NAME____________________________________ SCHOOL__________________________________ TEST 3 Friday, November 14, 2008 ___________________________________________ //-----------------------------------------\\ // \\ || FAX your answers to: || || || || Irvin Roy Hentzel at Fax 515-294-5454 || || || \\___________________________________________// \\_________________________________________// ___________________________________________ //-----------------------------------------\\ // \\ || Snail Mail your answers to: || || || || Irvin Roy Hentzel || || Department of Mathematics || || 432 Carver Hall || || Iowa State University || || Ames, Iowa 50011-2064 || \\___________________________________________// \\_________________________________________// Each problem has two parts. First write the integral as INT f(u) du and then integrate it. 1. INT Sqrt(5x+2) dx (a) (b) (a) 1/5 INT Sqrt(5x+2) 5 dx 3/2 (5x+2) (b) 1/5 -------- + C 3/2 2 4 2. INT x (x + 4) dx (a) (b) 2 4 (a) 1/2 INT (x + 4) 2x dx 2 5 (x + 4) (b) 1/2 ------------ + C 5 2 3 3. INT x Cos( 5 x + 5 ) dx (a) (b) 3 2 (a) 1/15 Cos( 5 x + 5 ) 15 x dx 3 (b) 1/15 Sin( 5 x + 5 ) + C 3/2 5/2 4. INT x Sin( Pi x + 2 ) dx (a) (b) 2 5/2 (a) ------- INT Sin( Pi x + 2 ) (5/2) Pi dx 5 Pi 2 5/2 (b) -------- (- Cos( Pi x + 2) + C 5 Pi 2 2 5. INT x Cos( Sin[ x ] ) Cos( x ) dx (a) (b) 2 2 (a) 1/2 INT Cos( Sin[x ] ) Cos(x ) 2x dx 2 (b) 1/2 ( Sin( Sin[x ] ) ) + C 3 6. INT Sin ( 5 x ) Cos( 5 x) dx (a) (b) 3 (a) 1/5 INT Sin ( 5x ) Cos(5x) 5 dx 4 Sin (5x) (b) 1/5 ------------ + C 4 7. INT Sec(x) Tan(x) dx (a) (b) (a) INT Sec(x) Tan(x) dx (b) Sec[x] + C 3 8. INT Cos[x] (Sin[x]+1) dx (a) (b) 3 (a) INT (Sin[x] + 1) Cos[x] dx 4 (Sin[x] + 1) (b) ----------------- + C 4 3 x 9. INT ----------- dx 4 2 (x + 9) (a) (b) 4 -2 3 (a) 1/4 INT (x + 9 ) 4 x dx 4 -1 (x + 9) (b) 1/4 --------- + C -1 1/5 3/5 10. INT x( x + 3 x + 7) dx (a) (b) 6/5 8/5 (a) INT x + 3 x + 7x dx 11/5 13/5 2 x x x (b) ------ + 3 ----- + 7 ----- + C 11/5 13/5 2 11. Use Newton's method to approximate a root of y = f(x) / given that f(8) = 39.6 and f (8) = 13.2. n(8) = 8 - f(8)/fp(8) = 8 - 39.6/13.2 = 5.0 12. The strength of a beam is proportional to its width and the square of its depth. Find the dimensions of the strongest beam that can be cut from a cylindrical log of radius a feet. _ _ 2 | 2 2 | s(x) = k (2x) |2 Sqrt( a - x )| |_ _| 2 2 s(x) = 8 k x(a - x ) 2 3 s(x) = 8 k ( a x - x ) 2 2 s'(x) = 8k( a - 3 x ) 2 2 s'(x) = 0 when a = 3 x So x = a/Sqrt[3]. 2 2 2 2 y = Sqrt[a - x ] = Sqrt[ a - a /3 ] = a Sqrt[2/3] The width is 2 a/Sqrt[3]. The depth is 2 a Sqrt[2/3] 13. Use Newton's method to approximate the smallest positive root of x Tan(x) + Cos(x) = 2 n[x_] := x - (x Tan[x] + Cos[x] - 2)/(Tan[x] + x Sec[x]^2 -Sin[x]) a = 1.0; Do[ (Print[i," ",a];a = n[a]),{i,1,20}] 1 1. 2 0.976407 3 0.975201 4 0.975199 5 0.975199 6 0.975199 7 0.975199 8 0.975199 9 0.975199 10 0.975199 11 0.975199 12 0.975199 13 0.975199 14 0.975199 15 0.975199 16 0.975199 17 0.975199 18 0.975199 19 0.975199 20 0.975199 14. A car can accelerate from 45 to 60 miles an hour in 11 seconds. How far will it travel as it accelerate from 0 to 75 miles/hr. 66 to 88 ft/s in 11 seconds means a = 2 feet/sec/sec a = 2 v = 2 t 2 s = t 75 mph is 110 ft/sec It takes 55 seconds the distance is 2 s = 55 = 3025 feet <================ 15. A car can accelerate from 0 mph to 60 mph in 176 feet. What is its acceleration. 0 ft/sec to 88 ft/sec in 176 feet. v = at 2 s = 1/2 a t 2 s = 176 = 1/2 a t v = 88 = a t 2 = 1/2 t t = 4 seconds 88 = 4 a 2 a = 22 ft/sec / \/ a = 22 v = 22 t v = 88 when t = 4 sec 2 2 s = 11 t s(4) = 11 4 = 176 feet _10_ \ 16. Evaluate /__ (i-1)(i-2) i=1 10 2 SUM i - 3 i + 2 i=1 10 11 21 10 11 ---------- -3 ------- + 20 6 2 = 240 Sum[ (i-1)(i-2),{i,1,10}] 17. Why must the area under y=f(x) from x=a to x=b equal F[b]-F[a] where F is any function whose derivative is f(x)? The area function and F has the same derivative. Therefore The area function must be A(x) = F(x) - F(a) Since A(b) is the area, then A(b) = F(b)-F(a) 18. The cross sections of a pond taken 18 feet apart are: 2 5 8 12 14 8 7 5 1 What is the surface area of the pond? Work the problem with Simpson's method and also with the Trapezoid method. T T S S 2 1 2 1 2 5 2 10 4 20 8 2 16 2 16 12 2 24 4 48 14 2 28 2 28 8 2 16 4 32 7 2 14 2 14 5 2 10 4 20 1 1 1 1 1 ------ -------- 121 181 x9 x6 ---- ------ 1089 1086 <====== Answer.