MATHEMATICS 54
Finals Exercise Set 2
1. Evaluate the following integrals. ˆ (a)
ˆ
x3 + 1 √ dx x2 1 + x2
2. Given the 2 lines:
1
x ln x dx
(b) 0
x = 1 − t `1 : y = 2t − 2 z=3
`2 : x + 1 = y + 1 =
z−1 2
(a) Find the point of intersection of `1 and `2 . (b) Find the equation of the plane containing both `1 and `2 . (c) Find cos θ, where θ is the angle between `1 and `2 . 3. Given f (u, v) = u3 + v2 , where u = (a) Find
x and v = xy. x+y
∂f ∂f and . ∂x ∂y
(b) Using linear approximation, estimate the value of f (2.02, 1.9). 4. Let S be the surface formed by revolving the curve x = y3 about the y−axis. (a) Find the equation of the surface S. (b) Sketch the surface S. 5. Given the quadric surface with equation
x2 y2 z2 + + = 1. 49 24 49
(a) Identify the quadric surface and sketch the graph. (b) The trace of the given quadric surface on the xy−plane is the ellipse hyperbola with the same foci as the ellipse and with one vertex at (3, 0).
x2 y2 + = 1. Find the equation of the 49 24
ˆ 6. The velocity of a moving object in space at any time t is given by ~v(t) = (3 cos 3t)iˆ + (3 sin 3t) jˆ + 4k. (a) Find the distance traveled by the particle from t = 0 to t =
π . 2
π . 2 (c) Find the curvature of the object’s path when t = π.
(b) Find the acceleration ~a(t) of the motion at t =
(d) Find the position vector ~r (t) of the object’s path at any time t, if its initial position is at (1, 0, 0). 7. Given the polar curves C1 : r = 4 cos θ and C2 : r = 4 sin 2θ. (a) Find the intersection points of C1 and C2 . (b) Set-up the integral for the area of the region of intersection of C1 and C2 . π (c) Find the slope of the tangent line to the curve C2 at θ = . 4 8. Evaluate the following limits. x 3 y2 (a) lim ( x,y)−→(0,0) 2x9 + y3
(b)
lim
( x,y)−→(1,0)
sin−1
x2 + xy − x x2 + 2xy + y2 − 1