MTH 121 - Summer - 2005 Essex County College - Division of Mathematics Test # 11 - Created May 31, 2005 Name: Signature: Show all work clearly and in order, and box your final answers. Justify your answers alge- braically whenever possible. You have at most 110 minutes2 to take this 100 point exam. No cellular phones allowed. 1. For what values of x does the graph of f (x) = x + 2sinx have a horizontal tangent? 9 points Solution: Take the derivative, set it equal to zero and solve for x. fprime(x) = 1 + 2cosx 0 = 1 + 2cosx ?12 = cosx The solution is: x = 2pi3 + 2pik or x = 4pi3 + 2pik where k ? Z . 2. Find fprime(a) using the definition. 9 points f (z) = 2z + 1z + 3 Solution: You must use the definition to get credit. fprime(a) = lim h?0 f (a + h)?f (a) h = lim h?0 2a + 2h + 1 a + h + 3 ? 2a + 1 a + 3 h = lim h?0 (2a + 2h + 1)(a + 3)?(2a + 1)(a + h + 3) h(a + h + 3)(a + 3) = lim h?0 5h h(a + h + 3)(a + 3) = 5 (a + 3)2 1This document was prepared by Ron Bannon using LATEX...