MATH 2500: TAKE HOME 02 (30 points.) NAME: DUE: Wednesday, June 3rd by 8 AM. DIRECTIONS: To receive full credit, make sure your work is neat and complete. 1. Let f (x) = x + sin(x) 1. (a) Explain why f is continuous. (b) Find f (0) and f (). (c) What theorem guarantees a solution to f (x) = 0 between x = 0 and x = ? 5x = 3. x1 2 2. Use the precise definition of limit to prove lim SCRATCHWORK: FORMAL PROOF: 10 3. Use the graph of y = f (x) below to answer the following questions and explain your reasoning. (a) Find f 0 (4). (b) Where is f 0 (x) = 0? (c) Explain why f is not differentiable at x = 2 nor at x = 4. (d) Is f (2) positive or negative? What about f 0 (2)? (e) Explain why f is not differentiable at x = 4. (f) Which is larger: f 0 (1) or f 0 (0)? Explain. 10 4. Let f (x) = 1 . x (a) Use the limit definition of derivative: f 0 (x) = lim h0 f (x + h) f (x) to find a formula for f 0 (x). h (b) Use part (a) to find f 0 (2) and the equation of the tangent line to y = f (x) when x = 2. (c) Use part (b) to help you approximate 1 . 1.9 10