HW4sol - -f(2 x-2 = lim x → 2-f x-f(2 x-2 ⇒ m = 4...

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M408C, Homework 4 — Solution Instructor: Guoliang Wu 1. Find f 000 ( x ) where f ( x ) = 3 x + 4 x 5 . Solution. Since f ( x ) = x 1 / 3 + x 5 / 4 , f 0 ( x ) = 1 3 x - 2 / 3 + 5 4 x 1 / 4 f 00 ( x ) = - 2 9 x - 5 / 3 + 5 16 x - 3 / 4 f 000 ( x ) = 10 27 x - 8 / 3 - 15 64 x - 7 / 4 . 2. Let f ( x ) = ( x 2 if x 2 mx + b if x > 2 . Find the values of m and b that make f differentiable everywhere. Solution. Since f is differentiable, it must be continuous at x = 2. Thus, lim x 2 + f ( x ) = lim x 2 - f ( x ) 2 m + b = 4 . Moreover, f is differentiable at x = 2, lim x
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Unformatted text preview: -f (2) x-2 = lim x → 2-f ( x )-f (2) x-2 ⇒ m = 4 . Therefore, b =-4. 3. Differentiate y = csc θ ( θ + cot θ ). Solution. Use the product rule, y =-csc θ cot θ ( θ + cot θ ) + csc θ (1-csc θ cot θ ) =-θ csc θ cot θ-csc θ cot 2 θ + csc θ-csc 2 θ cot θ 1...
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