MA 6 FINAL EXAM PRACTICE PROBLEMS Spring. Find the angle between the vectors v = i + j + k and w = i + j k. cos 8 cos 5 cos D. cos 7 E. cos. Find a such that u = i j + ak and v = i + j + k are perpendicular. D. E.. If w = w i + w j + w k is perpendicular to u = i + j k and v = i + j + k, and if w =, then w = D. E.. If v = i + j + k and w = i k, find proj v w. / /5 D. E. / 5. Find the area of the triangle with vertices P =,,, Q =,,, and R =,,. 7 7 D. E. 6. The radius of the sphere x + y + z + x + y 6z = is + 65 D. + 56 E. 7 7. The area of the region enclosed by the curves y = x + and y = x + is given by x + x dx x + x dx E. x + x dx x + x dx x + x dx D. 8. Let R be the region between the graphs of y = x and y = x. Find the volume of the solid generated by revolving R about the x axis. 6 D. E.. If the region in problem 8 is revolved about the y axis, then the volume of the solid is 6 D.. If R is the region bounded by the curves x = and x = y y, and if R is revolved around the y axis, then the volume of the solid is D. E. E. 6
. A force of lb. is required to stretch a spring / ft. beyond its natural length. How much work is required to stretch the spring from its natural length to ft. 8 ft lbs. ft lbs. 6 ft lbs. D. ft lbs. E. ft lbs.. A cylindrical tank of height feet and radius foot is filled with water. How much work is required to pump all but foot of water out of the tank. Density = 6.5 lbs./ft 6.5 ft-lbs. 6.5 ft-lbs. E. 66.5 ft-lbs. 6.5 ft-lbs. D. 86.5 ft-lbs.. Let fx = x. Find c in [, ] such that fc = f avg, where f avg is the average value of fx = x on the interval [, ].. c = c =.5 c = 5 D. c =. E. c = 6. xln x dx = x ln x I, where I = ln x dx ln x dx ln x dx D. x ln x dx E. xln x dx. Evaluate xe x dx. e +e D. E. e 6. / sin xdx = 7. / / D. / E. / / sec x tan x dx = / / D. / E. / dx 8. In order to compute + x / we make the substitution x = tan θ. This gives an integral in θ whose value is θ + sin θ cos θ + C lnsec θ + C θ + tan θ + C D. cos θ + C E. sin θ + C
. dx x =. sec x + C E. x + tan x x + x x + x dx = sin x + C + C tan x + C D. sin x + C ln x + ln x + C ln x ln x + C ln x x + C D. ln x + C E. ln x ln x x x + C. A partial fraction decomposition of A x + B x + Cx + D x + E. A x + B x + x + has the form x + x A x + Bx + C x + A x + B x + C x + D. A x + B x +. x + x + dx = ln + ln ln + D. ln + E. ln +. Use the Trapezoidal Rule with n = to approximate 5 6 5 5 D. 7 6 x + x dx E. 7. Indicate convergence or divergence for each of the following improper integrals: I dx II x ln x dx III x x dx I converges, II and III diverge. II converges, I and III diverge. I and III converge, II diverges. D. I and II converge, III diverges. E. I, II and III diverge. 5. Find the length of the curve y = x/, x. D. E.
6. If the curve y = e x, x, is revolved about the y axis, then the area of the surface obtained is + e x dx e x + e x dx x + e x dx D. e x + e x E. e x + e x dx 7. Find the centroid x, y of the region bounded by the x axis and the semicircle y = x. 8 8,,, D., E., 8. Evaluate lim n + n n. D. E. The limit does not exist.. Evaluate lim n /n +. n n!. e D. /e E. The limit does not exist. n= 5 5 n = / 5/ 5/ D. 5 E. 5. If L = n= + n n= n n, then L = / / D. / E. 5/. Find all values of p for which. n= n + p converges. p > p p D. p > / E. p / n= + n p converges for: p p > p < D. p > E. No values of p.. Which of the following series converge conditionally? n n n I II III n ln n n= n= n= n n e n II only. I and III only. I and II only. D. All three. E. None of them.
5. Which of the following series converge? I n= n n / II n= n! 5 n III n= n II only. I and III only. I and II only. D. All three. E. None of them. 6. Find the interval of convergence of the power series n= n x n n ln n. x < < x x D. x < E. < x < 7. Find the interval of convergence of the power series n= n 5 n x n. 5 < x < 5 < x < 7 < x < D. x < 7 E. < x < 7 8. Find the first three terms of the Maclaurin series of fx = ln + x x + x + x x x + x x + x! + x! D. x x! + x! E. x + x! + x!. If fx = n x n, then f = n + n= 7 D. 7 E.. x te t dt = n= x n n! n= x n nn! x n+ n +! n= D. n= x n+ n + n! E. x n+ n + n! n=. Use the power series representation of sin x to find the first three terms of the Maclaurin series of fx = x sinx x + x7! + x 5! x + x + x5 5 x x7! + x 5! D. x x + x5 5 E. x x7 + x 5. Find the fourth term of the Maclaurin series of fx = x + x. x x x D. x E. x 5
. The fourth term of the Taylor series of fx = ln x, centered at a =, is 6 x x x D. x E. x. Using Maclaurin series and the Alternating Series Estimation Theorem, we can obtain the approximation. e x dx, with error E, where the value of E is 5 6 6 D. 7 7 E. 5 5. Parametric equations of a curve C are The curve C is: x = cos t, y = sin t, t. A quarter of a circle. An ellipse. Half of an ellipse. D. Half of a circle. E. A quarter of an ellipse. 6. Find the slope of the tangent line at the point /, for the curve parameterized by x = t /, y = t + t. / / D. E. 7. Find the length of the parametric curve x = t, y = + t, t. / 7/ 7/ D. / E. 8/ 8. A point P has polar coordinates, /. Which of the following are also polar coordinates of P? I, / II, 5/ III, 7/ IV, 5/ I and II only. I and III only. I and IV only. D. II and III only. E. II and IV only.. The polar graph of r = sin θ + cos θ is: a parabola. a line. a cardioid. D. a rose. E. an ellipse. 5. The graph of y = x is a parabola whose focus is the point,. The point P =, lies on the parabola. Find the distance from P to the directrix. 8 5 6 D. E. 6
5. The ellipse x y + = has one vertex at, 5 5,, D., E., 5. Find an equation for the hyperbola with foci ±,, and asymptotes y = ± x. y 5x = 6 5x y = 6 x y = D. y x = E. 5x y = 5. Write the complex number i + i in the form a + bi. i + i i D. i E. + i 5. Write the complex number cos + i sin D. cos + i sin 6 6 i in polar form with argument between and. cos 5 6 + i sin 5 6 E. cos 6 + i sin 6 cos + i sin Answers. D;. C;. D;. A; 5. B; 6. E; 7. B; 8. E;. A;. B. C;. C;. A;. E;. B; 6. A; 7. D; 8. E;. B;. E. A;. A;. C;. A; 5. D; 6. C; 7. A; 8. B;. B;. C. E;. D;. E;. E; 5. D; 6. A; 7. E; 8. B;. B;. E. C;. D;. C;. B; 5. E; 6. B; 7. C; 8. D;. B; 5. D 5. D; 5. B; 5. A; 5. D 7