all positive integers is called an infinite sequence, or simply a sequence, and the values of fat n = 1,2, ... 44 Differential Calculus Chap. 2 are called the terms of the sequence. More informally, a sequence is a rule or procedure assigning a number to every positive integer. We can write a sequence by listing some of its terms f(1),j(2), ... ,j(n), ... , (4) where f(n) is called the general term of the sequence. " It is customary to save a lot of parentheses by simply writing (5) instead of (4).

This simply means that for every point P of the curve, there is another point Q of the curve such that the origin is the midpoint of the line segment PQ 50 Differential Calculus Chap. 2 y x Figure 6. (thus P and Q are, so to speak, on "opposite sides" of 0). To see that this is true, we note that changing the sign of x also changes the sign of y = x3, since (_X)3 = - x3• But the points P = (x, y), Q = (- x, - y) and 0 = (0,0) are collinear, as follows at once from the observation that the slope of the line through 0 and P has the same value y/x as the slope of the line through Q and O.

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