Mathematics 54 Final Exam
Reviewer 1st Sem AY2012-2013
*************************************************************************************************** General directions: Provide what is asked for in each item. 1. Evaluate Evaluate the following integrals. integrals. x3 + 1 (d) dx x2 1 + x + x2
∞
(a) e dx x (b) dx e 2x + 1 x
−
√
0
2
(e)
x
−
1
1
x ln x dx
0
∞
(c)
x(x2 + 1)
dx
2. Find the limit, limit, if it exists, exists, or show that the limit does not exist. 3
(a)
li m
(x,y) x,y)→(0, (0,0)
x − 2xy x + y 6
(c)
2
3yx 2/3 (b) li m (x,y) x,y) (0, (0,0) x2 + y 3 →
x3 y (0, (0,0) 2x6 + y 2
li m
(x,y) x,y)→
x2 + xy x (d) li m 2xy + + y y 2 1 (x,y) x,y) (1, (1,0) x2 + 2xy
−
||
→
−
3. Problems Problems Involving Involving Conic Sections. (y + 2) 2 (x ( x 1)2 (a) Sketch Sketch the the graph of = 11.. Indicate Indicate the coordinates coordinates of the cente center, r, foci, vertice vertices, s, and 9 16 end points of the conjugate axis. (x 3)2 (y + 1) 2 (b) Sketch Sketch the graph graph of + = 1. Indicate Indicate the coordinates coordinates of the the center center,, foci, and and vertice verticess 16 9 in your graph.
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(c) Find an equation in polar coordinates if a conic with eccentric eccentricity ity e e = = 32 and corresponding directrix to the focus at the pole is r sin θ = 2 (d) Find the polar equation equation of a conic with focus at the pole corresponding to the directrix directrix below the focus with eccentricity 3 and containing the point (2 , π/3). π/ 3). 4. Let C Let C 1 : r = r = 4 cos θ and C and C 2 : r : r = = 4 sin 2θ. (a) Find the intersec intersection tion points of C C 1 and C and C 2 . (b) Set-up Set-up the integral integral for the area of the region common common to both C both C 1 and C 2 . (c) Find Find the slope slope of the tangent tangent line line to the curve curve C 2 at θ = π = π//4. 5. Problems Problems Involving Involving Surfaces of Revolution Revolution (a) Condsider Condsider tbe surface of revolution revolution defined by y by y + + 2 = 2x 2 x2 + 2z 2z 2 . i. Determine Determine the traces on the co ordinate ordinate planes and the intercepts. intercepts. ii. Find a generating curve curve and axis of revolution revolution.. iii. Identify Identify the type of quadric and sketch sketch its graph. (b) Given Given the surface of revolution revolution with equation x equation x 2
− 9y2 + z2 = 1.
i. Determine Determine the traces and the intercepts. intercepts. ii. Find a generating curve curve and axis of revolution revolution.. iii. Sketch Sketch and identify identify the surface surface formed. Indicate Indicate the traces and intercepts intercepts in your graph. x2 y 2 z2 (c) Given Given the quadric surface with equation equation + + = 1. 49 24 49 i. Identify Identify the quadric and sketch sketch the graph. 2
2
ii. The trace the given quadric on the xy-plane xy -plane is the ellipse x49 + y24 = 1. Find Find the equation equation of the hyperbola with the same foci as the ellipse and with one vertex at (3 , 0).
6. Problems Involving Vectors (a) Given the curve defined by R(t) =< 2et cos t, 2et sin t, et >, t curvature at t = 0.
C
∈ [0, 1]. Find the length of C and the
(b) The velocity of a moving object in space at any time t is given by V (t) =< 3 cos 3t, 3sin3t, 4 >. i. ii. iii. iv.
−
Find the distance travelled by the particle from t = 0 to t = π/2. Find the acceleration A(t) of the motion at t = π/2. Find the curvature of the object’s path when t = π. Find the position vector R(t) of the object at any time t, if the initial position of the object is at (1, 0, 0).
7. Problems Involving Lines and Planes (a) Given the lines l 1 : x = 1
− t, y = 2t − 2, z = 3, and l 2 : x + 1 = y + 1 = z −2 1 .
i. Find the point of intersection of l 1 and l2 . ii. Find the equation of the plane containing both lines l 1 and l 2 . iii. Find cos θ, where θ is the angle between l 1 and l 2 . y 2 1 (b) Given the lines l1 : 1 x = = z 1 and l2 : x = 1 + 2t, y = 2t, z = + t, find the equation 3 2 for the plane containing them both. x 1 (c) Given the lines l 1 : = 2y = z 1 and l 2 : x = 2t + 1, y = t, z = t 1. 3 i. Find the point of intersection of two lines. ii. Find an equation of the plane containing the two lines.
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8. Problems Involving Multivariable Functions (a) Consider the surface defined by f (x, y) = x 3 y
− 2yx + 4y2.
i. Find an equation for the tangent plane to the surface S : z = f (x, y) at the point ( 1, 2, 18). ii. Use linear approximation to estimate the value of f ( 0.99, 1.98). ∂z (b) Find if 3x2 y3 + exyz + sin(ln yz) = 0. ∂x x (c) Given f (u, v) = u 3 + v 3 , where u = and v = xy. x + y ∂f ∂f i. Find and . ∂x ∂y ii. Use linear approximation to estimate the value of f at (2.02, 1.9).
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−
(d) Given the surface defined by f (x, y) = xe x
2
3y .
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−
i. Find the equation for the tangent plane to the surface at the point (2, 1, 0). ii. Using linear approximation, estimate the value of (1.99, 1.1).
*************************************** “I have fought the good fight, I have finished the race, I have kept the faith.” 2 Timothy 4:7
Compiled from previous final exams kmboydon051012
