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Mu Alpha Theta 2004 State Bowl Questions - Mu Round 1 1. Find all value(s) of t , such that tt b 1067 . 2. The length of a rectangle is increasing at 3 times the rate that its width is increasing. The diagonal of the rectangle is increasing how many times faster than the width when the length is 12 feet and the width is 6 feet? Round 2 3. Let f ( x ) be a linear function such that 2 0 f(x)dx 10 . Find the sum of the slope and the y-intercept of the graph of f. 4. Find the average value of f (x) = sin x on [ 0, ]. 2 Round 3 5. x x 3dx ax 3 6x; if x 1 then a 6. If the function f x is continuous and differentiable and f x = 2 bx 4; if x>1 Round 4 7. Let P = number of grams of the radioactive element petetonium. The rate of change in P is directly proportional to the amount present. At t = 1, there are 16 grams of P present. At t = 5, there are 4 grams present. At what time t will there be exactly 1 gram of P present? x2 y2 1 with sides parallel to the 100 50 coordinate axes. What is the maximum area of the rectangle? 8. A rectangle is to be inscribed in the ellipse Round 5 9. If y dy 4 y 2 x 2 , and y 0 12, find y. dx 10. If cos (9x) – cos (7x) = 0, find the number of solutions for x, where 0 < x < . 2 Round 6 11. The radius of convergence of n1 an x 2 n ; a 0 is what? 12. Consider the set of all possible four-digit numbers consisting only of the digits 3, 4, 5, 6, and 7 with no repetition of the digits in a given number. Find the sum of all such numbers. Round 7 13. Find lim x 1 x x 3 1 2 14. Find the length of the arc of y 2 x from x 4 to x 16. 9 Round 8 15. Find the area enclosed by the set of points defined by | x | + | y | + | x + y | = 2. 16. A slice of pie in the form of a sector of a circle is to have a perimeter of 18 inches. What should be the diameter of the circle in order to make the piece of pie largest? Round 9 e 17. Evaluate lnx dx. 2 1 3 18. Given that g x is an even function and that g x dx 5, evaluate 0 4g x 5dx. 3 3 Round 10 sin y tan y y0 3y 3 19. Evaluate lim 20. Find the value of b, greater than 1, such that the average value of f x x 2 on the interval 1,b equals 7. Mu Alpha Theta National Convention 2004 Mu State Bowl Answers # Answer # Answer 1-1 1, 5 7-13 1 1-2 7 5 5 7-14 112 2-3 5 8-15 3 8-16 9 9-17 e-2 2 2-4 x 3 5 2 3-5 5 2 3 2 x3 2 C 3-6 -14 9-18 70 4-7 9 10-19 1 6 4-8 100 2 10-20 4 5-9 148e 2x 3 3 4 5-10 4 6-11 x a 2 or x -a - 2 6-12 666,600 1. Find all value(s) of t , such that tt b 1067 Answer: 1, 5 Solution: 106 7 55 . Hence, tb t t(b 1) 55 . Since t must be an integer, can equal 1, 5, or 11. But it must also be a digit so t 5 , 1 are the only solutions. 2. The length of a rectangle is increasing at 3 times the rate that its width is increasing. The diagonal of the rectangle is increasing how many times faster than the width when the length is 12 feet and the width is 6 feet? 7 5 5 Answer: Solution: d 2 w2 l 2 180 d 2 =36+144 d 180 dw dd dw dw 6 12 3 42 dt dt dt dt 3. Let f(x) be a linear function such that 2d 2 0 dd dw dl 2w 2l dt dt dt d dd dw dl =w l dt dt dt dd 42 dw 42 dw 7 dw 7 5 dw dt 5 dt 180 dt 6 5 dt 5 dt f(x)dx 10 . Find the sum of the slope and the y-intercept of the graph of f(x). Answer: 5 2 mx 2 bx]20 2m 2b 2m + 2b = 10 so m + b = 5 2 0 4. Find the average value of f ( x ) = sin x on [ 0, ]. 2 Solution: Let f(x) = mx + b (mx b)dx 2 Ans: sin x dx 2 Solution: 0 2 5. x 0 cos x ] 2 0 2 (0 1) 2 2 x 3dx Answer: 5 3 2 x3 2 2 x3 2 C 5 x x 3dx Solution: Using u-substitution, let u x 3, then du dx and x u 3 1 3 5 3 3 2 52 2 2 2 2 2 2 u 3 u du u 3u du u 2u C x 3 2 x 3 C 5 5 1 2 3 ax 6x; if x 1 then a 6. If the function f x is continuous and differentiable and f x = 2 bx 4; if x>1 Answer: -14 Solution: To be both differentiable and continuous, the functions and the derivatives must be equal at x 1. a 6 b 4 and 3ax 2 6 2bx 3a 6 2b 3a 2b 6 b a 10 3a 2 a 10 6 a 14 7. Let P = number of grams of the radioactive element petetonium. The rate of change in P is directly proportional to the amount present. At t = 1, there are 16 grams of P present. At t = 5, there are 4 grams present. At what time t will there be exactly 1 gram of P present? Answer: 9 p' kp p p0 e kt Solution: add-multiply property of exponential functions. t P(t) 1 16 add four to t and divide P(t) by 4. 