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 and the minimum cost 2 Find the radius of the right circular cylinder of largest volume that can be inscribed in a right circular cone with radius 6 inches and height 10 inches 3 The manager of a 100 unit apartment complex knows from experience that all units will be occupied if the rent is 800 per month A market survey suggests that on average one additional unit will remain vacant for each 10 increase in rent What rent should the manager charge to maximize revenue 8 Solving Differential Equations 1 Find if 2 Find if x 3 Find when g x 2 3 e g 0 4 g 1 2 9 Integration a Evaluate the following algebraically 4 e 2 dx 2x e 6 b 0 5 0 2 dx 2 1 x Find the following indefinite integral 5 sin 2 x d x co s x 6 7 10 Applications of definite Integrals 1 The velocity function in m s for a particle moving along a line is given by Find the displacement and the distance traveled during 2 The acceleration function in m s2 and the initial velocity are a t 2 t 3 v 0 4 0 t 3 Find a the displacement and b the distance traveled by the particle during the given time interval 11 Applications of the Fundamental Theorem of Calculus Let be the continuous function graphed below and let note the curved piece is a semicircle 1 Evaluate 2 Evaluate 3 Over what intervals shown on the graph is F concave up Explain your answer 4 Over what intervals shown on the graph is F concave down Explain your answer 5 Find the values if any of x where F has local extrema and classify the extrema on 8 6 FINAL EXAM REVIEW ANSWERS Note There is a reasonable assumption that most of these answers are not incorrect 1 True False Questions 1 F 2 T 3 T 4 T 5 F 6 T 7 F 8 F 9 T 10 F 2 Limits 1 5 2 1 6 3 3 4 0 5 6 1 3 Proving derivative formulas 1 y csc x sin x 1 so y 1 sin x 2 cos x 1 cos x csc x cot x sin x sin x 2 y tan x sin x so cos x y cos x cos x sin x sin x 2 1 sec x 2 2 cos x cos x 3 y sec x 1 cos x 1 so cos x 2 y 1 cos x sin x sin x 2 cos x 1 sin x cos x cos x sec x tan x 4 Differentiation 1 1 7 1 2 3 x e 4 x 2 6 2 3 4x 1 2 4 sin x x 2 cos x 1 2 ln 2 x 2 x x x 2 x 3 5 cos 5 x cos 1 x 4 2 sin 5 x 1 x 2 x e sec x 2 2 x 5 sec x 5 x 5 ln sec x 5 x tan x 6 2 ln 5 5 2 x tan x 5 2 x sec 2 x 7 4 5x 3 2x 1 5 x 8 4 sin 3 1 x 2 26 x 6 ye 9 6 xe xy xy 2 2y 4 x 2 sin 2 x 10 4 ln 6 x 5 Related Rates 1 2 dA 0 4 cm dt 4 cm 2 min 3 3 65 mph 2 min 6 Applications of Derivatives 1 Critical point at x 2 Interval of Increase 0 2 Intervals of Decrease 0 2 Local max at x 2 Concave up on 3 Concave down on 0 0 3 Point of inflection at x 3 2 Critical points at x 1 and x 5 Intervals of Increase 1 5 Interval of Decrease 1 5 Local max at x 1 local min at x 5 Concave up on 2 Concave down on 2 Point of inflection at x 2 3 Critical point at x Interval of increase Local max at x 1 3 1 3 1 3 Interval of Decrease 1 3 2 2 3 3 Concave up on Concave down on Point of inflection at x 2 3 7 Optimization 1 Width 1 m Length 2 m Height 5 m Cost 18 2 4 3 900 8 Solving Differential Equations 1 f x 2 x 2 x 1 2 f x 1 x cos x x 1 3 6 3 g x x 2 3 e x 3 e x 1 9 Integration part a 1 21 2 1 3 4 5 …