(a) Use Exercise 55 to prove that sin a sin b a b for all a, b (b) Let (x) sin(x a) sin x Use part (a) to show that the graph of lies between the horizontal lines y a (c) Plot y (x) and the lines y a to verify (b) for a 0 5 and a 0 2 Data From Exercise 55 Use FTC I to prove that if '(x) K for x a, ...

a Let fx sin x so that fx cos x and for all x From Exercise 55 we get sin a si ...

(a) Use Exercise 55 to prove that | sin a sin b| |a b|...

Question:

(a) Use Exercise 55 to prove that | sin a − sin b| ≤ |a − b| for all a, b.
(b) Let (x) = sin(x + a) − sin x. Use part (a) to show that the graph of ƒ lies between the horizontal lines y = ±a.
(c) Plot y = ƒ(x) and the lines y = ±a to verify (b) for a = 0.5 and a = 0.2.

Data From Exercise 55

Use FTC I to prove that if |ƒ'(x)| ≤ K for x ∈ [a, b], then |ƒ'(x) − ƒ(a)| ≤ K|x − a| for x ∈ [a, b].

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