MTH 133 Solutions to Exam 1 Feb. 25th 2015
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1 MTH 133 Solutions to Exam 1 Feb. 5th 15 Name: Section: Recitation Instructor: READ THE FOLLOWING INSTRUCTIONS. Do not open your exam until told to do so. No calculators, cell phones or any other electronic devices can be used on this exam. Clear your desk of everything excepts pens, pencils and erasers. If you need scratch paper, use the back of the previous page. Without fully opening the exam, check that you have pages 1 through 9. Fill in your name, etc. on this first page. Show all your work. Write your answers clearly! Include enough steps for the grader to be able to follow your work. Don t skip its or equal signs, etc. Include words to clarify your reasoning. Do first all of the problems you know how to do immediately. Do not spend too much time on any particular problem. Return to difficult problems later. If you have any questions please raise your hand and a proctor will come to you. There is no talking allowed during the exam. You will be given exactly 9 minutes for this exam. I have read and understand the above instructions:. SIGNATURE Page 1 of 9
2 MTH 133 Solutions to Exam 1 Feb. 5th 15 Multiple Choice. Circle the best answer. No work needed. No partial credit available. 1. (5 points) Compute the derivative of y ln(x + ) A. dx 1 x + B. dx x + x + C. dx x + D. dx x x + E. None of the above. (5 points) Compute the derivative of y tan 1 (3x) A. dx x B. dx tan (3x) C. dx 3 tan (3x) sec (3x) D. dx x E. None of the above 3. (5 points) Compute the derivative of y cosh(1 + x 3 ) A. dx sinh(1 + x3 )(3x ) B. dx sinh(1 + x3 ) C. dx sinh(1 + x3 )(3x ) D. dx sinh(1 + x3 ) E. None of the above 4. (5 points) Compute the derivative of y x A. ( x)1 x dx B. ( x)1 x dx C. dx ln() x D. dx ln() x E. None of the above Page of 9
3 MTH 133 Solutions to Exam 1 Feb. 5th 15 Fill in the Blanks. No work needed. No partial credit available. 5. Evaluate the integrals. You do not need to simplify your answer. (a) (5 points) (b) (5 points) 1 e x + 1 dx ln(1 + e) ln(1) x + ex x 1 x 4 dx sin 1 (x )/ + C (c) (5 points) 1 + ln x x dx (1 + ln x) / + C Extra Work Space. Page 3 of 9
4 MTH 133 Solutions to Exam 1 Feb. 5th 15 Standard Response Questions. Show all work to receive credit. Please BOX your final answer. 6. Let R be the region bounded by the parabola y x x and the line y x. (a) (7 points) Sketch the region R y x (b) (8 points) Set up (DO NOT EVALUATE) an integral to find the volume of the solid formed when R is rotated about the line y 1. V π π(r(x) r(x) ) dx (1 + x) (1 (x x )) ] dx Page 4 of 9
5 MTH 133 Solutions to Exam 1 Feb. 5th (4 points) Evaluate the following its: (a) x x x x x x x x x x x ln() x x (ln()) x (L Hopitals) (L Hopitals) (b) x +(x)3x Consider ln(x)3x x + x +(3x) ln(x) Now x +(x)3x ( 1 (c) x 1 ln x 1 ) x 1 x + eln(x)3x 3 ln(x) x +(3x) ln(x) x + 1/x 3/x x + 1/x e x + ln(x)3x First lets get a common denominator x + 3x ] e 1. 1 ln x 1 x 1 (x 1) ln x (x 1) ln x. (L Hopitals) (x 1) ln x x 1 (x 1) ln x 1 1/x x 1 ln x + 1 1/x (L Hopitals) 1/x 1/ (L Hopitals) x 1 1/x + 1/x Page 5 of 9
6 MTH 133 Solutions to Exam 1 Feb. 5th (1 points) Solve the initial value problem: 1x 6y x + 1 dx, y() 1x 6y x + 1 dx 1x x + 1 dx 6y 1x x + 1 dx 6y 1 x C y x C y Since y() this gives C giving C 3 for the final solution y 3 5 x (1 points) A population of fruit flies grows exponentially. Initially there are 1 flies and after 1 days there are 5 flies. After how many days were there flies? y 1e kt 5 1e 1k 5 e 1k ln 5 1 k Now we can plug in and solve for t t(ln 5)/1 1e t(ln 5)/1 e ln t(ln 5)/1 1 ln /(ln 5) t 1 1. (1 points) Make the appropriate trigonometric substitution needed to integrate (x + 1) dx. That is, change the integral to the form: f(θ) dθ. DO NOT EVALUATE THE INTEGRAL. Take x tan θ then we get dx sec θ for the final solution: sec θ (tan θ + 1) dθ Page 6 of 9
7 MTH 133 Solutions to Exam 1 Feb. 5th Evaluate the following integrals. Show all work. 9x (a) (1 points) x x 6 dx Consider giving us the system of equations 9x x x 6 9x (x 3)(x + ) A x 3 + B x + A x 3 + B x + 9x x x 6 A(x + ) + B(x 3) 9x (x 3)(x + ) x x 6 A(x + ) + B(x 3) 9x (A + B)x + (A 3B) 9x A + B 9 A 3B for correct A and B. Which can be solved for A 5 and B 4. Giving the final solution 9x 5 x x 6 dx x ] dx 5 ln x ln x + + C x + (b) (1 points) xe x dx This is a standard integration by parts u x du dx dv e x dx v e x / Giving us: xe x dx xex e x dx xex ex 4 + C Page 7 of 9
8 MTH 133 Solutions to Exam 1 Feb. 5th (1 points) A tank is in the shape of a downward-pointing cone (vertex at the bottom) which has height 5 feet and radius feet. The tank is full of oil weighing 7 lb/ft 3. Find the work it would take to pump the oil from the tank to an outlet 3 feet above the top of the tank. From the picture we can see that the height and radius are related by r 5 h. So we have W b a 7d(h)A(h) dh 3 Where A(h) is the area of the cross section at height h and d(h) is distance the cross section has to travel to get 3 ft above the tank π 5 8π 5 8π 5 7(8 h)πr ] dh 7(8 h) 5 5 π ( ) ] 5 h (8 h)h dh (8h h 3 ) dh dh 8h 3 /3 h 4 /4 ] 5 8π 5 8(5) 3 3 ] (1 points) Evaluate the improper integral convergent. xe x dx if it is convergent or determine it does not xe x dx xe x e u ] e u dx xe u dx (u x ) u du ] e u e (du x dx) ] + 1 1/ Page 8 of 9
9 MTH 133 Solutions to Exam 1 Feb. 5th 15 Congratulations you are now done with the exam! Go back and check your solutions for accuracy and clarity. Make sure your final answers are BOXED. When you are completely happy with your work please bring your exam to the front to be handed in. Please have your MSU student ID rea so that is can be checked. DO NOT WRITE BELOW THIS LINE. Page Points Score Total: 15 Page 9 of 9
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