Fall 2002 Math 152 Exam 2 Test Form A PRINT Surname: Rest of name: Signature: Student ID: Instructor: Section #: Calculators may be used only during the last 30 minutes of the test. The ScanTron forms will be collected at the end of the test. Calculators may not be used to perform \calculus" operations, such as nding inde nite integrals! Aggies do not lie, cheat, or steal, nor tolerate those who do. In Part I, mark the correct choice on your ScanTron with a #2 pencil. For your own records, also mark your choices on your test paper, because your ScanTron will not be returned. Do not use the ScanTron as scratch paper. Remember to write your name, section, and test form (A or B) on the ScanTron! In Part II, all work to be graded must be shown in the space provided, or clearly pointed to therefrom, and your nal answer must be clearly indicated. You may use the back of any page for scratch work; any other paper used should be turned in with the test. POSSIBLY USEFUL FORMULAS M n = x f x 0 +x 1 2 +f x 1 +x 2 2 + +f x n+1 + x n 2 T n = x 2 [f(x 0 )+2f(x 1 )+ +2f(x n−1 )+f(x n )] S n = x 3 [f(x 0 )+4f(x 1 )+2f(x 2 )+ +2f(x n−2 )+4f(x n−1 )+f(x n )] jE M j K(b−a) 3 24n 2 if jf 00 (x)j Kfor all x in [a; b]. jE T j K(b−a) 3 12n 2 if jf 00 (x)j Kfor all x in [a; b]. jE S j K(b−a) 5 180n 4 if jf (4) (x)j Kfor all x in [a; b]. x lnx 5 1.60944 6 1.79176 7 1.94591 8 2.07944 9 2.19722 10 2.30259 152B-F02-A Page 2 Part I: Multiple Choice (5 points each) There is no partial credit. Do not use a calculator for symbolic operations, such as evaluating integrals and derivatives. 1. The improper integral Z 1 2 2+cosx x 4 dx (A) diverges to +1. (B) diverges, but does not approach 1 because the integrand oscillates. (C) converges, by comparison with the integral Z 1 2 3 x 4 dx. (D) converges to the value 1 12 . (E) converges, because the integrand oscillates. 2. What integral represents the arc length of the parametric curve segment x =1+cos(2t);y=t−sin(2t); 0 t ? (A) R 0 p 2−4cos(2t)+4cos 2 (2t)dt (B) R 0 p 6−4cos(2t)dt (C) R 0 p 2+2cos(2t)+t 2 −2 sin(2t)dt (D) R 0 p 5−4cos(2t)dt (E) R 0 p 3+t 2 + 2 cos(2t)− 2 sin(2t)dt 152B-F02-A Page 3 3. If dy dt = y 2 and y(0) = −2, what is y(1)? (A) − 2 3 (B) −1 (C) − 1 4 (D) 1 (E) 4 3 4. The integral Z 1 0 dx (x−2) 2 (A) diverges, because of the behavior of the integrand at in nity. (B) diverges, because of the behavior of the integrand at zero. (C) converges, by comparison with the integral Z 1 1 dx x 2 . (D) converges, because the integrand approaches a nite constant as x ! 0. (E) none of these. 5. A body of mass m 1 = 9 grams is located at point P 1 =(1;0) in the x{y plane. A body of mass m 2 = 1 gram is at P 2 =(0;1). Where is the center of mass? (A) (9:0; 1:0) (B) (1:0; 9:0) (C) (4:5; 0:5) (D) (0:5; 0:5) (E) (0:9; 0:1) 152B-F02-A Page 4 6. The improper integral Z 1 3 lnx p x dx (A) converges to the value −ln3. (B) diverges, by comparison with the integral Z 1 3 dx p x . (C) converges, by comparison with the integral Z 1 3 lnxdx. (D) converges to the value p 3ln3. (E) converges, by comparison with the integral Z 1 3 dx p x . 7. Find the arc length of the curve segment y = x 3 6 + 1 2x ; 1 x 3. (A) 14 3 (B) 4 (C) 11 2 (D) 5 (E) 10 3 152B-F02-A Page 5 8. The vertical ends of a tank have the shape of the region above the hyperbola y = p 1+x 2 (dimensions in meters). The tank is 5 meters long, and it is lled with liquid to a depth of 3 meters (at the deepest point). The hydrostatic force on each vertical end is (A) Z 3 10 p y 2 −1(3−y) g dy (B) Z 4 1 2 p y 2 −1(4−y) g dy (C) Z 3 0 2 p y 2 −1y gdy (D) Z 3 1 5 p y 2 −1y gdy (E) Z 4 1 5 p y 2 −1(4−y) g dy 9. Consider evaluating Z 7 1 dx x by the trapezoidal rule with n subintervals (n + 1 function evaluations). Find the smallest n in the list below that is large enough to guarantee (according to the appropriate error formula) that the error in the result is at most 10 −4 . (A) 6 (B) 60 (C) 100 (D) 600 (E) 10,000 152B-F02-A Page 6 10. To solve the di erential equation dy dx +3x 2 y=e 2x you would (A) treat it as a separable equation. (B) nd one solution by inspection and multiply it by an arbitrary constant. (C) multiply the equation by the integrating factor I(x)=e x 3 . (D) multiply the equation by the integrating factor I(x)=3x 2 . (E) multiply the equation by the integrating factor I(x)=e −2x . Part II: Write Out (10 points each) Give complete solutions (\show work"). Appropriate partial credit will be given. Do not use a calculator for symbolic operations, such as evaluating integrals and derivatives. 11. The curve segment y 3 =3x,0 y 3, is revolved about the y (vertical) axis. Find the surface area of the resulting surface. 152B-F02-A Page 7 12. Evaluate Z x 2 +1 x 3 +2x 2 +x dx. 152B-F02-A Page 8 13. When Jason got his rst job, he opened a savings account and started depositing $200 per month. The account earns 5% interest, compounded continuously. (This means that an invested amount A [dollars] grows at the rate 0:05A [dollars per year], in addition to any deposits or withdrawals.) (a) Write the di erential equation and the initial condition satis ed by the value of the account, A(t). Measure t in years, and make the approximation that the deposits are also continuous (i.e., the money dribbles in at the rate of $2400 per year). (b) Solve the problem, obtaining a formula for A(t). 152B-F02-A Page 9 14. A region is bounded on the left by the y axis and on the top and bottom by the curves y = p 4−x (which meet at x = 4). Find the centroid (x; y ) of this region. 152B-F02-A Page 10 15. (a) Use Simpson’s rule with n = 6 to approximate Z 7 1 dx x . (Do not just write one number; show where the number came from!) (b) Estimate the error in your answer by as many methods as you can. Compare the situation with that of Question 9. (What would happen if you used n =60?)