Mid Term # 2 Question 1 [10 marks] Choose the correct answer: (i) The surface area of the cone frustum generated by revolving the line segment y = x 2 + 12 , 1 x 3 about the x axis is (a) 4 5 (b) 2 (c) 3 5 (d) 98/81 (ii) The curve y = (a) 4 5 (b) 2 (c) 3 5 (d) 98/81 x3 9 ,0 x 2 is revolved about the x axis. The area of the resulting surfaces is (iii) The volume of a solid generated by revolving a region between the y axis and the curve x = R(y), c y d about the x axis, is given by Rd (a) c [R(y)]2 dy Rd (b) c 2[R(y)]2 dy Rd (c) c [R(y)]dy Rd (d) c 2 [R(y)]2 dy (iv) The volume of the solid generated by revolving the region bounded by x = the x axis is (a) 16 15 (3 2 + 5) (b) 40 3 (c) 8 3 (d) 6 (v) limx0 xsin x x3 =? (a) 0/0 (b) 2 (c) 1/2 (d) 1/6 (vi) limx0+ sin x x2 =? (a) 0/0 1 y, x = y and y = 2 about (b) (c) 2 (d) 1/2 (vii) limx(/2) sec x 1+tan x =? (a) / (b) 0/0 (c) 1 (d) 0 (viii) limx ln x 2 x =? (a) 0 (b) (c) 0/0 (d) / (ix) A parameterized curve x = f (t), y = g(t) is smooth if (a) f and g are differentiable (b) the curve is differentiable at every parameter value (c) f 0 and g 0 are continuous and not simultaneously zero (d) none of the above (x) If a smooth curve x = f (t), y = g(t), a t b, is traversed exactly once as t increases from a to b, the curve's length is Rbq dy 2 2 (a) L = 2y a ( dx dt ) + ( dt ) dt q Rb dy 2 2 (b) L = 2x a ( dx dt ) + ( dt ) dt q Rb dy 2 2 (c) L = a ( dx dt ) + ( dt ) dt (d) none of the above (xi) The tangent line to the curve x = 2 cos t, y = 2 sin t at the point where t = /4 is 2 (a) y = x + 2 3 (b) y = 3x 3 + 2 (c) y = x + 2 3 1 (d) y = 3+2x (xii) The tangent line to the curve x = 4 sin t, y = 2 cos t at the point where t = /4 is (a) y = x + 2 (b) y = 21 x + 2 2 (c) y = x + 41 (d) y = 2x 3 2 (xiii) Which of the following describes a second order homogeneous linear differential equation? 2 d y dy (a) P (x) dx 2 + Q(x) dx + R(x)y = G(x) 2 d y dy (b) P (x) dx 2 + Q(x) dx + R(x)y = 0 dy + P (x)y = Q(x) (c) dx (d) none of the above (xiv) A differential equation of the form ay 00 + by 0 + cy = 0 has the characteristic equation (a) y = erx (b) x2 + y 2 = 1 2 2 (c) xa2 + yb2 = 1 (d) ar2 + br + c = 0 (xv) If the auxiliary equation has only one real root r, then the general solution of ay 00 + by 0 + cy = 0 is given by (a) y = c1 x + c2 (b) y = c1 er1 x + c2 er2 x (c) y = c1 erx + c2 xerx (d) y = ex (c1 cos x + c2 sin x) (xvi) The general solution to a second order homogeneous linear differential equation, given two linearly independent particular solutions y1 and y2 , is (a) y(x) = c(y1 (x) + y2 (x)) (b) y(x) = y1 (x)ey2 (x) (c) y(x) = c1 y1 (x) + c2 y2 (x) (d) y(x) = y1 (x)/y2 (x) (xvii) The nth term of the infinite sequence 1, 4, 9, 16, 25, ... is (a) (1)n+1 ; n 1 (b) n2 1; n 1 (c) (1)n+1 n2 ; n 1 n+1 (d) 1+(1) ;n 1 2 (xviii) The nth term of the infinite sequence 1, 5, 9, 13, 17, ... is (a) 4n 3; n 1 (b) n 4; n 1 (c) 3n+2 n! ; n 1 n3 (d) 5n+1 ;n 1 (xix) The first four terms of the infinite sequence an = (a) 0, 1/4, 2/9, 3/16 (b) 1, 1/3, 1/5, 1/7 (c) 1/2, 1/2, 1/2, 1/2 3 1n n2 are (d) none of the above (xx) The first four terms of the infinite sequence an = 2n 2n+1 are (a) 0, 1/4, 2/9, 3/16 (b) 1, 1/3, 1/5, 1/7 (c) 1/2, 1/2, 1/2, 1/2 (d) none of the above Question 2 [4 marks] (a) [2 marks] Find the volumes using the disk method of the solids generated by revolving the regions bounded by the lines and curves given below about the x axis: 2 (i) y = x , y = 0, x = 2 (ii) y = 9 x2 , y = 0 (iii) y = cosx, 0 x /2, y = 0, x = 0 (iv) y = sec x, y = 0, x = /4, x = /4 (b) [2 marks] Use the shell method to find the volumes of the solids generated by revolving the regions bounded by the curves and lines given below about the y axis. (i) y = x, y = x/2, x = 2 (ii) y = 2x, y = x/2, x = 1 (iii) y = x2 , y = 2 x, x = 0, for x 0 (iv) y = 2 x2 , y = x2 , x = 0 Question 3 [4 marks] (a) [1 mark] Prove L'Hopital's Rule: limxa f (x) g(x) = f 0 (a) g 0 (a) (b) [3 marks] Use L'Hopital's Rule to find the limits below: x (i) limx0 1cos x2 2 +3x (ii) limx x2x 3 +x+1 ln(x+1) log2 x 2x (iv) limx loglog(x+3) 3 1sin (v) lim/2 1+cos 2 (iii) limx (vi) limt0 t(1cos t) tsin t Question 4 [4 marks] (a) [2 marks] Find the tangent to the right-hand hyperbola branch: x = sec t, y = tan t, 2 < t < /2 at the point ( 2, 1), where t = /4. 4 (b) [2 marks] Find d2 y/dx2 if x = t t2 and y = t t3 . Question 5 [4 marks] Solve the following second order linear homogeneous differential equations: (i) y 00 y 0 6y = 0 (ii) y 00 4y 0 + 13y = 0 2 (iii) 2 ddt2y + 2 dy dt y = 0 2 d y dy (iv) 9 dx 2 12 dt + 4y = 0 Question 6 [4 marks] Which of the following sequences converge and diverge? Find the limits of each convergent sequence. (i) an = 12n 1+2n 2n+1 (ii) an = 13 n (iii) an = (iv) an = 1n3 704n2 n 2n 5