5 4 9 1 x2 y2 1 with sides parallel to the 8. A rectangle is to be inscribed in the ellipse 100 50 coordinate axes. What is the maximum area of the rectangle? Answer: 100 2 Solution: Solving for y in the ellipse we get y 2 100 x 2 . The area of the rectangle = 2 1 2 2 2 2 2 (2x)(2 y) A 2x 100 x A 2 2x 100 x 2 x 2 x 2 100 x 2 A 2 2 2 2 100 x 2 0 0 2x 2 100 x 2 50 2 2 100 x 100 x x 50 5 2 9. If y y 2 50 5 A 10 10 2 100 2 2 dy 4 y 2 x 2 , and y 0 12, find y 2 . dx Answer: 148e 2x3 3 4 ydy Solution: By separation of variables 4 y2 x 2 dx 1 x3 ln 4 y 2 C 2 3 2x 2x 4 y 2 2x 3 1 2x 3 4 y2 2 3 3 ln 148 C ln 4 y 2 ln148 ln e y 148e 4 2 3 3 148 148 3 3 10. If cos (9x) – cos (7x) = 0, find the number of solutions for x, where 0 < x < . 2 Ans: 4 Solution: cos (9x) – cos (7x) = 0 cos (8x + x) – cos (8x – x) = 0 cos 8x cos x – sin 8x sin x – [cos 8x xos x + sin 8x sin x] = 0 3 , -2sin 8x sin x = 0 sin 8x sin x = 0 8x = , 2, 3, 4 x = , , 8 4 8 2 an ; a 0 is what? 11. The radius of convergence of n n1 x 2 Answer: x a 2 or x -a - 2 n Solution: a The infinite series of will converge when x 2 a 1 means x2 x2 a 1 1 x+2 >a x 2 a or x 2 a x a 2 or x -a - 2 a x2 12. Consider the set of all possible four-digit numbers consisting only of the digits 3, 4, 5, 6, and 7 with no repetition of the digits in a given number. Find the sum of all such numbers. Ans: 666,600 Solution: There are 120 possible four-digit numbers. 5 is the average of the digits therefore the average of the four-digit numbers is 5555. 120 x 5555 = 666,600 1 13. Find lim x x x Answer: 1 Solution: For this limit of indeterminate form, we need to get it into the 1 L’Hopital’s rule y x x ln y 1 ln x lim lim x 0 ln y 0 x x x 1 form to use g x f x 1 ln x ln x now L'Hopital's rule can be applied x x y e0 1 3 1 2 14. Find the length of the arc of y 2 x from x 4 to x 16. 9 Answer: 112 Solution: 1 dy 1 2 3 x dx 9 16 4 2 dy 1 dx 9 x 9 9x 1 2 23 1 9x 1 dx 3 x dx 3 x 3 16 1 2 4 16 4 2 16 4 2 64 8 2 56 112 3 2 3 2 15. Find the area enclosed by the set of points defined by | x | + | y | + | x + y | = 2. Ans: 3 Solution: Area = 1 + .5 + .5 + 1 = 3 (1, 0) (-1, 0) 16. A slice of pie in the form of a sector of a circle is to have a perimeter of 18 inches. What should be the diameter of the circle in order to make the piece of pie largest? Answer: 9 18 2r A r2 r 2 9 A 9r r 2 A 9 2r 0 r diameter 9 2 Perimeter of a sector = r r r 18 Solution: 1 18 2r 2 r 2 r e 17. Evaluate lnx dx. 1 Answer: e 2 Solution: 2 Using integration by parts, let u ln x e 2 ln x dx x ln x 2 1 1 and let dv dx. Then du 2 ln x dx and v x. x e 1 2 x ln x dx x ln x 1 x 1 e again with u ln x and dv dx 2 giving e 2 1 e 2 ln xdx Using integration by parts 1 = x lnx 2 e 1 2 x ln x e 1 1 2 x dx x 1 e e 1 0 2e 0 2 e 1 e 2 3 18. Given that g x is an even function and that g x dx 5, evaluate 0 4g x 5dx. 3 3 Answer: 70 0 Since g x is even, g x dx 5, Solution: -3 0 3 3 3 -3 3 3 3 4g x 5dx 4 g x dx 5 dx 3 4 g x dx 4 g x dx 5dx 4 5 4 5 5 3 3 20 20 30 70 3 3 0 sin y tan y y0 3y 3 19. Evaluate lim Answer: 1 6 sin y 2sec 2 y tan y y0 18y Solution: Using L’Hopital’s rule 3 times: lim cos y 2sec 4 y 4sec 2 y tan 2 y 1 2 1 y0 18 18 6 20. Find the value of b, greater than 1, such that the average value of f x x 2 on the lim interval 1,b equals 7. Answer: 4 Solution: f av b 1 1 x3 2 x dx b 1 1 b 1 3 b 1 b3 1 b2 b 1 7 b2 b 1 21 3 3 b1 b2 b 20 0 b 5 b 4 0 b 5 and b4 only b 4 satisfies the conditions