Math 100 Worksheet 2 September 26, 2020 Question 1. Use the definition of derivative to find f 0 (0) where 2 x sin x1 if x 6= 0, f (x) = 0 if x = 0. Question 2. Use the limit definition of derivative to prove the reciprocal rule 1 g 0 (x) d =− , if g(x) 6= 0. dx g(x) (g(x))2 Question 3. Use the reciprocal rule to find the derivative of h(x) = x−n , where n is a positive integer. Question 4. Use the reciprocal rule to prove the quotient rule. Question 5. Suppose that we know that for every x and h, a function f satisfies f (x + h) − f (x) = −hx2 − 6hx − 3h2 x + 2h2 + 4h3 . Find f 0 (x). Do you know of any function f satisfying equation (1) for every x and h? Question 6*. (Optional, just for fun) Show that 100n n20 √ = 0. n→∞ n! lim 1 (1)