Math 10A Name Total Points 50 Sample Exam 3 PID Print your NAME on every page and write your PID in the space provided above Show all of your work in the spaces provided No credit will be given for unsupported answers even if correct No calculators tablets phones or other electronic devices are allowed during this exam You may use one page of handwritten notes but no books or other assistance 1 pt 0 Follow the instructions on this exam and any additional instructions given during the exam 4 pt 1 The following is the graph of y f x where f x is a third degree polynomial a On which interval or intervals is the rst derivative f cid 48 x positive b On which interval or intervals is the second derivative f cid 48 cid 48 x negative NAME Sample Exam 3 Page 2 of 10 5 pt 2 Find the value of a and b that make the function g is continuous 1 x2 x ax b g x if x 2 if 2 x 4 if x 4 NAME Sample Exam 3 Page 3 of 10 5 pt 3 For the curve y arctan ex nd an equation for the tangent line where x 0 NAME 5 pt 4 Compute f cid 48 x and f cid 48 cid 48 x if f x ln x2 1 What is the domain of f Sample Exam 3 Page 4 of 10 NAME 5 pt 5 Let f x x6 3 x4 2 2 3 a Find the critical points for the function f and identify each as a local maximum local minimum or neither b Find all points of in ection for the function f Sample Exam 3 Page 5 of 10 NAME Sample Exam 3 Page 6 of 10 5 pt 6 In the theory of relativity the mass of a particle with speed v is m f v m0 cid 112 1 v2 c2 where m0 is the rest mass of the particle and c is the speed of light in a vacuum Suppose that a particle having mass 1 at rest is moving according to the position function s t t sin t 4 a Compute the instantaneous velocity of the particle at time t 2 b Compute the instantaneous acceleration of the particle at time t 2 c Use parts a and b and the Chain Rule to compute dm dt at time t 2 5 pt 7 Let f be a one to one di erentiable function such that f 3 6 and f cid 48 3 Calculate the derivatives NAME a g cid 48 3 if g x cid 0 f x cid 1 1 b h cid 48 6 if h x f 1 x Sample Exam 3 Page 7 of 10 NAME Sample Exam 3 Page 8 of 10 5 pt 8 If f x a function that is continuous everywhere x arctan x for x cid 54 0 then what value should be assigned to f 0 in order to make f NAME Sample Exam 3 Page 9 of 10 5 pt 9 A box with square base and no lid has volume 1 What is the minimum surface area the box can have NAME 5 pt 10 Use implicit di erentiation to nd the slope of the line tangent to the ellipse the point 4 2 Sample Exam 3 Page 10 of 10 y2 4 at x2 2