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1. Let f(x) = 3 x2 sin; (x # 0) 0; (x = 0) Calculate f'(0) using the definition. Hint: Use Example 2 below on the derivative of x sin1/x. ii. Calculate f'(0) for x # 0. Hint: Use the product rule and the chain rule. iii. Is f'(x) continuous at x = 0? Hint: Use Example 2 below on the derivative of x sin 1/x. 2. Example 2 Let asin+; ifa 40 f(z) = { 0; ifa=0 e Show that f(x) is continuous. If 40, f(a) has no bad point. In fact, it is the product of two continuous functions x and sin i, both of which are continuous away from 0. At 0, there is a problem due to the denominator being 0. We show that the function, as defined, is still continuous at 0, using sequential continuity: 0 = f(a) f(0). In fact, Zn : x 1 lim f(z) = lim xsin z +0 r0 xr = 0, because it is the product of a bounded function (namely sin =) and a function that ap- proaches 0 (namely x). Since f(0) = 0, we have him, f(z) =0 = f(0), and so the function is continuous at 0