MATH 2411 Spring 2011 Practice Exam #1 Tuesday, March 1 st Sections: Sections ; 6.8; Instructions:

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1 MATH 411 Sprig 011 Practice Exam #1 Tuesday, March 1 st Sectios: Sectios ; 6.8; Name: Score: = 100 Istructios: 1. You will have a total of 1 hour ad 50 miutes to complete this exam.. A No-Graphig Calculator may be used o the exam. 3. Show ALL of your work. I give out partial credit. 4. Cheatig of ay kid o the exam will ot be tolerated ad will result i a grade of 0%. 5. If you have a questio, please come up ad ask it is much better to ask right away tha to stay stuck for a log period of time. 6. Take a deep breath, relax, ad good luck!! A mathematicia, a physicist, ad a egieer are all give idetical rubber balls ad told to fid the volume. They are give aythig they wat to measure it, ad have all the time they eed. The mathematicia pulls out a measurig tape ad records the circumferece. He the divides by two times pi to get the radius, cubes that, multiplies by pi agai, ad the multiplies by four-thirds ad thereby calculates the volume. The physicist gets a bucket of water, places gallos of water i the bucket, drops i the ball, ad measures the displacemet to six sigificat figures. Ad the egieer? He writes dow the serial umber of the ball, ad looks it up.

2 Short Aswer Sectio The followig problems will be graded as right or wrog. No eed to show work. ( poits each) a) Give the velocity fuctio for a object is v( t) = 6 t ad s ( 0) = 0. Determie the positio fuctio for the object. s(t) = b) If you have a itegral that ivolves the term x 49 you cosider that may simplify it?, what substitutio would x = c) Give the appropriate form of the partial fractio decompositio for the followig ratioal fuctio (No eed to solve for ay ukows A, B, C, etc.; just set it up properly). x x 8 = 1 d) True or False: = l x + c x e) Draw the regio bouded by y = l(x), y = 1, ad x = 1 (You ca chage the scale if eeded).

3 Show Your Work Questios The followig questios should be aswered to the best of your ability. Show work where you ca as partial credit will be awarded i some cases. 1. For the followig, assume t is time measured i secods ad that the velocity has uits of m / s. Give the velocity fuctio v( t) = 3si( π t) ad s ( 0) = 0 fid: a) The displacemet over the iterval [ 0,4]. b) The distace traveled over the iterval [ 0,4].

4 . Fid the area betwee the curves o the give iterval. Give a rough sketch of the regio. a. y = x 1, y = 1 x, 0 x b. x = y, x = y, x = 1

5 3. Compute the volume of the solid formed by revolvig the regio about the give lie. You should also draw a rough sketch of the resultig figure. Regio bouded by y = x, y = 0, ad x = 0 a. About the lie y = 3 b. About the x-axis

6 4. Give the regio bouded by the fuctios y = x ad y = 1 x ; a. Fid the area of the regio. b. Fid the volume of the solid of revolutio whe the regio is rotated about the y-axis. Draw a rough sketch of the resultig solid.

7 5. Set up the itegral that you would use for fidig the arc legth of the graph of x y = e from = 3 x to x = 5. Do ot evaluate the itegral. 6. Fid the mass of the followig thi bar with the give desity fuctio: x ρ ( x) = for 0 x 7. How much work is required to move a object from x = 1 to x = 5 (measured i meters) i the presece of a costat force of 5 N actig alog the x-axis?

8 8. Uraium-38 has a half-life of 4.5 billio years. Gelogists fid a rock cotaiig a mixture of U-38 ad lead, ad determie that 85% of the origial U-38 remais; the other 15% has decayed ito lead. How old is the rock? 9. Evaluate the followig itegrals usig ay techique you have available. t a) t e dt b) x l( x)

9 10. Evaluate the followig itegrals usig ay method you kow. 3 a. e x cos(x) π b. x si(x) Evaluate the followig itegrals usig ay method you kow. a) si 5 (x)

10 b) si ( x )cos ( x) π 3 c) π 3 sec θ 1dθ 1. Evaluate the followig itegral usig ay method you kow. 64 x

11 13. Trasform the followig itegrals ito a itegral ivolvig trigoometric fuctios (No eed to evaluate the itegral, just set it up properly). a) x 81 for x > 9 b) x 4 + x 14. Evaluate the itegral by the method of partial fractios. 1 x + 1 x + 1 ( ) ( )

12 My Gift To You Importat Itegrals 1. ta( x ) = l cos( x) + C. sec( x ) = l sec( x) + ta( x) + C 3. cot( x ) = l si( x) + C 4. csc( x ) = l csc( x) + cot( x) + C Reductio Formulas si ( x)cos( x) 1 1 = si ( x) 1. si ( x) + cos ( x)si( x) 1 1 = cos ( x). cos ( x) + ta ( x) 1 1 = ta ( x) 3. ta ( x) sec ( x) ta( x) 1 1 = sec ( x) 4. sec ( x) +

1 Cabin. Professor: What is. Student: ln Cabin oh Log Cabin! Professor: No. Log Cabin + C = A Houseboat!

1 Cabin. Professor: What is. Student: ln Cabin oh Log Cabin! Professor: No. Log Cabin + C = A Houseboat! MATH 4 Sprig 0 Exam # Tuesday March st Sectios: Sectios 6.-6.6; 6.8; 7.-7.4 Name: Score: = 00 Istructios:. You will have a total of hour ad 50 miutes to complete this exam.. A No-Graphig Calculator may