ASU MAT 265 - mat265_test_3_rev_12 (3 pages)

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mat265_test_3_rev_12

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School:
Arizona State University
Course:
Mat 265 - Calculus for Engineers I
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Last updated 11 30 12 MAT 265 Review Problems for Test 3 Absolute Maximum and Minimum on 1 Algebraically find absolute maximum and absolute minimum of 2 Algebraically find all critical numbers of the function Mean Value Theorem 3 Verify that the function satisfies the hypothesis of the Rolle s Theorem over the interval Then find all numbers that satisfy the conclusion of the Rolle s Theorem 4 Verify that the functions satisfy the hypotheses of the Mean Value Theorem on the given interval Then find all numbers that satisfy the conclusion of the Mean Value Theorem a b Derivative Shape of Graph 5 Determine whether the following statements are true or false a If at each x of an open interval then is constant on b Suppose that is twice differentiable on an open interval I If increases on I then the graph of is concave up c Suppose that is twice differentiable on an open interval I If decreasing on I then the graph of is concave down d For c in I if and then has a local min at e If the point is a point of inflection then 6 Let f x x e 2 x use f x to algebraically find all critical point s the interval s on which f is increasing or decreasing and the local maximum and minimum values of f 7 Let use f x to algebraically find all critical point s the interval s on which f is increasing or decreasing and the local maximum and minimum values of f 8 Algebraically find the intervals of concavity and the inflection points of the following functions a b c Graph The Function Summary of Curve Sketching 9 Use the guidelines on page 225 226 of the text book to sketch the graphs of the following functions a b f x sin 2 x 2 sin x x 10 Sketch the graph of a function f x that satisfies all of the following conditions f 0 f 2 f 4 0 f x 0 if x 0 or 2 x 4 f x 0 if 0 x 2 or x 4 f x 0 if 1 x 3 f x 0 if x 1 or x 3 Optimization 11 A box is to be made out of 25 square feet of material The Marketing department wants the length of the box to be twice its width What dimensions will maximize the volume of the box 12 Find the volume of largest right circular cone that can be inscribed in a sphere of radius 3 cm 13 3 meters of wire is supposed to be cut into two pieces Then one piece will be reshaped as a square and the other into a circle Find the lengths of each piece to minimize the sum of the areas of the resulting shapes 14 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 Anti derivatives 15 Find all antiderivatives of the following functions a b c 16 Find Solve the following Differential Equations a b 17 Suppose that a particle initially at rest is accelerating according to a t 5 sin t Find the position of the particle at time t Area and Distance 3 18 Estimate x dx using Riemann sums Both Left and Right with n 5 rectangles round 1 your answer to 4 decimal places 5 19 Estimate ln 2 x dx using Riemann sums Both Left and Right with n 5 rectangles 1 round your answer to 4 decimal places 20 The speed of a runner increased steadily during the first three seconds of a race Her speed at half second intervals is given in the table Find left hand and right hand estimates for the distance that she traveled during these three seconds Time s 0 0 5 1 0 1 5 2 0 2 5 3 0 Velocity ft s 0 6 2 10 8 14 9 18 1 19 4 20 2 21 Determine whether the following statements are true or false a If F and G are both antiderivatives of g then b where c is a constant c d e 22 If following a b where C is a constant c and find the d 23 Find by interpreting the integral as the area of a region that can be calculated using geometric methods 24 Find by interpreting the integral as the area of a region that can be calculated using geometric methods Note There is a reasonable assumption that most of these answers are not incorrect 1 Absolute Max Absolute Min 2 3 f is continuous on 6 0 and differentiable on 6 0 4 a f is continuous on 0 2 and differentiable 0 b f is continuous on and differentiable 5 See book for examples counterexamples properties and theorems 6 Critical point at f is decreasing on is minimum of 8 a inflection point concave up b inflection point and increasing on concave down concave up c inflection points The local f is decreasing on 7 Critical point at is and increasing on concave down concave up The local max of f concave down 9 10 Answer will vary 11 Width 1 4433756 ft Length 2 886751346 ft Height 1 924500898 ft 12 13 Circle circumference 1 319702540 m and the square side 14 900 90 units will be occupied 15 a b c 16 a 17 b 18 19 20 left 34 7 feet right 44 8 feet 21 See book for examples counterexamples properties and theorems 22 A 5 B 3 C 10 D 0 23 57 2 24 0 4200743650 m


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