3 Now if x 2 x 1 \u03b4 then f x 2 f x 1 f prime c x 2 x 1 3 x 2 x 1 3 \u03b4 3 epsilon1

# 3 now if x 2 x 1 δ then f x 2 f x 1 f prime c x 2 x

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3 Now, if | x 2 - x 1 | < δ , then | f ( x 2 ) - f ( x 1 ) | = | f prime ( c ) | | x 2 - x 1 | ≤ 3 | x 2 - x 1 | < 3 δ = 3 epsilon1 3 = epsilon1. Therefore, f is uniformly continuous on I . (b) What is the estimated error if p 5 is used to approximate f (2 / 3)? error = | f (2 / 3) - p 5 (2 / 3) | max | f (6) | 6! parenleftbigg 2 3 parenrightbigg 6 3 6! parenleftbigg 2 3 parenrightbigg 6 = 2 6 6! 3 5 0 . 0000366 (c) Find the least integer n such that p n approximates f (1 / 2) with error less than 0 . 0001. We want the least n such that error = | f (1 / 2) - p n (1 / 2) | ≤ max | f ( n +1) | ( n + 1)! parenleftbigg 1 2 parenrightbigg n +1 3 ( n + 1)! parenleftbigg 1 2 parenrightbigg n +1 < 0 . 0001 3 ( n + 1)! parenleftbigg 1 2 parenrightbigg n +1 < 0 . 0001 is equivalent to ( n + 1)! 2 n +1 > 30 , 000 Try n = 4: 5! 2 5 > 30 , 000? 120 · 32 = 3840 No Try n = 5: 6! 2 6 > 30 , 000? 720 · 64 = 46 , 080 Yes The least n is n = 5. 2. Let f ( x ) = cos x . Let p n denote the Taylor polynomial of degree n for f . (a) Find p 4 in powers of ( x - π/ 4). p 4 ( x ) = 2 2 - 2 2 ( x - π/ 4) - 2 / 2 2! ( x - π/ 4) 2 + 2 / 2 3! ( x - π/ 4) 3 + 2 / 2 4! ( x - π/ 4) 4 and p 4 ( x ) = 2 2 bracketleftbigg 1 - ( x - π/ 4) - 1 2! ( x - π/ 4) 2 + 1 3! ( x - π/ 4) 3 + 1 4! ( x - π/ 4) 4 bracketrightbigg 4 (b) Use p 3 to approximate cos 48 o and estimate the error in your approximation. 48 o = 45 o + 3 o = π 4 + π 60 . p 3 (48 o ) = p 3 parenleftBig π 4 + π 60 parenrightBig = 2 2 bracketleftbigg 1 - π 60 - 1 2 parenleftBig π 60 parenrightBig 2 + 1 6 parenleftBig π 60 parenrightBig 3 bracketrightbigg 0 . 6691303 error = vextendsingle vextendsingle vextendsingle cos parenleftBig π 4 + π 60 parenrightBig - p 3 parenleftBig π 4 + π 60 parenrightBigvextendsingle vextendsingle vextendsingle max | f (4) | 4! parenleftBig π 60 parenrightBig 4 1 24 · parenleftBig π 60 parenrightBig 4 0 . 000000313 Calculator value: 0 . 6691306 3. Let f ( x ) = ln(1 + x ). Let p n denote the Taylor polynomial of degree n in powers of x for f . (a) Use p 3 to approximate ln 1 . 2. Estimate the error in your approximation.  #### You've reached the end of your free preview.

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