New PDF release: A treatise on differential equations

By George Boole

This Elibron Classics publication is a facsimile reprint of a 1877 variation by way of Macmillan and Co., London.

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The limit as x goes to 6 from the right is 4. Since the two are different, the limit does not exist. The function is not continuous at 6 (see the jump). We do not have to test the second part of the definition since part 1 fails. EXAMPLE 23— Suppose 2x fsxd ϭ • ax ϩ b x3 xϽ1 1рxр4 xϾ4 If a and b are numbers to make f(x) continuous, we must find a and b. If x is continuous at x ϭ 1, limϪ2x ϭ limϩ(ax ϩ b). xS1 xS1 So a ϩ b ϭ 2. If x is continuous at x ϭ 4, limϪ (ax ϩ b) ϭ limϩ x3. xS4 xS4 So 4a ϩ b ϭ 64.

If y ϭ tan x, yЈ ϭ sec2 x. D. If y ϭ cot x, yЈ ϭ Ϫcsc2 x. E. If y ϭ sec x, yЈ ϭ tan x sec x. F. If y ϭ csc x, yЈ ϭ Ϫcot x csc x. Once you prove A, B is proved by using the identities cos x ϭ sin(␲/2 Ϫ x), sin x ϭ cos(␲/2 Ϫ x), and rule 9. Once A and B are proved, C through F are proved by writing those four in terms of sin x and cos x, by using rule 8 and other basic trig identities. However, I always like to prove that the derivative of the sine is the cosine. It’s not the proof that is important but the parts of the proof.

So will we. RULE 6 A. If y ϭ sin x, yЈϭ cos x. B. If y ϭ cos x, yЈ ϭ Ϫsin x. C. If y ϭ tan x, yЈ ϭ sec2 x. D. If y ϭ cot x, yЈ ϭ Ϫcsc2 x. E. If y ϭ sec x, yЈ ϭ tan x sec x. F. If y ϭ csc x, yЈ ϭ Ϫcot x csc x. Once you prove A, B is proved by using the identities cos x ϭ sin(␲/2 Ϫ x), sin x ϭ cos(␲/2 Ϫ x), and rule 9. Once A and B are proved, C through F are proved by writing those four in terms of sin x and cos x, by using rule 8 and other basic trig identities. However, I always like to prove that the derivative of the sine is the cosine.

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