NORTHEASTERN UNIVERSITY Department of Mathematics MATH 1342 (Calculus 2 for Engineering and Science) Final Exam Spring 2010 Do not write in these boxes: pg1 pg2 pg3 pg4 pg5 pg6 pg7 pg8 Total (100 points) Instructor: Instructions: Write your name and your instructor s name in the blanks above. Circle or underline your final answers to make them clear. SHOW YOUR WORK. No credit will be given unless work is shown. Answers directly from your calculator will receive no credit. If there is not enough room to show your work, use the back of the page. For your convenience, there is a table of formulas at the end of the exam. 1. Evaluate each of the following integrals. x 2 (a) dx (8 points) (x + 1)(x + 2) 1
(b) (3x + 1)e 2x dx (6 points) (c) cos 3 (x) sin 2 (x) dx (6 points) 2
2. Evaluate the improper integral and STATE whether the integral converges or diverges. 6 1 (x 4) 3 dx. 3. The region R in the first quadrant is bounded by the curves y = x 2 and y = x. SET UP and SIMPLIFY a definite integral which gives the volume of the solid obtained when the region R is rotated around the line y = 1. DO NOT EVALUATE THE INTEGRAL THAT YOU SET UP. (7 points) 3
4. Find the radius of convergence (R) and the interval of convergence of the following power series. 3 n (x 4) n (n + 2)! (6 points) radius of convergence (R) = interval of convergence = 5. Evaluate the integral as a power series in x: x 2 e x2 dx. Your answer must be expressed in proper sigma notation. 4
6. Given the vectors a = 4i j + k and b = 2i 2k, find: (a) 3a 4b (3 points) (b) a b (the dot product of a and b) (2 points) (c) the angle (to the nearest degree) between a and b (4 points) (d) the vector projection of a onto b (4 points) 5
7. Use the definition of the Taylor series to find the Taylor series centered at 1 for: f(x) = ln(x). Your answer must be expressed in proper sigma notation. (7 points) 8. A tank in the shape of a rectangular solid is filled with water weighing 62.5 pounds per cubic foot. The tank is 20 feet long, 10 feet wide, and 10 feet high. Suppose that all the water in the tank is pumped to a height 8 feet above the top edge of the tank. SET UP a definite integral giving the work done in ft lbs. Define the variable of integration clearly. DO NOT EVALUATE THE INTEGRAL you set up. 6
9. Determine whether each of the following series converges or diverges. Indicate the test used and how you apply it. Be clear and precise. (a) n=1 cos(n 3 ) n 4 (6 points) (b) ( ) 2n ( 1) n 3n 1 n=1 10. Find the sum of the series: ( 2 n + 3 n 1 ) 4 n n=1 (6 points) 7
11. The path of an object in the xy-plane is given by the vector function:. r(t) = 3t i + (4 2t 2 ) j, for 0 t 2 (a) Draw x and y-axes, sketch the path of the object and indicate with arrows the direction of increasing t. (b) Find the unit tangent vector T to the path in part (a) at t = 1. Draw and label T in your sketch in part (a). (c) SET UP AND SIMPLIFY a definite integral which gives the length of the path you sketched in part (a) over the interval 0 t 2. DO NOT EVALUATE THE INTEGRAL THAT YOU SET UP. 8
f(g(x))g (x)dx = udv = uv vdu du u 2 + a 2 = 1 ( u ) a arctan + C; a List of facts and formulas f(u)du = F (u) + C, if f(x)dx = F (x) + C du ( u ) a2 u = arcsin + C 2 a sin 2 x + cos 2 x = 1, cos 2 x = 1 2 (1 + cos 2x), sin2 x = 1 2 (1 cos 2x) degree P (x) < degree Q(x), all factors of Q(x) have the form (x a) or (x 2 + b) (b > 0) and no factor repeats, then partial fractions of P (x) have the form A Bx + C and Q(x) x a x 2 + b. 1 lim t t = 0, lim arctan t = π f(x) ; if lim t 2 x g(x) = or 0 0 Area disk = πr 2, Area washer = π(r 2 r 2 ) L = b a f(x) then lim x g(x) = lim f (x) x g (x). (x (t)) 2 + (y (t)) 2 + (z (t)) 2 dt, if r(t) = x(t), y(t, z(t), a t b Work = (Force)(Distance) if force is constant and parallel to displacement; If given weight density, multiply by volume of slice to get weight. lbs = weight, kg = mass: kg 9.8 = weight. If force = F (x), Work = b a n=1 F (x) dx. 1 converges for p > 1 diverges for p 1 np ar n converges for r < 1 to the sum lim n a n 0 = a n diverges first term 1 r, diverges for r 1 positive series a n diverges if it s bigger than a divergent series; converges if it s smaller than a convergent series positive series a n, a n b n either both conv or both div if lim = L 0 n b n ( 1) n b n, alt series (b n > 0) conv if b n 0 AND b n+1 b n series/power series conv abs if/when lim a n+1 n a n < 1, div if/when limit > 1 or DNE, test fails when limit = 1. f (n) (a) Taylor series for f(x) at x = a : (x a) n. Maclaurin series is the Taylor series at a = 0. n! e x = x n n!, cos x = ( 1) n x 2n, sin x = (2n)! ( 1) n x 2n+1 (2n + 1)! a 1, a 2, a 3 b 1, b 2, b 3 = a 1 b 1 + a 2 b 2 + a 3 b 3 = a b cos θ, a 1, a 2, a 3 = a 2 1 + a2 2 + a2 3, vector proj of b onto a = proj a b = a b a 2 a, scalar proj of b onto a = comp a b = a b a unit tangent vector T(t) = r (t) r, if r(t) = x(t), y(t, z(t). (t) 9