By George Boole
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Extra info for A Treatise on Differential Equations
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.
Lim f(x)g(x) ϭ LM xSa D. lim f(x)/g(x) ϭ L/M if M 2 0 xSa E. lim f(x)n ϭ Ln xSa n F. lim 2f(x) ϭ L1/n xSa Note 1:We took /2 for each for the limits so that the limit we were trying to prove has in it, as in the technical definition of limits. Note 2:All of you will get this proof technique with a little work; however, to truly understand that we have proved something, you must understand the technical definition of a limit. We will prove A and C. Proof of A: Given an ε Ͼ 0, we must find a ␦ Ͼ 0 such that if 0 Ͻ |x Ϫ a| Ͻ ␦, then |f(x) ϩ g(x) Ϫ (L ϩ M)| Ͻ ε.
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.