MAT 128B Midterm: MATH 127B Midterm iI Solutions Winter
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1. (20 pts) let f (x) = ( x sin 1. Explain: is f di erentiable at x 6= 0. Explain your answer and compute f (x) if possible: is f di erentiable at x = 0. Explain your answer and compute f (x) if possible. 2. (20 pts) suppose that f is di erentiable and 2 f (x) 3 on ir. 2x f (x) f (0) 3x. 3. (20 pts) find the following limits if they exist. lim x 0 lim x 1 sin x x cos x x sin x. 4. (20 pts: suppose f (x) = 1 + x show that f (k)(x) = ( 1)k 1. 2: find the taylor series of f (x) where the remainder is of degree n, what is the remainder rn(x). 5. (20 pts) assume f and f are di erentiable and f is continuous on ir.