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MAT 265 Final Exam Review 1 True False Questions 1 If the left and right hand limits of f at a both exist and are equal then f is continuous at a 2 For a differentiable function y f x f 2 0 means that the tangent line to the graph of f at x 2 is horizontal 3 The derivative of a function at a number is if this limit exists 4 If a function f is differentiable at x a then f is continuous at x a 5 If a function f is continuous at x a then f is differentiable at x a 6 If f is a one to one differentiable function and f f 1 a 0 Then the inverse function is differentiable at 7 The function f x 2x 3 x 1 and f 1 a 1 f f 1 a satisfies the hypotheses of the Mean Value Theorem on the interval 0 2 8 If f c 0 then f has a local maximum or local minimum at c 9 If f x increasing on an interval then f is concave upward on that interval 10 The equation of the vertical asymptote for the function is 2 Limits Algebraically find the following limit 6 lim 3 x x 3 Proving derivative formulas 1 3x 4 Differentiation Find of the following 5 Related Rates 1 A puddle is evaporating in such a way that its diameter is decreasing at a rate of 0 1 cm min At what rate is the area of the puddle decreasing when the diameter is 8 cm 2 The volume of a cube is increasing at a rate of 10 cm3 min How fast is the surface area increasing when the length of the edge is 30 cm 3 Two cars start moving from the same point One travels South at 60 mi h and the other travels West at 25mi h At what rate is the distance between the two cars changing two hours later 6 Applications of Derivatives Algebraically find all critical points intervals of increase decrease local minimums and maximums intervals of concavity up down and inflection points 1 f x x 1 x 2 7 Optimization 1 A rectangular storage container with an open top is to have a volume of 10 The length of its base is twice the width Material for the base costs 3 per Material for the sides costs 0 4 per Find the dimensions of the container which will minimize cost