(Last updated 11/30/12) MAT 265 Review Problems for Test 3 Absolute Maximum and Minimum 1. Algebraically find absolute maximum and absolute minimum of ( ) ( ) on . 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 xexxf2)( , use )(xf 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 )(xf 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)  xxxxf ),sin(2)2sin()( 10. Sketch the graph of a function f(x) that satisfies all of the following conditions: '(0) '(2) '(4) 0'( ) 0 0 2 4'( ) 0 0 2 4"( ) 0 1 3"( ) 0 1 3f f ff x if x or xf x if x or xf x if xf x if x or x           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 18. Estimate 31dxx using Riemann sums (Both Left and Right) with n = 5 rectangles, round your answer to 4 decimal places. 19. Estimate 51)2ln( dxx using Riemann sums (Both Left and Right) with n = 5 rectangles, 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 ( ) ( ) where C is a constant. b) ∫ ( ) ∫ ( ) , where c is a constant. c) ∫ ( ) ∫ ( ) d) ∫ ( ) ∫ ( ) e) ∫ ( ) ∫ ( ) ∫ ( ) 22. If ∫ ( ) ∫ ( ) ∫ ( ) and ∫ ( ) , find the following: a) ∫ ( ) b) ∫ ( ) c) ∫ ( ) , 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 ( ) and increasing on ( ) . The local minimum of is ( ) . 7. Critical point at ( ). f is decreasing on ( ) ( ) and increasing on ( ). The local max of f is ( ) 8. a. inflection point ( ); concave up ( ); concave down ( ) b. inflection point ( ); concave up ( ); concave down ( ) c. inflection points ( ); concave up ( ); 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 = 0.4200743650 m 14. $900 (90 units will be occupied) 15. a. ( ) b. ( ) ( ) ( ) ( ) c. ( ) ( ) 16. a. ( ) ( ) b. ( ) ( ) 17. ( ) ( ) 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